1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.6 Chain Rule

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

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

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

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 5 Quick Review Solutions

6. 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.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 7 Rule 8 The Chain Rule

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 8 Rule 8 The Chain Rule

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 9 Example Derivatives of Composite Functions

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 10 “Outside-Inside” Rule

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 11 Example “Outside-Inside” Rule

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 12 Example Repeated Use of the Chain Rule

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 13 Slopes of Parametrized Curves

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 14 Finding dy/dx Parametrically

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 15 Power Chain Rule

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 16 Quick Quiz Sections 3.4 – 3.6

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 17 Quick Quiz Sections 3.4 – 3.6

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 18 Quick Quiz Sections 3.4 – 3.6

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 19 Quick Quiz Sections 3.4 – 3.6

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 20 Quick Quiz Sections 3.4 – 3.6

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 21 Quick Quiz Sections 3.4 – 3.6

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

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 23 Quick Review

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 24 Quick Review

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 25 Quick Review

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 26 Quick Review Solutions

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 27 Quick Review Solutions

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 28 Quick Review Solutions

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 29 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.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 30 Implicitly Defined Functions

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 31 Implicitly Defined Functions

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 32 Example Implicitly Defined Functions

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 33 Implicit Differentiation Process

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 34 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.

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 35 Lenses, Tangents and Normal Lines

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 36 Example Lenses, Tangents and Normal Lines

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 37 Example Lenses, Tangents and Normal Lines

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 38 Example Derivatives of a Higher Order

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 39 Rule 9 Power Rule For Rational Powers of x

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.8 Derivatives of Inverse Trigonometric Functions

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 41 Quick Review

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 42 Quick Review

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 43 Quick Review Solutions

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 44 Quick Review Solutions

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 45 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.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 46 Derivatives of Inverse Functions

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 47 Derivative of the Arcsine

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 48 Example Derivative of the Arcsine

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 49 Derivative of the Arctangent

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 50 Derivative of the Arcsecant

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 51 Example Derivative of the Arcsecant

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 52 Inverse Function – Inverse Cofunction Identities

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 53 Calculator Conversion Identities

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 54 Example Derivative of the Arccotangent

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.9 Derivatives of Exponential and Logarithmic Functions

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 56 Quick Review

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 57 Quick Review

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 58 Quick Review Solutions

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 59 Quick Review Solutions

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 60 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.

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 61 Derivative of e x

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 62 Example Derivative of e x

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 63 Derivative of a x

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 64 Derivative of ln x

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 65 Example Derivative of ln x

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 66 Derivative of log a x

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 67 Rule 10 Power Rule For Arbitrary Real Powers

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 68 Example Power Rule For Arbitrary Real Powers

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 69 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.

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 70 Example Logarithmic Differentiation

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 71 Quick Quiz Sections 3.7 – 3.9

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 72 Quick Quiz Sections 3.7 – 3.9

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 73 Quick Quiz Sections 3.7 – 3.9

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 74 Quick Quiz Sections 3.7 – 3.9

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 75 Quick Quiz Sections 3.7 – 3.9

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 76 Quick Quiz Sections 3.7 – 3.9

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 77 Chapter Test

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 78 Chapter Test

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 79 Chapter Test Solutions

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 80 Chapter Test Solutions

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 81 Chapter Test Solutions