Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.6 Chain Rule
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 Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 5 Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 6 What you’ll learn about Derivative of a Composite Function “Outside-Inside” Rule Repeated Use of the Chain Rule Slopes of Parametrized Curves Power Chain Rule … and why The chain rule is the most widely used differentiation rule in mathematics.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 7 Rule 8 The Chain Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 8 Rule 8 The Chain Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 9 Example Derivatives of Composite Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide “Outside-Inside” Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example “Outside-Inside” Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Repeated Use of the Chain Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Slopes of Parametrized Curves
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Finding dy/dx Parametrically
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Power Chain Rule
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.4 – 3.6
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.7 Implicit Differentiation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 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 Implicitly Defined Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Implicitly Defined Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Implicitly Defined Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Implicit Differentiation Process
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 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 Lenses, Tangents and Normal Lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Lenses, Tangents and Normal Lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Lenses, Tangents and Normal Lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivatives of a Higher Order
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Rule 9 Power Rule For Rational Powers of x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.8 Derivatives of Inverse Trigonometric Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Derivatives of Inverse Functions Derivatives of the Arcsine Derivatives of the Arctangent Derivatives of the Arcsecant Derivatives of the Other Three … and why The relationship between the graph of a function and its inverse allows us to see the relationship between their derivatives.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivatives of Inverse Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of the Arcsine
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivative of the Arcsine
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of the Arctangent
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of the Arcsecant
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivative of the Arcsecant
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Inverse Function – Inverse Cofunction Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Calculator Conversion Identities
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivative of the Arccotangent
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.9 Derivatives of Exponential and Logarithmic Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Derivative of e x Derivative of a x Derivative of ln x Derivative of log a x Power Rule for Arbitrary Real Powers … and why The relationship between exponential and logarithmic functions provides a powerful differentiation tool called logarithmic differentiation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of e x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivative of e x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of a x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of ln x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Derivative of ln x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Derivative of log a x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Rule 10 Power Rule For Arbitrary Real Powers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Power Rule For Arbitrary Real Powers
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Logarithmic Differentiation Sometimes the properties of logarithms can be used to simplify the differentiation process, even if logarithms themselves must be introduced as a step in the process. The process of introducing logarithms before differentiating is called logarithmic differentiation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Logarithmic Differentiation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 3.7 – 3.9
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions