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ImplicitDifferentiationand RelatedRates 2.7 OBJECTIVE • Differentiate implicitly. • Solve related rate problems. 2012 Pearson Education, Inc. All rights reserved Slide 2.7-2 2.7ImplicitDifferentiationand RelatedRates 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 2012 Pearson Education, Inc. All rights reserved dy/dx. Slide 2.7-3 2.7ImplicitDifferentiationand RelatedRates Example 1: For y3 + x2y5 x4 = 27 a) Find dy/dx using implicit differentiation. b) Find the slope of the tangent line to the curve at the point (0, 3). a) d dx y3 + x2 y5 x4 = dx(27) 3y2 �y + x2 �5y4 �y + y5 �2x 4x= 0 dx(3y2 +5x2 y4 ) = 4x3 2xy5 2012 Pearson Education, Inc. All rights reserved Slide 2.7-4 2.7ImplicitDifferentiationand RelatedRates Example 1 (concluded): dy dx = 4x3 2xy5 3y2 + 5x2y4 b) The slope of the tangent line at (0, 3) is dy/dx at (0, 3). dy dx dy dx = = 4(0)3 2(0)(3)5 3(3)2 +5(0)2(3)4 0 27 dy dx = 0 2012 Pearson Education, Inc. All rights reserved Slide 2.7-5 ... - tailieumienphi.vn
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