Chapter 4 More Derivatives Section 4 2 Implicit

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Chapter 4 More Derivatives Section 4. 2 Implicit Differentiation Copyright © 2015, 2012, and

Chapter 4 More Derivatives Section 4. 2 Implicit Differentiation Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 2

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 3

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4

Quick Review Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 4

Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5

Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 5

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Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 6

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Quick Review Solutions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 7

What you’ll learn about Implicitly defined functions n Using the Chain Rule to find

What you’ll learn about Implicitly defined functions n Using the Chain Rule to find derivatives of functions defined implicitly n Tangent and normal lines to implicitly defined curves n Finding higher order derivatives of implicitly defined functions n Extending the Power Rule from integer powers to rational powers … 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. n Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 8

Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9

Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 9

Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10

Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 10

Example Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11

Example Implicitly Defined Functions Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 11

Implicit Differentiation Process (Take Note) Copyright © 2016, 2012, and 2010 Pearson Education, Inc.

Implicit Differentiation Process (Take Note) Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 12

Lenses, Tangents and Normal Lines In the law that describes how light changes direction

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 © 2016, 2012, and 2010 Pearson Education, Inc. 13

Lenses, Tangents and Normal Lines (Take Note) Copyright © 2016, 2012, and 2010 Pearson

Lenses, Tangents and Normal Lines (Take Note) Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 14

Example Lenses, Tangents and Normal Lines Copyright © 2016, 2012, and 2010 Pearson Education,

Example Lenses, Tangents and Normal Lines Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 15

Example Lenses, Tangents and Normal Lines Copyright © 2016, 2012, and 2010 Pearson Education,

Example Lenses, Tangents and Normal Lines Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 16

Example Derivatives of a Higher Order Copyright © 2016, 2012, and 2010 Pearson Education,

Example Derivatives of a Higher Order Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 17

Rule 9 Power Rule For Rational Powers of x (Take Note) Copyright © 2016,

Rule 9 Power Rule For Rational Powers of x (Take Note) Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 18

Homework n 3 -51 Multiples of 3, 54, 58 Copyright © 2016, 2012, and

Homework n 3 -51 Multiples of 3, 54, 58 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. Slide 4. 2 - 19 19

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 20

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 21

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 22

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 23

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 24

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010

Quick Quiz Sections 4. 1 – 4. 2 Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 25

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