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TheChainRule 1.7 OBJECTIVE
• Find the composition of two functions.
• Differentiate using the Extended Power Rule or the Chain Rule.
2012 Pearson Education, Inc.
All rights reserved Slide 1.7-2
1.7TheChainRule
THEOREM 7: The Extended Power Rule
Suppose that g(x) is a differentiable function of x. Then, for any real number k,
dx ég(x)ùk = këg(x)ûk 1 × dx g(x)
2012 Pearson Education, Inc. All rights reserved Slide 1.7-3
1.7TheChainRule
Example 1: Differentiate f (x)= 1+ x3 ) .
dx(1+
1
x3 2 =
1(1+
1 x3 2
1 3x2
= 3x2 (1+ x3 )
1
2
=
2
3x2
1+ x3
2012 Pearson Education, Inc. All rights reserved Slide 1.7-4
1.7TheChainRule
Example 2:
Differentiate
f (x)= (3x 5) (7 x 10 .
Combine Product Rule and Extended Power Rule
f¢(x)= (3x 5) 10(7 x) ( 1)+
4(3x 5) (7 x)0 (3) Simplified:
f¢(x)= 2(3x 5) (7 x) (67 21x)
2012 Pearson Education, Inc. All rights reserved Slide 1.7-5
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