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Copyright © 2016, 2012 Pearson Education, Inc

2.8 Copyright © 2016, 2012 Pearson Education, Inc Implicit Differentiation and Related Rates OBJECTIVE Differentiate implicitly. Solve related rate problems.

Copyright © 2016, 2012 Pearson Education, Inc 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 (the Power, Product, Quotient, and Chain Rules) 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. 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc To differentiate implicitly : d) Factor out dy/dx. e) Divide both sides of the equation to isolate dy/dx. 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc 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) 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Example 1 (concluded): b) The slope of the tangent line at (0, 3) is dy/dx at (0, 3). 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Implicit Differentiation and Related Rates Quick Check 1 For, find using implicit differentiation.

Copyright © 2016, 2012 Pearson Education, Inc Example 2: For the demand equation differentiate implicitly to find dp/dx. 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc 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? 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Implicit Differentiation and Related Rates Quick Check 2 A spherical balloon is deflating, losing of air per minute. At the moment when the radius of the balloon is, how fast is the radius decreasing (Hint: )?

Copyright © 2016, 2012 Pearson Education, Inc 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? 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Example 4 (continued): a) b) 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Example 4 (concluded): c) 2.8 Implicit Differentiation and Related Rates

Copyright © 2016, 2012 Pearson Education, Inc Implicit Differentiation and Related Rates Section Summary If variables x and y are related to one another by an equation but neither variable is isolated on one side of the equation, we say that x and y have an implicit relationship. To find dy/dx without solving such an equation for y, we use implicit differentiation. Whenever we implicitly differentiate y with respect to x, the factor dy/dx will appear as a result of the Chain Rule. To determine the slope of a tangent line at a point on the graph of an implicit relationship, we may need to evaluate the derivative by inserting both the x-value and the y-value of the point of tangency.