1. 1.7 Copyright © 2014 Pearson Education, Inc. The Chain Rule OBJECTIVE Find the composition of two functions. Differentiate using the Extended Power Rule or the Chain Rule.

2. Slide 1- 2 Copyright © 2014 Pearson Education, Inc. THEOREM 7: The Extended Power Rule Suppose that g(x) is a differentiable function of x. Then, for any real number k, 1.7 The Chain Rule

3. Slide 1- 3 Copyright © 2014 Pearson Education, Inc. Example 1: Differentiate 1.7 The Chain Rule

4. Slide 1- 4 Copyright © 2014 Pearson Education, Inc. Example 2: Differentiate Combine Product Rule and Extended Power Rule Simplified: 1.7 The Chain Rule

5. Slide 1- 5 Copyright © 2014 Pearson Education, Inc. 1.7 The Chain Rule Quick Check 1 Differentiate: We will combine both the quotient rule and the chain rule:

6. Slide 1- 6 Copyright © 2014 Pearson Education, Inc. DEFINITION: The composed function, the composition of f and g, is defined as 1.7 The Chain Rule

7. Slide 1- 7 Copyright © 2014 Pearson Education, Inc. Example 3: For and Find and 1.7 The Chain Rule

8. Slide 1- 8 Copyright © 2014 Pearson Education, Inc. Example 4: For and Find and 1.7 The Chain Rule

9. Slide 1- 9 Copyright © 2014 Pearson Education, Inc. 1.7 The Chain Rule Quick Check 2 For the functions in Example 4, find: a.) b.)

10. Slide 1- 10 Copyright © 2014 Pearson Education, Inc. THEOREM 8: The Chain Rule The derivative of the composition is given by 1.7 The Chain Rule

11. Slide 1- 11 Copyright © 2014 Pearson Education, Inc. 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,

12. Slide 1- 12 Copyright © 2014 Pearson Education, Inc. 1.7 The Chain Rule Section Summary Concluded The Chain Rule is used to differentiate a composition of functions. If Then