1. Copyright © Cengage Learning. All rights reserved. 2 Differentiation

2. Copyright © Cengage Learning. All rights reserved. The Chain Rule 2.5

3. 33  Find derivatives using the Chain Rule.  Find derivatives using the General Power Rule.  Write derivatives in simplified form.  Use derivatives to answer questions about real- life situations.  Review the basic differentiation rules for algebraic functions. Objectives

4. 44 The Chain Rule

5. 55 In this section, you will study one of the most powerful rules of differential calculus—the Chain Rule. This differentiation rule deals with composite functions and adds versatility to the rules. The Chain Rule

6. 66 One advantage of the notation for derivatives is that it helps you remember differentiation rules, such as the Chain Rule. For instance, in the formula you can imagine that the du’s divide out. The Chain Rule

7. 77 When applying the Chain Rule, it is helpful to think of the composite function as having two parts—an inside and an outside—as illustrated below. The Chain Rule tells you that the derivative of is the derivative of the outer function (at the inner function u) times the derivative of the inner function. That is, The Chain Rule

8. 88 Example 2 – Using the Chain Rule Find the derivative of Solution: To apply the Chain Rule, you need to identify the inside function u. The inside function is

9. 99 Example 2 – Solution By the Chain Rule, you can write the derivative as shown. cont’d

10. 10 The General Power Rule

11. 11 The function in Example 2 illustrates one of the most common types of composite functions—a power function of the form The rule for differentiating such functions is called the General Power Rule, and it is a special case of the Chain Rule. The General Power Rule

12. 12 The General Power Rule

13. 13 Example 4 – Finding an Equation of a Tangent Line Find an equation of the tangent line to the graph of at x = 2. Solution: Begin by rewriting the function in rational exponent form. Then, using the inside function, apply the General Power Rule.

14. 14 Example 4 – Solution When x = 2, y = 4 and the slope of the line tangent to the graph at (2, 4) is Using the point-slope form, you can find the equation of the tangent line to be cont’d

15. 15 Example 4 – Solution The graph of the function and the tangent line is shown in Figure 2.30. cont’d Figure 2.30

16. 16 Simplification Techniques

17. 17 Throughout this chapter, writing derivatives in simplified form has been emphasized. The reason for this is that most applications of derivatives require a simplified form. The next two examples illustrate some useful simplification techniques. Simplification Techniques

18. 18 Example 6 – Simplifying by Factoring Out Least Powers Find the derivative of Solution:

19. 19 Example 6 – Solution cont’d

20. 20 Example 7 – Differentiating a Quotient Raised to a Power Find the derivative of Solution:

21. 21 Example 7 – Solution cont’d

22. 22 Application

23. 23 Example 8 – Finding Rates of Change From 2000 through 2009, the revenue per share R (in dollars) for U.S. Cellular can be modeled by where t is the year, with t = 0 corresponding to 2000. Use the model to approximate the rates of change in the revenue per share in 2001, 2002, and 2005. Would U.S. Cellular stockholders have been satisfied with the performance of this stock from 2000 through 2009?

24. 24 The rate of change in R is given by the derivative dR/dt. You can use the General Power Rule to find the derivative. In 2001, the revenue per share was changing at a rate of Example 8 – Solution

25. 25 Example 8 – Solution cont’d In 2002, the revenue per share was changing at a rate of In 2005, the revenue per share was changing at a rate of

26. 26 Example 8 – Solution cont’d The graph of the revenue per share function R is shown in Figure 2.31. So, most stockholders would have been satisfied with the performance of this stock. Figure 2.31

27. 27 Review of Basic Differentiation Rules

28. 28 You now have all the rules you need to differentiate any algebraic function. For your convenience, they are summarized in the next slide. Review of Basic Differentiation Rules

29. 29 Review of Basic Differentiation Rules