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Section 2.4 – The Chain Rule
Example 1
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
Example 2If each function below represents ,
define and .
f x
g x
f g x
DECOMPOSITION OF FUNCTIONS
1. y 3 x 3 4x
2. y 1
2x 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 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
df df du
dx du dx
Other ways to write the Rule:
( ( )) ' ( ) '( )d
f g x f g x g xdx
f ' u f ' u u'
Instructions for The Chain Rule
For , 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
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
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
x
x 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
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
Now try the Chain Rule in combination with all of our other
rules.
Example 1Differentiate .
Chain Rule Twice
f x cos x 2 5 3x 4 6
u1
f1 u1
u1'
f1' u1
x 2
cosu1
2x
sinu1
f ' x ddx cos x 2 5 3
x 4 6
ddx cos x 2 d
dx 5 3x 4 6
ddx cos x 2 5 d
dx3x 4 6
u2
f2 u2
3x 1 4
u26
u2 '
f2 ' u2
3x 2
6u25
sin u1 2x 5 6u25 3x 2
sin x 2 2x 5 6 3x 1 4 5 3x 2
2
52 90 32 sin 4xxx x
Use the old derivative rules
Example 2Find the derivative of the function .
Chain Rule
922 1ttg t
u
f u
u'
f ' u
t 22t1
9u
5
2t1 289u
g' t ddt
t 22t1 9
u' 2t1 ddt t 2 t 2 d
dt 2t1 2t1 2
g' t 9u8 5
2t1 2
9 t 22t1 8
5
2t1 2
Quotient Rule
2t1 1 t 2 22t1 2
2t1 2t4
2t1 2
5
2t1 2
91
t 2 8
2t1 8 5
2t1 2
45 t 2 8
2t1 10
Example 3Differentiate .
Chain Rule Twice
y 2x 1 5x 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 4ddx 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 42
34 3 2 32 2 1 1 2 1 2 3 1 1 5x x x x x x x
2 2x 1 4x 3 x 1 3
17x 3 6x 2 9x 3
Example 4Differentiate .
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
sec u2
3x 2
sec u2tanu2
u1'sec u2tanu23x 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
sec x 3 1 2 sec x 3tan x 33x 2
3x 2 sec x 3 tan x 3
2 sec x 3
Example 5Find 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