© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 33 Chapter 3 Techniques of Differentiation

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 2 of 33  The Product and Quotient Rules  The Chain Rule and the General Power Rule  Implicit Differentiation Chapter Outline

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 3 of 33 The Product Rule

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 4 of 33 The Product RuleEXAMPLE SOLUTION Differentiate the function. Let and. Then, using the product rule, and the general power rule to compute g΄(x),

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 5 of 33 The Quotient Rule

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 6 of 33 The Quotient RuleEXAMPLE SOLUTION Differentiate. Let and. Then, using the quotient rule Now simplify.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 7 of 33 The Quotient Rule Now let’s differentiate again, but first simplify the expression. Now we can differentiate the function in its new form. CONTINUED Notice that the same answer was acquired both ways.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 8 of 33 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

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 9 of 33 The Chain Rule

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 10 of 33 The Chain RuleEXAMPLE SOLUTION Use the chain rule to compute the derivative of f (g(x)), where and. Finally, by the chain rule,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 11 of 33 The Chain RuleEXAMPLE SOLUTION Compute using the chain rule. Since y is not given directly as a function of x, we cannot compute by differentiating y directly with respect to x. We can, however, differentiate with respect to u the relation, and get Similarly, we can differentiate with respect to x the relation and get

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 12 of 33 The Chain Rule Applying the chain rule, we obtain It is usually desirable to express as a function of x alone, so we substitute 2x 2 for u to obtain CONTINUED

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 13 of 33 Implicit Differentiation

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 14 of 33 Implicit DifferentiationEXAMPLE SOLUTION Use implicit differentiation to determine the slope of the graph at the given point. The second term, x 2, has derivative 2x as usual. We think of the first term, 4y 3, as having the form 4[g(x)] 3. To differentiate we use the chain rule: or, equivalently,

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 15 of 33 Implicit Differentiation On the right side of the original equation, the derivative of the constant function -5 is zero. Thus implicit differentiation of yields Solving for we have CONTINUED At the point (3, 1) the slope is

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 16 of 33 Implicit Differentiation This is the general power rule for implicit differentiation.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 17 of 33 Implicit DifferentiationEXAMPLE SOLUTION Use implicit differentiation to determine This is the given equation. Differentiate. Eliminate the parentheses. Differentiate all but the second term.