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

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

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

If find y’. Example 2: Answer: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Differentiation Techniques: The Product and Quotient Rules Example 3: Differentiate Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Find the derivative of Answer: Example 4: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Differentiation Techniques: The Product and Quotient Rules 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 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Differentiation Techniques: The Product and Quotient Rules 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. 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Differentiation Techniques: The Product and Quotient Rules Example 6 (continued): b) First we must find Then we can substitute 3 (hundred) into NOTE: In the textbook, on p. 164, they substitute in 300 (instead of 3). 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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Do p. 258 ex. 5 Do p.260 #40 (a & b) Example 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example 8: Average Profit The total profit (in tens of dollars) from selling x self-help books is Find the average profit from each sales level. 8 books B) 15 books Answers: A) $0.224 tens or $2.24 B) $1.39 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley