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

Implicit Differentiation and Related Rates OBJECTIVES  Differentiate implicitly.  Solve related rate problems. 2.7

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates To differentiate implicitly : a) Differentiate both sides of the equation with respect to x (or whatever variable you are differentiating with respect to). b) Apply the rules of differentiation as necessary. Any time an expression involving y is differentiated, dy/dx will be a factor in the result. c) Isolate all terms with dy/dx as a factor on one side of the equation. d) Factor out dy/dx. e) Divide both sides of the equation to isolate dy/dx.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: For a) Find dy/dx using implicit differentiation. b) Find the slope of the tangent line to the curve at the point (0, 3). a) 2.7 Implicit Differentiation and Related Rates

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 1 (concluded): b) The slope of the tangent line at (0, 3) is dy/dx at (0, 3).

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 2: For the demand equation differentiate implicitly to find dp/dx.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 3: A restaurant supplier services the restaurants in a circular area in such a way that the radius r is increasing at the rate of 2 mi. per year at the moment when r = 5 mi. At that moment, how fast is the area increasing?

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 4: For Luce Landscaping, the total revenue from the yard maintenance of x homes is given by and the total cost is given by Suppose that Luce is maintaining 10 homes per day at the moment when the 400 th customer is signed. At that moment, what is the rate of change of (a) total revenue, (b) total cost, and (c) total profit?

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 4 (continued): a) b)

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Implicit Differentiation and Related Rates Example 4 (concluded): c)