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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
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