section 2.4 – the chain rule
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Section 2.4 – The Chain Rule. Warm-Up. Wrong answer: The exponent of 30. Although the exponent makes the derivative difficult, we could use the product rule to find the derivative. Right answer: The function inside of the secant function. - PowerPoint PPT PresentationTRANSCRIPT
Section 2.4 – The Chain Rule
Warm-UpExplain why we can not differentiate the function
below:
30 3sec 4 2y x x Wrong answer: The
exponent of 30.Although the exponent makes the derivative difficult, we could use the product rule to find
the derivative.
Right answer: The function inside of the
secant function.We have not taken the
derivative of a composition of
functions.
Composition of Functions
If and , find .
f x tan x
g x 3x 2
f g x
f g x
f 3x 2
tan 3x 2
COMPOSITION OF FUNCTIONS
Decomposition of FunctionsIf each function below represents , define
and .
f x
g x
f g x
DECOMPOSITION OF FUNCTIONS
1. y 3 x 3 4x
2. y 12x 3 3
3. y sin 3x
4. y 3x 8 4
5. y csc2 2x
g x
f x
g x
f x
g x
f x
g x
f x
g x
f x
x 3 4x
3 x
2x 3
1x 3
3x
sin x
3x 8
x 4
2x
csc2 x
The Chain RuleIf 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
df df dudx du dx
Other ways to write the Rule: ( ( )) ' ( ) '( )d f g x f g x g x
dx
f ' u f ' u u'
Instructions for The Chain RuleFor , to find :• Decompose the function
• Differentiate the MOTHER FUNCTION
• Differentiate the COMPOSED FUNCTION
• Multiply the resultant derivatives
• Substitute for u and Simplify
f u x 'f
Find: and f u u x
Find: 'f u
Find: 'u x
' ' 'f x f u u x
Make sure each function can be differentiated.
Example 1Find if and .
y u3 3u2 1
dydx f ' u u'
23 6 2u u x
22 23 2 6 2 2x x x
4 2 23 12 12 6 12 2x x x x
4 23 6 2x x x
Define f and u:
Find the derivative of f and u:
dydx
u x 2 2
u
f u
u'
f ' u
x 2 2
u3 3u2 1
2x
3u2 6u
6x 5 12x 3
Use the Chain Rule:
Substitute:
Substitute for u:
Simplify:
Example 2Differentiate .
' ' 'f x f u u
cosu 6x 5 2cos 3 5 7 6 5x x x
26 5 cos 3 5 7x x x
Define f and u:
Find the derivative of f and u:
f x sin 3x 2 5x 7
u
f u
u'
f ' u
3x 2 5x 7
sinu
6x 5
cosu
Use the Chain Rule:
Substitute:
Substitute for u:
Simplify:
Example 3If f and g are differentiable, , ,
and ; find . ' 3 2g
Define h and u:
Find the derivative of h and u:
f ' 6 3
h x g f x
u
h u
u'
h' u
f x
g u
f ' x
g' u
h' 6
h' x h' u u' ' 'g u f x
' 6 ' 6 ' 6h g f f
23
6 ' 'g f x f x
' 3 3g
6 3f
Example 4Find if .
F x x 2 1
' ' 'F x f u u
12 u 1 2 2x
12 x 2 1 1 2
2x
2x
2 x 2 1 1 2
xx 2 1
Define f and u:
Find the derivative of f and u:
F ' x
u
f u
u'
f ' u
x 2 1
u
2x
12 u 1 2
u1 2
White Board ChallegeFind f '(-2) if:
421f x x
32' 8 1f x x x
' 2 432f
Example 5Differentiate .
dydx f ' u u'
2ucos x
2sin xcos x
sin 2x
Define f and u:
Find the derivative of f and u:
f x sin2 x
u
f u
u'
f ' u
sin x
u2
cos x
2u
OR
Example 6Differentiate .
Chain Rule Twice
f x cos x 2 5 3x 4 6
u1
f1 u1
u1'
f1' u1
x 2
cosu1
2x
sin u1
62 3' cos 5 4ddx xf x x
62 3cos 5 4d ddx dx xx
ddx cos x 2 5 d
dx3x 4 6
u2
f2 u2
3x 1 4
u26
u2 '
f2 ' u2
3x 2
6u25
5 21 2sin 2 5 6 3u x u x
52 1 2sin 2 5 6 3 4 3x x x x
2
52 90 32 sin 4xxx x
Use the old derivative rules
Example 7Find the derivative of the function .
Chain Rule
922 1ttg t
u
f u
u'
f ' u
t 22t 1
9u
52t1 2
89u
g' t ddt
t 22t1 9
u' 2t1 ddt t 2 t 2 d
dt 2t 1 2t1 2
g' t 9u8 52t 1 2
9 t 22t1 8
52t1 2
Quotient Rule
2t1 1 t 2 22t1 2
2t1 2t 42t 1 2
52t 1 2
91
t 2 8
2t1 8 52t1 2
45 t 2 8
2t1 10
Example 8Differentiate .
Chain Rule Twice
y 2x 1 5 x 3 x 1 4
u1
f1 u1
u1'
f1' u1
x 3 x 1
u14
3x 2 1
4u13
45 3' 2 1 1ddxy x x x
2x 1 5 ddx x 3 x 1 4
x 3 x 1 4 ddx 2x 1 5
u2
f2 u2
2x 1
u25
u2 '
f2 ' u2
2
5u24
2x 1 54u13 3x 2 1 x 3 x 1 4
5u242
2x 1 54 x 3 x 1 3 3x 2 1 x 3 x 1 4
5 2x 1 4 2
34 3 2 32 2 1 1 2 1 2 3 1 1 5x x x x x x x
2 2x 1 4 x 3 x 1 317x 3 6x 2 9x 3
Product Rule
White Board ChallegeFind the equation of the tangent to the curve
y=3sin(2x) at the point:
2 ,0
6 3y x
Example 9Differentiate .
Chain Rule
y sec x 3
u1
f1 u1
u1'
f1' u1
sec x 3
u1
sec x 3tan x 33x 2
12 u1
1 2
y' ddx sec x 3
Chain Rule Again
u2
f2 u2
u2 '
f2 ' u2
x 3
secu2
3x 2
sec u2tan u2
u1'sec u2tan u23x 2
sec x 3tan x 33x 2
u11 2
y'12 u1
1 2sec x 3tan x 33x 2
12 sec x 3 1 2
sec x 3tan x 33x 2
12
1
secx 3 1 2 sec x 3tan x 33x 2
3x 2 secx 3 tan x 3
2 secx 3
Example 10Find an equation of the tangent line to at .
y'cosucos x
cos sin x cos x
y sin sin x
u
f u
u'
f ' u
sin x
sinu
cos x
cosu
,0
y' ddx sin sin x
Find the Derivative
Evaluate the Derivative at x = π
y'cos sin cos
cos 0 1
1 1
1
Find the equation of the line
y 0 1 x
y x