math 112 lecture 13: differentiation -- derivatives of ...€¦ · derivatives of the inverse...
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Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Lecture 13: Differentiation – Derivatives of Trigonometric Functions
Derivatives of the Basic Trigonometric FunctionsDerivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Applying the Trig Function Derivative RulesExample 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Derivatives of the Inverse Trigonometric FunctionsThe arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions1/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
sin (x ± y)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
sin (x ± y) = sin x cos y ± cos x sin y
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
sin (x ± y) = sin x cos y ± cos x sin y
Though we don’t need it right away, the corresponding formula for cos is
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
sin (x ± y) = sin x cos y ± cos x sin y
Though we don’t need it right away, the corresponding formula for cos is
cos (x ± y)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin
Recall that in Example 31(c) we guessed that
ddx
sin x = cos x
by considering the graphs of sin and cos. We will now prove this using thedefinition of the derivative and some basic trigonometric identities.
First recall the sum and difference formulas for sin
sin (x ± y) = sin x cos y ± cos x sin y
Though we don’t need it right away, the corresponding formula for cos is
cos (x ± y) = cos x cos y ∓ sin x sin y
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions2/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
= limh→0
sin x cos h + cos x sin h − sin xh
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
= limh→0
sin x cos h + cos x sin h − sin xh
= limh→0
sin x (cos h − 1) + cos x sin hh
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
= limh→0
sin x cos h + cos x sin h − sin xh
= limh→0
sin x (cos h − 1) + cos x sin hh
= sin x(
limh→0
cos h − 1h
)
+ cos x(
limh→0
sin hh
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
= limh→0
sin x cos h + cos x sin h − sin xh
= limh→0
sin x (cos h − 1) + cos x sin hh
= sin x(
limh→0
cos h − 1h
)
+ cos x(
limh→0
sin hh
)
Recall that using the Squeeze Theorem we proved that
limx→0
sin xx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
With f (x) = sin x, using Formula 3 we have
f ′(x) = limh→0
sin (x + h) − sin xh
= limh→0
sin x cos h + cos x sin h − sin xh
= limh→0
sin x (cos h − 1) + cos x sin hh
= sin x(
limh→0
cos h − 1h
)
+ cos x(
limh→0
sin hh
)
Recall that using the Squeeze Theorem we proved that
limx→0
sin xx
= 1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions3/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x = sin x (0) + cos x (1)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x = sin x (0) + cos x (1) = cos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x = sin x (0) + cos x (1) = cos x
as we predicted.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x = sin x (0) + cos x (1) = cos x
as we predicted.
You use the sum formula for cos to prove the corresponding differentiationformula for cos x, which is
ddx
cos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of sin – continued
Further, using the same approach as used in Example 13 we can show that
limh→0
cos h − 1h
= 0
Thus we have
f ′(x) =ddx
sin x = sin x (0) + cos x (1) = cos x
as we predicted.
You use the sum formula for cos to prove the corresponding differentiationformula for cos x, which is
ddx
cos x = − sin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions4/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
=
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.Thus, using the Chain Rule gives
ddx
cos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.Thus, using the Chain Rule gives
ddx
cos x =d
dxsin(
π
2 − x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.Thus, using the Chain Rule gives
ddx
cos x =d
dxsin(
π
2 − x)
= cos(
π
2 − x) d
dx(
π
2 − x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.Thus, using the Chain Rule gives
ddx
cos x =d
dxsin(
π
2 − x)
= cos(
π
2 − x) d
dx(
π
2 − x)
= sin x(−1)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of cos Using the Chain Rule
We can prove the formula for the derivative of cos in a different way. Twobasic trigonometric identities are
sin(
π
2 − x)
= cos xcos
(
π
2 − x)
= sin x
since x and π
2 − x are complementary angles in a right triangle.Thus, using the Chain Rule gives
ddx
cos x =d
dxsin(
π
2 − x)
= cos(
π
2 − x) d
dx(
π
2 − x)
= sin x(−1) = − sin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions5/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
=(cos x) cos x − sin x (− sin x)
cos2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
=(cos x) cos x − sin x (− sin x)
cos2 x
=cos2 x + sin2 x
cos2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
=(cos x) cos x − sin x (− sin x)
cos2 x
=cos2 x + sin2 x
cos2 x=
1cos2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
=(cos x) cos x − sin x (− sin x)
cos2 x
=cos2 x + sin2 x
cos2 x=
1cos2 x
=
(
1cos x
)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule
Recall thattan x =
sin xcos x
Thus, using the Quotient Rule gives
ddx
tan x =ddx
sin xcos x
=
(
ddx
sin x)
cos x − sin x(
ddx
cos x)
(cos x)2
=(cos x) cos x − sin x (− sin x)
cos2 x
=cos2 x + sin2 x
cos2 x=
1cos2 x
=
(
1cos x
)2= sec2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions6/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x = sec2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x = sec2 x = 1 + tan2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x = sec2 x = 1 + tan2 x
The equality above can also be proved using the Pythagorean identity
1 + tan2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x = sec2 x = 1 + tan2 x
The equality above can also be proved using the Pythagorean identity
1 + tan2 x = sec2 x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivative of tan Using the Quotient Rule – continued
An alternative way to simplify the previous expression is
ddx
tan x =cos2 x + sin2 x
cos2 x=
cos2 xcos2 x
+sin2 xcos2 x
= 1 +
(
sin xcos x
)2= 1 + tan2 x
So thatd
dxtan x = sec2 x = 1 + tan2 x
The equality above can also be proved using the Pythagorean identity
1 + tan2 x = sec2 x
Most text books use the sec2 x formula for the derivative of tan x, but Mapleand other symbolic differentiating programs use the 1 + tan2 x formula.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions7/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u = sec u tan ududx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u = sec u tan ududx
ddx
csc u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u = sec u tan ududx
ddx
csc u = − csc u cot ududx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u = sec u tan ududx
ddx
csc u = − csc u cot ududx
ddx
cot u
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Derivative of sinDerivative of cos Using the Chain RuleDerivative of tan Using the Quotient RuleDerivatives the Six Trigonometric Functions
Derivatives the Six Trigonometric Functions
Using basic differentiation rules as in the derivation of the derivative formulafor tan we can find derivative formulas for all of the other trigonometricfunctions. Also, recall that when we derived the General Rule for theExponential Function we stated that we would give all derivative formulas ina general form using the Chain Rule. In this form we introduce anintermediate variable u assumed to represent some function of x. With thisassumption the derivative rules for all six basic trigonometric functions are:
ddx
sin u = cos ududx
ddx
cos u = − sin ududx
ddx
tan u = sec2 ududx
=(
1 + tan2 u) du
dxddx
sec u = sec u tan ududx
ddx
csc u = − csc u cot ududx
ddx
cot u = − csc2 ududx
= −(
1 + cot2 u) du
dx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions8/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Example 46 – Differentiating with Trig Functions
Find and simplify the indicated derivative(s) of each function.
(a) Find f ′(x) and f ′′(x) for f (x) = x2 cos (3x).
(b) Finddsdt
for s =cos t
sin t + cos t.
(c) Find C′(x) for C(x) = tan(
e√
1+x2)
.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions9/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
= 2x cos (3x) − 3x2 sin (3x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
= 2x cos (3x) − 3x2 sin (3x)
= x [2 cos (3x) − 3x sin (3x)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
= 2x cos (3x) − 3x2 sin (3x)
= x [2 cos (3x) − 3x sin (3x)]
For f ′′(x) use the expression in the second line. Again using the Product andChain Rules gives
f ′′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
= 2x cos (3x) − 3x2 sin (3x)
= x [2 cos (3x) − 3x sin (3x)]
For f ′′(x) use the expression in the second line. Again using the Product andChain Rules gives
f ′′(x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(a)
Using the Product Rule followed by the Chain Rule (for cos (3x)) gives
f ′(x) = 2x cos (3x) + x2 [−3 sin (3x)]
= 2x cos (3x) − 3x2 sin (3x)
= x [2 cos (3x) − 3x sin (3x)]
For f ′′(x) use the expression in the second line. Again using the Product andChain Rules gives
f ′′(x) = 2 cos (3x) − 6x sin (3x) − 6x sin (3x) − 9x2 cos (3x)
=(
2 − 9x2)
cos (3x) − 12x sin (3x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions10/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
=− sin t (sin t + cos t) − cos t (cos t − sin t)
(sin t + cos t)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
=− sin t (sin t + cos t) − cos t (cos t − sin t)
(sin t + cos t)2
=− sin2 t − sin t cos t − cos2 t + cos t sin t
(sin t + cos t)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
=− sin t (sin t + cos t) − cos t (cos t − sin t)
(sin t + cos t)2
=− sin2 t − sin t cos t − cos2 t + cos t sin t
(sin t + cos t)2
=−(
sin2 t + cos2 t)
(sin t + cos t)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
=− sin t (sin t + cos t) − cos t (cos t − sin t)
(sin t + cos t)2
=− sin2 t − sin t cos t − cos2 t + cos t sin t
(sin t + cos t)2
=−(
sin2 t + cos2 t)
(sin t + cos t)2
= − 1
(sin t + cos t)2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(b)
Using the Quotient Rule gives
dsdt
=− sin t (sin t + cos t) − cos t (cos t − sin t)
(sin t + cos t)2
=− sin2 t − sin t cos t − cos2 t + cos t sin t
(sin t + cos t)2
=−(
sin2 t + cos2 t)
(sin t + cos t)2
= − 1
(sin t + cos t)2
This example illustrates the fact that when simplifying derivatives involvingtrig functions, you sometimes need to use standard trigonometric identities.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions11/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x,
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex,
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2,
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x) = 1 + x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x) = 1 + x2
Using the Chain Rule three times gives
C′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x) = 1 + x2
Using the Chain Rule three times gives
C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x) = 1 + x2
Using the Chain Rule three times gives
C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)
= sec2(
e√
1+x2) (
e√
1+x2) (
12
) (
1 + x2)−1/2
(2x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 46(c)
This is a composite function with
C(x) = f (g (h (k(x))))
where
f (x) = tan x, g(x) = ex, h(x) =√
x = x1/2, k(x) = 1 + x2
Using the Chain Rule three times gives
C′(x) = f ′ (g (h (k(x)))) g′ (h (k(x))) h′ (k(x)) k′(x)
= sec2(
e√
1+x2) (
e√
1+x2) (
12
) (
1 + x2)−1/2
(2x)
=xe
√1+x2 sec2
(
e√
1+x2)
√1 + x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions12/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Example 47 – Damped Oscillations
Consider the functionq(t) = e−7t sin (24t)
This function describes damped simple harmonic motion. It gives theposition of a mass attached to a spring relative to the equilibrium (resting)position of the spring. A frictional force acts to gradually slow the mass.
(a) Find q′(t) and q′′(t) and explain their meaning in terms of the dampedoscillatory motion.
(b) Note that q(0) = 0. This means that the initial position of the mass is atthe equilibrium position of the spring. Find the initial velocity of themass. Also find the velocity when the mass first returns to theequilibrium position.
(c) Draw a graph of the function q(t).(d) Find the first two times when the oscillating mass turns around. Show
the corresponding points on the graph of q(t).(e) Show that the function q(t) satisfies the differential equation
d2qdt2 + 14
dqdt
+ 625q = 0
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions13/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
q′′(t)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t[
−242 sin (24t) − 7(24) cos (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t[
−242 sin (24t) − 7(24) cos (24t)]
= −e−7t[
14(24) cos (24t) +(
242 − 72)
sin (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t[
−242 sin (24t) − 7(24) cos (24t)]
= −e−7t[
14(24) cos (24t) +(
242 − 72)
sin (24t)]
= −e−7t [336 cos (24t) + 527 sin (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(a)
Using the Product and Chain Rules gives
q′(t) = −7e−7t sin (24t) + e−7t [24 cos (24t)]
= e−7t [24 cos (24t) − 7 sin (24t)]
From our interpretation of the derivative as a rate of change, we know thatthis is the velocity of the mass at time t.Taking the derivative of the expression above for q′(t) gives
q′′(t) = −7e−7t [24 cos (24t) − 7 sin (24t)] + e−7t[
−242 sin (24t) − 7(24) cos (24t)]
= −e−7t[
14(24) cos (24t) +(
242 − 72)
sin (24t)]
= −e−7t [336 cos (24t) + 527 sin (24t)]
The rate of change of velocity is acceleration, so this gives the acceleration ofthe mass at time t.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions14/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when .
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
The velocity at this time is
v(
π
24
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
The velocity at this time is
v(
π
24
)
= e−7π/24 [24 cos (π) − 7 sin (π)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
The velocity at this time is
v(
π
24
)
= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
The velocity at this time is
v(
π
24
)
= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24 = −9.6
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(b)
Substituting t = 0 into the expression for the velocity v(t) = q′(t) gives
v(0) = e0 [24 cos (0) − 7 sin (0)] = 24
The mass returns to the equilibrium position when q(t) = 0. The first timeafter t = 0 when this happens is when
24t = π ⇒ t =π
24
The velocity at this time is
v(
π
24
)
= e−7π/24 [24 cos (π) − 7 sin (π)] = −24e−7π/24 = −9.6
This velocity is less than the initial velocity and in the opposite direction.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions15/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(c)
The graph of the function q(t) looks like this.The amplitude of the motion decreasesfollowing an envelope given by the decayingexponential function e−7t, as shown.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions16/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(c)
The graph of the function q(t) looks like this.The amplitude of the motion decreasesfollowing an envelope given by the decayingexponential function e−7t, as shown. As wesaw in the last part, not only does theamplitude decrease, but so does the velocity.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions16/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(c)
The graph of the function q(t) looks like this.The amplitude of the motion decreasesfollowing an envelope given by the decayingexponential function e−7t, as shown. As wesaw in the last part, not only does theamplitude decrease, but so does the velocity.Further, note that where the graph is concavedown, q′′(t) < 0, the mass is decelerating. Thevelocity is getting less positive or morenegative.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions16/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(c)
The graph of the function q(t) looks like this.The amplitude of the motion decreasesfollowing an envelope given by the decayingexponential function e−7t, as shown. As wesaw in the last part, not only does theamplitude decrease, but so does the velocity.Further, note that where the graph is concavedown, q′′(t) < 0, the mass is decelerating. Thevelocity is getting less positive or morenegative. And, where the graph is concaveup, q′′(t) > 0, the mass is accelerating. Thevelocity is getting more positive or lessnegative.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions16/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
⇒24t = arctan(
247
)
+ kπ
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
⇒24t = arctan(
247
)
+ kπ
where k is any integer.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
⇒24t = arctan(
247
)
+ kπ
where k is any integer. The first two positive tvalues are t1 = 0.0536 and t2 = 0.1845.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
⇒24t = arctan(
247
)
+ kπ
where k is any integer. The first two positive tvalues are t1 = 0.0536 and t2 = 0.1845. Atthese values the tangent line to the graph ofq(t) is horizontal, as shown.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(d)
The mass turns around when v(t) = q′(t) = 0.This happens when
24 cos (24t) − 7 sin (24t) = 0
⇒ tan (24t) =247
⇒24t = arctan(
247
)
+ kπ
where k is any integer. The first two positive tvalues are t1 = 0.0536 and t2 = 0.1845. Atthese values the tangent line to the graph ofq(t) is horizontal, as shown.
0.1 0.2 0.3 0.4
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions17/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(e)
From part (a) we have
d2qdt2 + 14
dqdt
+ 625q
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions18/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(e)
From part (a) we have
d2qdt2 + 14
dqdt
+ 625q
= −e−7t [336 cos (24t) + 527 sin (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions18/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(e)
From part (a) we have
d2qdt2 + 14
dqdt
+ 625q
= −e−7t [336 cos (24t) + 527 sin (24t)]
+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions18/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(e)
From part (a) we have
d2qdt2 + 14
dqdt
+ 625q
= −e−7t [336 cos (24t) + 527 sin (24t)]
+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)
= e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)]
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions18/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
Example 46 – Differentiating with Trig FunctionsExample 47 – Damped Oscillations
Solution: Example 47(e)
From part (a) we have
d2qdt2 + 14
dqdt
+ 625q
= −e−7t [336 cos (24t) + 527 sin (24t)]
+ 14e−7t [24 cos (24t) − 7 sin (24t)] + 625e−7t sin (24t)
= e−7t [(−336 + 14 × 24) cos (24t) + (−527 − 14 × 7 + 625) sin (24t)]= 0
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions18/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one
π
2−π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one, and sodoes not have an inverse function.
π
2−π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one, and sodoes not have an inverse function. However,restricting the domain of the tangent function, π
2−π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one, and sodoes not have an inverse function. However,restricting the domain of the tangent function,as shown in the graph, to the interval
− π
2< x <
π
2
π
2−π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one, and sodoes not have an inverse function. However,restricting the domain of the tangent function,as shown in the graph, to the interval
− π
2< x <
π
2
gives a one-to-one function.
π
2−π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arctan Function
Recall that the graph of the tangent functionlooks like this. From this graph we realize thatthe tangent function is not one-to-one, and sodoes not have an inverse function. However,restricting the domain of the tangent function,as shown in the graph, to the interval
− π
2< x <
π
2
gives a one-to-one function.
π
2−π
2
So the inverse of the tangent function is defined as
arctan x = tan−1 x = the angle between − π
2 and π
2 whose tangent is x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions19/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain is
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x) =(
1 + tan2 (arctan x)) d
dxarctan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x) =(
1 + tan2 (arctan x)) d
dxarctan x
=(
1 + x2) d
dxarctan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x) =(
1 + tan2 (arctan x)) d
dxarctan x
=(
1 + x2) d
dxarctan x =
ddx
x = 1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x) =(
1 + tan2 (arctan x)) d
dxarctan x
=(
1 + x2) d
dxarctan x =
ddx
x = 1
Solving forddx
arctan x gives
ddx
arctan x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arctan Function
With this definition of the inverse tangent function, we see that its domain isall real numbers and its range is
(
−π
2 , π
2)
. Further, we have
tan (arctan x) = x and arctan (tan x) = x for − π
2 < x <π
2
Taking the derivative of the first of the formulas above, as we did to find thederivative of the ln function, gives
ddx
tan (arctan x) =(
1 + tan2 (arctan x)) d
dxarctan x
=(
1 + x2) d
dxarctan x =
ddx
x = 1
Solving forddx
arctan x gives
ddx
arctan x =1
1 + x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions20/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one
π
2−π
2
1
−1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function.
π
2−π
2
1
−1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function,
π
2−π
2
1
−1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
π
2−π
2
1
−1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
gives a one-to-one function.
π
2−π
2
1
−1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
gives a one-to-one function.
π
2−π
2
1
−1
So the inverse of the sine function is defined as
arcsin x = sin−1 x = the angle between − π
2 and π
2 whose sin is x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
gives a one-to-one function.
π
2−π
2
1
−1
So the inverse of the sine function is defined as
arcsin x = sin−1 x = the angle between − π
2 and π
2 whose sin is x
The domain of inverse sine function, so defined, is
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
gives a one-to-one function.
π
2−π
2
1
−1
So the inverse of the sine function is defined as
arcsin x = sin−1 x = the angle between − π
2 and π
2 whose sin is x
The domain of inverse sine function, so defined, is [−1, 1] and its range is
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The arcsin Function
Recall that the graph of the sine function lookslike this. From this graph we realize that thesine function is not one-to-one, and so doesnot have an inverse function. However,restricting the domain of the sine function, asshown in the graph, to the interval
− π
2≤ x ≤ π
2
gives a one-to-one function.
π
2−π
2
1
−1
So the inverse of the sine function is defined as
arcsin x = sin−1 x = the angle between − π
2 and π
2 whose sin is x
The domain of inverse sine function, so defined, is [−1, 1] and its range is[
−π
2 , π
2]
.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions21/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x. So that
ddx
sin (arcsin x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x. So that
ddx
sin (arcsin x) =√
1 − sin2 (arcsin x)ddx
arcsin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x. So that
ddx
sin (arcsin x) =√
1 − sin2 (arcsin x)ddx
arcsin x =√
1 − x2 ddx
arcsin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x. So that
ddx
sin (arcsin x) =√
1 − sin2 (arcsin x)ddx
arcsin x =√
1 − x2 ddx
arcsin x
Solving forddx
arcsin x gives
ddx
arcsin x
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
The Derivative of the arcsin Function
As with the arctan function, we have
sin (arcsin x) = x and arcsin (sin x) = x for − π
2 ≤ x ≤ π
2
Taking the derivative of the first of the formulas above, as we just did for thearctan, gives
ddx
sin (arcsin x) = cos (arcsin x)ddx
arcsin x =ddx
x = 1
Now for −π
2 ≤ x ≤ π
2 we have cos x ≥ 0 , and using the basic Pythagorean
identity cos2 x + sin2 x = 1, gives cos x =√
1 − sin2 x. So that
ddx
sin (arcsin x) =√
1 − sin2 (arcsin x)ddx
arcsin x =√
1 − x2 ddx
arcsin x
Solving forddx
arcsin x gives
ddx
arcsin x =1√
1 − x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions22/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Example 48 – Differentiating with Inverse Trig Functions
Find and simplify the indicated derivative(s) of each function.
(a) Find f ′(x) and f ′′(x) for f (x) =(
1 + x2) arctan x.
(b) Finddydx
for y = arcsin(√
1 − x2)
.
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions23/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x) = 2x arctan x +(
1 + x2)
(
11 + x2
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x) = 2x arctan x +(
1 + x2)
(
11 + x2
)
= 2x arctan x + 1
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x) = 2x arctan x +(
1 + x2)
(
11 + x2
)
= 2x arctan x + 1
Taking the derivative of the expression above, using the Product Rule again,gives
f ′′(x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x) = 2x arctan x +(
1 + x2)
(
11 + x2
)
= 2x arctan x + 1
Taking the derivative of the expression above, using the Product Rule again,gives
f ′′(x) = 2 arctan x + 2x(
11 + x2
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(a)
Using the Product Rule gives
f ′(x) = 2x arctan x +(
1 + x2)
(
11 + x2
)
= 2x arctan x + 1
Taking the derivative of the expression above, using the Product Rule again,gives
f ′′(x) = 2 arctan x + 2x(
11 + x2
)
= 2 arctan x +2x
1 + x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions24/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
=
(
1√x2
)( −x√1 − x2
)
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
=
(
1√x2
)( −x√1 − x2
)
= − x|x|
√1 − x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
=
(
1√x2
)( −x√1 − x2
)
= − x|x|
√1 − x2
Thus, if 0 ≤ x < 1, then
ddx
arcsin(√
1 − x2)
= − 1√1 − x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
=
(
1√x2
)( −x√1 − x2
)
= − x|x|
√1 − x2
Thus, if 0 ≤ x < 1, then
ddx
arcsin(√
1 − x2)
= − 1√1 − x2
But you can show that
ddx
arccos x = − 1√1 − x2
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
Derivatives of the Basic Trigonometric FunctionsApplying the Trig Function Derivative Rules
Derivatives of the Inverse Trigonometric Functions
The arctan FunctionThe arcsin FunctionExample 48 – Differentiating with Inverse Trig Functions
Solution: Example 48(b)
Using the Chain Rule gives
dydx
=
1√
1 −(√
1 − x2)2
(
12
)
(
1 − x2)−1/2
(−2x)
=
(
1√
1 − (1 − x2)
)
( −x√1 − x2
)
=
(
1√x2
)( −x√1 − x2
)
= − x|x|
√1 − x2
Thus, if 0 ≤ x < 1, then
ddx
arcsin(√
1 − x2)
= − 1√1 − x2
But you can show that
ddx
arccos x = − 1√1 − x2
So is it true that arccos x = arcsin(√
1 − x2)
?
Clint LeeMath 112 Lecture 13: Differentiation – Derivatives of Trigonometric Functions25/25
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