1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Differentiation Techniques: The Product and Quotient Rules OBJECTIVES  Differentiate using the Product and the Quotient Rules.  Use the Quotient Rule to differentiate the average cost, revenue, and profit functions. 4.2

2. Slide 1.6- 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.2 Differentiation Techniques: The Product and Quotient Rules THEOREM 5: The Product Rule Let Then,

3. Slide 1.6- 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Find 4.2 Differentiation Techniques: The Product and Quotient Rules

4. Slide 1.6- 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

5. Slide 1.6- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 6: The Quotient Rule If then, Differentiation Techniques: The Product and Quotient Rules

6. Slide 1.6- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Differentiate Differentiation Techniques: The Product and Quotient Rules

7. Slide 1.6- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

8. Slide 1.6- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: If C(x) is the cost of producing x items, then the average cost of producing x items is If R(x) is the revenue from the sale of x items, then the average revenue from selling x items is If P(x) is the profit from the sale of x items, then the average profit from selling x items is Differentiation Techniques: The Product and Quotient Rules

9. Slide 1.6- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Paulsen’s Greenhouse finds that the cost, in dollars, of growing x hundred geraniums is given by If the revenue from the sale of x hundred geraniums is given by find each of the following. a) The average cost, the average revenue, and the average profit when x hundred geraniums are grown and sold. b) The rate at which average profit is changing when 300 geraniums are being grown. Differentiation Techniques: The Product and Quotient Rules

10. Slide 1.6- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): a) We let,, and represent average cost, average revenue, and average profit.

11. Slide 1.6- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): b) First we must find Then we can substitute 3 (hundred) into Differentiation Techniques: The Product and Quotient Rules

12. Slide 1.6- 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): Thus, at 300 geraniums, Paulsen’s average profit is increasing by about $11.20 per plant.