Chapter 3 Techniques of Differentiation

1. Chapter 3Techniques of Differentiation

2. Chapter Outline • The Product and Quotient Rules • The Chain Rule and the General Power Rule • Implicit Differentiation and Related Rates

3. §3.1 The Product and Quotient Rules

4. Section Outline • The Product Rule • The Quotient Rule • Rate of Change

5. The Product Rule

6. The Product Rule EXAMPLE Differentiate the function. SOLUTION Let and . Then, using the product rule, and the general power rule to compute g΄(x),

7. The Quotient Rule

8. The Quotient Rule EXAMPLE Differentiate. SOLUTION Let and . Then, using the quotient rule Now simplify.

9. The Quotient Rule CONTINUED Now let’s differentiate again, but first simplify the expression. Now we can differentiate the function in its new form. Notice that the same answer was acquired both ways.

10. Rate of Change EXAMPLE (Rate of Change) The width of a rectangle is increasing at a rate of 3 inches per second and its length is increasing at the rate of 4 inches per second. At what rate is the area of the rectangle increasing when its width is 5 inches and its length is 6 inches? [Hint: Let W(t) and L(t) be the widths and lengths, respectively, at time t.] SOLUTION Since we are looking for the rate at which the area of the rectangle is changing, we will need to evaluate the derivative of an area function, A(x) for those given values (and to simplify, let’s say that this is happening at time t = t0). Thus This is the area function. Differentiate using the product rule.

11. Rate of Change CONTINUED Now, since the width of the rectangle is increasing at a rate of 3 inches per second, we know W΄(t) = 3. And since the length is increasing at a rate of 4 inches per second, we know L΄(t) = 4. Now, we are determining the rate at which the area of the rectangle is increasing when its width is 5 inches (W(t) = 5) and its length is 6 inches (L(t) = 6). Now we substitute into the derivative of A. This is the derivative function. W΄(t) = 3, L΄(t) = 4, W(t) = 5, and L(t) = 6. Simplify.

12. The Product Rule & Quotient Rule Another way to order terms in the product and quotient rules, for the purpose of memorizing them more easily, is PRODUCT RULE QUOTIENT RULE