1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Differentiation Techniques: The Power and Sum-Difference Rules OBJECTIVES  Differentiate using the Power Rule  Differentiate using the Sum-Difference Rule.  Differentiate a constant or a constant times a function.  Determine points at which a tangent line has a specified slope. 4.1

2. Slide 1.5- 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Leibniz’s Notation: When y is a function of x, we will also designate the derivative,, as which is read “the derivative of y with respect to x.” 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

3. Slide 1.5- 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 1: The Power Rule For any real number k, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

4. Slide 1.5- 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Differentiate each of the following: a) b) c) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

5. Slide 1.5- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a)b) c) Example 1 - Answers

6. Slide 1.5- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2: Differentiate: a) b) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

7. Slide 1.5- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) b) Answers – Example 2

8. Slide 1.5- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 2: The derivative of a constant function is 0. That is, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

9. Slide 1.5- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 3: The derivative of a constant times a function is the constant times the derivative of the function. That is, 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

10. Slide 1.5- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: Find each of the following derivatives: a) b) c) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

11. Slide 1.5- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) b) Answers – Example 3 c)

12. Slide 1.5- 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 4: The Sum-Difference Rule Sum: The derivative of a sum is the sum of the derivatives. Difference: The derivative of a difference is the difference of the derivatives. 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

13. Slide 1.5- 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Find each of the following derivatives: a) b) 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

14. Slide 1.5- 14 a) b) Answers – Example 4

15. Slide 1.5- 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5: Find the points on the graph of at which the tangent line is horizontal. 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

16. Slide 1.5- 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley So, for Setting equal to 0: Answers – Example 5 Recall that the derivative is the slope of the tangent line, and the slope of a horizontal line is 0. Therefore, we wish to find all the points on the graph of f where the derivative of f equals 0.

17. Slide 1.5- 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (continued): To find the corresponding y-values for these x-values, substitute back into Thus, the tangent line to the graph of is horizontal at the points (0, 0) and (4, 32). 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

18. Slide 1.5- 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5 (concluded): 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

19. Slide 1.5- 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Find the x values of the points on the graph of at which the tangent line has slope 6. 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

20. Slide 1.5- 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Answer -Example 6 Here we will employ the same strategy as in Example 5, except that we are now concerned with where the derivative equals 6. Recall that we already found that

21. Slide 1.5- 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): If we were asked to find the corresponding y-values, we would substitute these x-values into

22. Slide 1.5- 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): 4.1 Differentiation Techniques: The Power Rule and Sum-Difference Rules

23. Slide 1.5- 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Problems

24. Slide 1.5- 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Answers: 1) y = 8x – 4 2) y = -8x - 18

25. Application 1)Suppose that the price of q units is and the cost is Find the marginal profit for a) 500 units b) 815 units 2) Often sales of a new product grow rapidly at first and then level off with time. This is the case with the sales represented by the function Find the rate of change of sales for the following number of years (a) 1 (b) 10. Answers: 1) (a) $30 (b)$4.80 2) (a) 100 (b) 1

26. Slide 1.5- 26 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3) If the price, in dollars, of a stereo system is given by where q represents the demand for the product, find the marginal revenue when the demand is 10. Answer: $990

27. Slide 1.5- 27 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Go back to problems in 3.3 and 3.4. Work using Theorems for Derivatives.