# Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Implicit Differentiation Section 3.7.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Implicit Differentiation Section 3.7

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 2 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 3 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 4 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 5 What you’ll learn about Implicitly Defined Functions Lenses, Tangents, and Normal Lines Derivatives of Higher Order Rational Powers of Differentiable Functions … and why Implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 6 Implicitly Defined Functions An implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable. Implicit differentiation is especially useful when dy/dx is needed, but it is difficult or inconvenient to solve the equation for y in terms of x.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 7 Implicit Differentiation Process

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 8 Example Implicitly Defined Functions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 9 Example Implicitly Defined Functions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 10

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 11 Lenses, Tangents and Normal Lines In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry (angles A and B in Figure 3.50). This line is called the normal to the surface at the point of entry. In a profile view of a lens, the normal is a line perpendicular to the tangent to the profile curve at the point of entry. Implicit differentiation is often used to find the tangents and normals of lenses described as quadratic curves.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 12 Lenses, Tangents and Normal Lines

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 13 Ex. Lenses, Tangents and Normal Lines

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 14 Example Derivatives of a Higher Order

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 15 Rule 9 Power Rule For Rational Powers of x

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Page 155 (1-21 odd) ======================== Page 155 (23-35 odd) Slide 3- 16