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Calculus and its applications: 1.7

"Calculus and its applications: 1.7" - The chain rule have 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- The Chain Rule 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- 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 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 1: Differentiate 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 2: Differentiate Combine Product Rule and Extended Power Rule Simplified: 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 1.7 The Chain Rule Quick Check 1 Differentiate: 2012 Pearson Education, Inc. All rights reserved Slide 1.7- DEFINITION: The composed function , the composition of f and g, is defined as 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 3: For and Find and 1.7 The Chain Rule p. 172 works example 5 like example 5 was posed with parts (a) and (b) (but it wasn’t). 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 4: For and Find and 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 1.7 The Chain Rule Quick Check 2 For the functions in Example 4, find: a.) b.) 2012 Pearson Education, Inc. All rights reserved Slide 1.7- THEOREM 8: The Chain Rule The derivative of the composition is given by 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 5: For and find and and 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 1.7 The Chain Rule Quick Check 3 If and , find . 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 6: For and find and 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 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, 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 1.7 The Chain Rule Section Summary Concluded The Chain Rule is used to differentiate a composition of functions. If Then | 2012 Pearson Education, Inc. All rights reserved Slide 1.7- The Chain Rule 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- 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 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 1: Differentiate 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- Example 2: Differentiate Combine Product Rule and Extended Power Rule Simplified: 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide 1.7- 1.7 The Chain Rule Quick Check 1 Differentiate: 2012 Pearson Education, Inc. All rights reserved Slide 1.7- DEFINITION: The composed function , the composition of f and g, is defined as 1.7 The Chain Rule 2012 Pearson Education, Inc. All rights reserved Slide .