1.7

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The Chain Rule

OBJECTIVE• Find the composition of two functions.• Differentiate using the Extended Power Rule or the Chain Rule.

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THEOREM 7: The Extended Power Rule

Suppose that g(x) is a differentiable functionof x. Then, for any real number k,

ddx

g x kk g x

k 1

ddx

g x

1.7 The Chain Rule

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Example 1: Differentiate 1

3 21 .f x x

1.7 The Chain Rule

1 1 13 3 22 211 1 3

2d x x xdx

12

3 23 12x x

2

3

32 1

xx

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Example 2:Differentiate

Combine Product Rule and Extended Power Rule

Simplified:

f x 3x 5 4 7 x 10 .

4 9 3 10

4 9 3 10

3 9

3 5 10 7 1 4 3 5 7 3

10 3 5 7 12 3 5 7

2 3 5 7 5(3 5) 6(7 )

f x x x x x

x x x x

x x x x

3 92 3 5 7 21 67f x x x x

1.7 The Chain Rule

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1.7 The Chain RuleQuick Check 1

Differentiate:

We will combine both the quotient rule and the chain rule:

2

24

(2 1)( )3 2

xf xx

4 2 2 2 4 2

4 2 2

(3 2) (2 1) (2 1) ((3 2) )( )

[(3 2) ]

d dx x x xdx dxf x

x

4 2 2 4 3

4 4

(3 2) (4 ) (2 1) (2(3 2)(12 )( )(3 2)

x x x x xf xx

4 2 3 2 4 4 3 2

4 4 4 3

4 (3 2) 24 (2 1)(3 2) 4 (3 2) 24 (2 1)( )(3 2) (3 2)

x x x x x x x x xf xx x

5 3

34

36 24 8

3 2

x x xf xx

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DEFINITION:

The composed function , the composition of f and g, is defined as

f g

( ( ))f g f g x

1.7 The Chain Rule

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Example 3: For andFind and

f (x) x3 g(x) 1 x2 ,( )( )f g x ( )( ).g f x

1.7 The Chain Rule

2 4 61 3 3x x x

2 3(1 )x

2(1 )f x

( )( ) ( ( ))f g x f g x ( )( ) ( ( ))g f x g f x3( )g x

3 21 ( )x 61 x

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Example 4: For andFind and

f (x) x g(x) x 1,( )( )f g x ( )( ).g f x

1.7 The Chain Rule

( )( ) ( ( ))f g x f g x

( 1)f x

( )( ) 1f g x x

( )( ) ( ( ))g f x g f x

( )g x

( )( ) 1g f x x

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1.7 The Chain Rule

Quick Check 2

For the functions in Example 4, find:

a.)

b.)

f f x

g g x

( )( ) ( ( ))f f x f f x ( )f x

11 22x x

4 x

g g x g g x ( 1)g x 1 1x 2x

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THEOREM 8: The Chain Rule

The derivative of the composition is given byf g

( )( ) ( ( )) '( ( )) '( )d df g x f g x f g x g xdx dx

1.7 The Chain Rule

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1.7 The Chain Rule

Section Summary

• The Extended Power Rule tells us that if then

• The composition of with and is written and is

defined as

• In general,

( ) ,ky f x

1[ ( )] [ ( )] ( ).k kdy f x k f x f xdx

( )f x ( )g x ( )( )f g x

( )( ) ( ( )).f g x f g x

( )( ) ( )( ).f g x g f x

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1.7 The Chain Rule

Section Summary Concluded

• The Chain Rule is used to differentiate a composition of functions.

If

Then

( ) ( )( ) ( ( )),F x f g x f g x

( ) [( )( )] .dF x f g x f g x g xdx