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3The Derivative
3.1 Introduction to the Derivative
Consider a function f and a line that passes through the points (c, f(c)) and (c + ∆x, f(c + ∆x)).y
x
Definition. If f is defined on an open interval containing c, and if
lim∆x→0
∆y
∆x=
exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of fat the point (c, f(c)).
Example 1. Find the slope of the graph of y = 3x + 2 at the point (1, 5).
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3 The Derivative
Example 2. Find the slope of the tangent lines to the graph of f(x) = x2 − 1 at the points (0,−1)and (1, 0).
Definition. The derivative of f at x is given by
f ′(x) =
provided the limit exists. For all x for which this limit exists, f ′ is a function of x.
Example 3. Find the derivative of f(x) = x2 + 3x + 2.
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3.1 Introduction to the Derivative
Example 4. Find f ′(x) for f(x) = 1x. Use your result to find the slope of the graph of f(x) at the
points (12, 2), (1, 1), and (2, 1
2)
Example 5. Find the derivative with respect to t if y =√
t.
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3 The Derivative
Notation. We may write the derivative with respect to x in the following ways:
• f ′(x)
• dy
dx
• y′
• d
dx
[
f(x)]
• Dx[y]
The Alternate Form of the Derivative
The following form of the derivative is useful for investigating the relationship between differentia-bility and continuity.
f ′(c) =
y
x
Example 6. Use the alternate form of the derivative to find f ′(c) if f(x) = x2.
4
3.1 Introduction to the Derivative
Example 7. Use the alternate form of the derivative to investigate the differentiability at x = 0 forf(x) = |x|.
Example 8. Use the alternate form of the derivative to investigate the differentiability at x = 0 forf(x) = x1/3.
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3.2 Basic Differentiation
3.2 Basic Differentiation
Theorem 3.2. If k is a real number, then
d
dx[k] =
y
x
Recall
From the binomial theorem that
(x + ∆x)n = xn + nxn−1(∆x) +n(n − 1)xn−2
2(∆x)2 + · · · + (∆x)n
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3 The Derivative
Theorem 3.3 (The Power Rule). If n is a rational number, then the function f(x) = xn is
differentiable and
d
dx[xn] =
For f to be differentiable at x = 0, n must be a number such that xn−1 is defined on an openinterval containing 0.
Example 1. Find the derivativeof f(x) = x5.
Example 2. Find the derivativeof g(x) = 4
√x.
Example 3. Find the derivativeof h(x) = 1
x3 .
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3.2 Basic Differentiation
Example 4. Find the slope of the graph of f(x) = x6 when x = −1, 0, and 1.
Example 5. Find an equation of the tangent line to the graph of f(x) = x3 when x = 2.
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3 The Derivative
Theorem 3.4. If f is differentiable and k is a real number, then kf is differentiable and
d
dx[kf(x)] =
Example 6. Findd
dx
[
3
x
]
. Example 7. Findd
dt
[
2
3t2
]
.
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3.2 Basic Differentiation
Example 8. Findd
dx[3√
x]. Example 9. Findd
dx
[
1
54√
x3
]
.
Theorem 3.5. Let f and g be differentiable functions. Then, f ± g is differentiable and
d
dx
[
f(x) ± g(x)]
=
Example 10. Findd
dx[x2 − 3x + 2].
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3 The Derivative
Theorem 3.6.
1.d
dx[sin x] = 2.
d
dx[cos x] =
Example 11. Findd
dx[√
2 sin x]. Example 12. Findd
dx[x2 + cos x].
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3.2 Basic Differentiation
Theorem 3.7.
d
dx[ex] =
Example 13. Findd
dx[x3 + ex]. Example 14. Find
d
dx[cos x + ex].
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3 The Derivative
Let s(t) be a position function with respect to time t. Then, the average velocity is given by
average velocity =
Example 15. If an object is dropped from a height of 50 feet, the height s of the object at time tis given by
s(t) = −16t2 + 50
where s is measured in feet and t is measured in seconds. Find the average velocity over each of thefollowing time intervals: [1, 2], [1, 1.5], and [1, 1.1].
The velocity of an object at time t is given by
v(t) =
where
s(t) =1
2gt2 + v0t + s0
is the position of the object at time t having the acceleration due to gravity g = −32 feet perseconds squared, v0 and s0 are the initial velocity and initial position respectively.
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3.2 Basic Differentiation
Example 16. At the time t = 0, an object is propelled upward from a height of 80 feet. The positionof the object is given by
s(t) = −16t2 + 64t + 80
where s is in feet and t is in seconds. When does the object hit the ground? What is the velocityof the object at impact?
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3 The Derivative
3.3 Differentiation Rules
In this section, we point out two facts:
• d
dx[f(x)g(x)] 6= d
dx[f(x)]
d
dx[g(x)] • d
dx[f(x) ÷ g(x)] 6= d
dx[f(x)] ÷ d
dx[g(x)]
That is, differentiation does not behave like the limit with respect to multiplication anddivision.
Theorem 3.8 (The Product Rule). Let f and g be differentiable functions. Then, fg is differen-
tiable and
d
dx
[
f(x)g(x)]
=
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3.3 Differentiation Rules
Example 1. Findd
dx[(3x2 − 4)(2x − 7)].
Example 2. Findd
dx[x2ex].
Example 3. Findd
dx[2x sin x + 2 cos x].
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3 The Derivative
Note. The product rule can be generalized to products of more than two factors. For example, forthree factors we have
d
dx[uvw] =
du
dxvw + u
dv
dxw + uv
dw
dx
Example 4. Findd
dx[x2 cos x sin x].
Next, we encounter the Quotient Rule and the Reciprocal Rule for derivatives. The ProductRule can be used to derive weak version of these rules. It is a “weak” version in that it does notprove that the quotient is differentiable, but only says what its derivative is if it is differentiable.
Theorem 3.9 (The Quotient Rule). Let f and g be differentiable functions. Then, f/g is differ-
entiable and
d
dx
[
f(x)
g(x)
]
=
and if f(x) = 1, we obtain the Reciprocal Rule
d
dx
[
1
g(x)
]
=
The Quotient Rule often memorized as a rhyme type song. “lo-dee-hi less hi-dee-lo, draw theline and square below”; Lo being the denominator, Hi being the numerator and “dee” being thederivative.
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3.3 Differentiation Rules
Example 5. Findd
dx
[
x − 4
x2 − 1
]
.
Example 6. Findd
dx
1 − 1
xx + 1
.
Example 7. Findd
dx
[
1
x − sin x
]
.
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3 The Derivative
Theorem 3.10.
1.d
dx[tan x] = 2.
d
dx[cot x] =
3.d
dx[sec x] = 4.
d
dx[sec x] =
Notation. We can take derivatives of derivatives, we call those higher-order derivatives.
Second Derivative nth Derivative
f ′′(x) f (n)(x)
d2y
dx2
dny
dxn
y′′ y(n)
d2
dx2
[
f(x)] dn
dxn
[
f(x)]
D2x[y] Dn
x[y]
Higher-order derivatives have many interesting applications. One application in particular is thatwe can calculate the the curvature of the graph of a function y = f(x). Intuitively, curvature isthe amount by which a geometric object deviates from being flat and is given by
κ =|y′′|
[1 + (y′)2]3/2
Example 8. Find the curvature of y = x3.
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3.4 Differentiating the Composition of Functions
3.4 Differentiating the Composition of Functions
Theorem 3.11 (The Chain Rule). If y = f(u) is a differentiable function of u and u = g(x) is a
differentiable function of x, then y = f(
g(x))
is a differentiable function of x and
dy
dx=
in other words,
d
dx
[
f(
g(x))]
=
Example 1. Findd
dx[cos(2x)]. Example 2. Find
d
dx[cot2 x].
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3 The Derivative
Example 3. Findd
dx[√
x2 + 2x + 3].
Theorem 3.12. If y =[
u(x)]n
, where u is a differentiable function of x and n is a rational number,
then
dy
dx=
in other words,
d
dx[un] =
Example 4. Findd
dx[(x3 + 27)4].
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3.4 Differentiating the Composition of Functions
Example 5. Find all points on the graph of f(x) = 3
√
(x2 − 4)2 for which f ′(x) = 0 and those forwhich f ′(x) does not exist.
Example 6. Findd
dt
[
− 5
(3t + 2)2
]
.
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3.4 Differentiating the Composition of Functions
Example 9. Finddy
dxif y =
(
2x + 1
x2 + 1
)10
.
Example 10. Finddy
dxif y = sin(cos(tanx)).
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3 The Derivative
Theorem 3.13. Let u be a differentiable function of x. Then
1.d
dx[ln x] =
2.d
dx[ln u] =
3.d
dx[ln |u|] =
Example 11. Finddy
dxif y = ln(3x). Example 12. Find
dy
dxif y = ln(x2 + 4).
Example 13. Finddy
dxif y = x2 ln x. Example 14. Find
dy
dxif y = (ln x)2.
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3.4 Differentiating the Composition of Functions
Example 15. Finddy
dxif y = ln(
√x + 4).
Example 16. Finddy
dxif y = ln
[
x(x2 + 4)2
√x3 − 1
]
.
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3 The Derivative
Recall that
• ax =
• loga x =
Theorem 3.15. Let u be a differentiable function of x. Then
1.d
dx[ax] = 2.
d
dx[au] =
3.d
dx[loga x] = 4.
d
dx[loga u] =
Example 17. Finddy
dxif y = 3x. Example 18. Find
dy
dxif y = 52x.
Example 19. Finddy
dxif y = log10(sin x).
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3.5 Implicit and Logarithmic Differentiation
3.5 Implicit and Logarithmic Differentiation
We wish to differentiate functions that are defined implicitly. For example, y = −2x + 1 is saidto be in explicit form as y is already solved for as opposed to 2x + y = 1, which is in implicit
form. We have to use the Chain Rule to differentiate function to differentiate implicit functionsbecause we assume that y is a differentiable function of x. For example,
• d
dx[x2]
• d
dx[y3]
Example 1. Findd
dx[x2y3].
Example 2. Finddy
dxgiven that x3 + y3 = 6xy. Use your result to find the slope at the point (3, 3).
y
1
2
3
4
5
−1
−2
−3
−4
−5
1 2 3 4 5−1−2−3−4−5
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3 The Derivative
Example 3. Find y′ if sin(x + y) = y2 cos x.
Example 4. Finddy
dxif y = arctan x.
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3.5 Implicit and Logarithmic Differentiation
Example 5. Finddy
dxusing logarithmic differentiation if y = xx.
Example 6. Finddy
dxusing logarithmic differentiation if y =
(
2x + 1
x2 + 1
)10
.
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3 The Derivative
3.6 Inverse Functions and Derivatives
Theorem 3.16. Let f be a function that has an inverse, whose domain is an interval I. Then,
1. If f is continuous on I, then f−1 is continuous on its domain.
2. If f is differentiable at c and f ′(c) 6= 0, then f−1 is differentiable at f(c).
Theorem 3.17 (The Inverse Function Theorem). Let f be a function that is differentiable on
an open interval I. If f has an inverse function g, then g is differentiable at any x for which
f ′(
g(x))
6= 0 and
g′(x) =
in other words, since f−1 = g,
d
dx
[
f−1(x)]
=
Example 1. Let f(x) = 12x3 + x − 1. What is f−1(x) when x = 5? What is the value of (f−1)′(x)
when x = 5?
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3.6 Inverse Functions and Derivatives
Note. A function and its inverse have reciprocal slopes, in other words
dy
dx=
1
dx
dy
Example 2. If f(x) = 3√
x, find the slopes of f and f−1 when x = 8.
Example 3. Suppose that f(x) = sin x and f−1(x) = arcsin x for −π/2 ≤ f(x) ≤ π/2. Use the
Inverse Function Theorem to findd
dx[f−1(x)].
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3 The Derivative
Theorem 3.18. Let u be a differentiable function of x. Then
1.d
dx[arcsin u] = 2.
d
dx[arccos u] =
3.d
dx[arctan u] = 4.
d
dx[arccot u] =
5.d
dx[arcsec u] = 6.
d
dx[arccsc u] =
Example 4. Differentiate y = arccos x − x√
1 − x2.
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3.7 Related Rates
3.7 Related Rates
Example 1. A spherical balloon is expanding Given that the radius is increasing at at rate of 2inches per minute, at what rate is the volume increasing when the radius is 5 inches?
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3 The Derivative
Example 2. Car A is traveling west at a rate of 50 miles per hour and car B is traveling north asa rate of 60 miles per hour/ Both are headed for the same intersection of the two roads. At whatrate are the cars approaching each other when car A is 0.3 miles and car B is 0.4 miles from theintersection respectively.
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3.7 Related Rates
Example 3. A 13-foot ladder leans against the side of a building, forming an angle θ with theground. Given that the foot of the ladder is being pulled away from the building at at rate of 0.1feet per second, what is the rate of change of θ when then top of the ladder is 12 feet above theground?
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3 The Derivative
Example 4. A conical paper cup of dimensions 8 inches across the top and 6 inches deep is full ofwater. The cup springs a leak and is losing water at a rate of 2 cubic inches per minute. How fastis the water level dropping when the water level is exactly 3 inches deep?
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3.7 Related Rates
Example 5. A balloon leaves the ground 500 feet away from an observer and rises vertically at arate of 140 feet per minute. At what rate is the inclination of the observer’s line of sight increasingwhen the balloon is exactly 500 feet above the ground?
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