The Chain Rule 1 7 OBJECTIVE Find the

The Chain Rule 1. 7 OBJECTIVE • • Find the composition of two functions. Differentiate using the Extended Power Rule or the Chain Rule. Copyright © 2014 Pearson Education, Inc.

1. 7 The Chain Rule THEOREM 7: The Extended Power Rule Suppose that g(x) is a differentiable function of x. Then, for any real number k, Copyright © 2014 Pearson Education, Inc. Slide 1 - 2

1. 7 The Chain Rule Example 1: Differentiate Copyright © 2014 Pearson Education, Inc. Slide 1 - 3

1. 7 The Chain Rule Example 2: Differentiate Combine Product Rule and Extended Power Rule Simplified: Copyright © 2014 Pearson Education, Inc. Slide 1 - 4

1. 7 The Chain Rule Quick Check 1 Differentiate: We will combine both the quotient rule and the chain rule: Copyright © 2014 Pearson Education, Inc. Slide 1 - 5

1. 7 The Chain Rule DEFINITION: The composed function and g, is defined as , the composition of f Copyright © 2014 Pearson Education, Inc. Slide 1 - 6

1. 7 The Chain Rule Example 3: For Find and Copyright © 2014 Pearson Education, Inc. Slide 1 - 7

1. 7 The Chain Rule Example 4: For Find and Copyright © 2014 Pearson Education, Inc. Slide 1 - 8

1. 7 The Chain Rule Quick Check 2 For the functions in Example 4, find: a. ) b. ) Copyright © 2014 Pearson Education, Inc. Slide 1 - 9

1. 7 The Chain Rule THEOREM 8: The Chain Rule The derivative of the composition Copyright © 2014 Pearson Education, Inc. is given by Slide 1 - 10

1. 7 The Chain Rule Section Summary • The Extended Power Rule tells us that if • The composition of with then and is written defined as • In general, Copyright © 2014 Pearson Education, Inc. Slide 1 - 11 and is

1. 7 The Chain Rule Section Summary Concluded • The Chain Rule is used to differentiate a composition of functions. If Then Copyright © 2014 Pearson Education, Inc. Slide 1 - 12