1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 3 Limits and Continuity

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.1 Derivative of a Function

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

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

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 8 What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f and f ' Graphing the Derivative from Data One-sided Derivatives … and why The derivative gives the value of the slope of the tangent line to a curve at a point.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 9 Definition of Derivative

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

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 11 Example Definition of Derivative

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 12 Derivative at a Point (alternate)

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 13 Notation

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 14 Relationships between the Graphs of f and f’ Because we can think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function f’ looks like by estimating the slopes at various points along the graph of f. We estimate the slope of the graph of f in y-units per x-unit at frequent intervals. We then plot the estimates in a coordinate plane with the horizontal axis in x-units and the vertical axis in slope units.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 15 Graphing the Derivative from Data Discrete points plotted from sets of data do not yield a continuous curve, but we have seen that the shape and pattern of the graphed points (called a scatter plot) can be meaningful nonetheless. It is often possible to fit a curve to the points using regression techniques. If the fit is good, we could use the curve to get a graph of the derivative visually. However, it is also possible to get a scatter plot of the derivative numerically, directly from the data, by computing the slopes between successive points.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 16 One-sided Derivatives

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 17 One-sided Derivatives Right-hand and left-hand derivatives may be defined at any point of a function’s domain. The usual relationship between one-sided and two-sided limits holds for derivatives. Theorem 3, Section 2.1, allows us to conclude that a function has a (two-sided) derivative at a point if and only if the function’s right-hand and left-hand derivatives are defined and equal at that point.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 18 Example One-sided Derivatives

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.2 Differentiability

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 20 Quick Review

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 21 Quick Review

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 22 Quick Review Solutions

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

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 24 What you’ll learn about How f’(a) Might Fail to Exist Differentiability Implies Local Linearity Derivatives on a Calculator Differentiability Implies Continuity Intermediate Value Theorem for Derivatives … and why Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 25 How f’(a) Might Fail to Exist

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 26 How f’(a) Might Fail to Exist

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 27 How f’(a) Might Fail to Exist

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 28 How f’(a) Might Fail to Exist

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 29 How f’(a) Might Fail to Exist

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 30 Example How f’(a) Might Fail to Exist

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 31 How f’(a) Might Fail to Exist Most of the functions we encounter in calculus are differentiable wherever they are defined, which means they will not have corners, cusps, vertical tangent lines or points of discontinuity within their domains. Their graphs will be unbroken and smooth, with a well-defined slope at each point.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 32 Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability.

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 33 Differentiability Implies Local Linearity

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

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 35 Example Derivatives on a Calculator

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 36 Derivatives on a Calculator Because of the method used internally by the calculator, you will sometimes get a derivative value at a nondifferentiable point. This is a case of where you must be “smarter” than the calculator.

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 37 Differentiability Implies Continuity The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point.

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 38 Intermediate Value Theorem for Derivatives Not every function can be a derivative.

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.3 Rules for Differentiation

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 40 Quick Review

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 Quick Review Solutions

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 46 What you’ll learn about Positive Integer Powers, Multiples, Sums and Differences Products and Quotients Negative Integer Powers of x Second and Higher Order Derivatives … and why These rules help us find derivatives of functions analytically in a more efficient way.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 47 Rule 1 Derivative of a Constant Function

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 48 Rule 2 Power Rule for Positive Integer Powers of x.

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 49 Rule 3 The Constant Multiple Rule

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 50 Rule 4 The Sum and Difference Rule

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 51 Example Positive Integer Powers, Multiples, Sums, and Differences

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 52 Example Positive Integer Powers, Multiples, Sums, and Differences

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 53 Rule 5 The Product Rule

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 54 Example Using the Product Rule

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 55 Rule 6 The Quotient Rule

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 56 Example Using the Quotient Rule

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 57 Rule 7 Power Rule for Negative Integer Powers of x

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 58 Example Negative Integer Powers of x

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 59 Second and Higher Order Derivatives

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 60 Second and Higher Order Derivatives

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 61 Quick Quiz Sections 3.1 – 3.3

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 62 Quick Quiz Sections 3.1 – 3.3

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 63 Quick Quiz Sections 3.1 – 3.3

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 64 Quick Quiz Sections 3.1 – 3.3

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 65 Quick Quiz Sections 3.1 – 3.3

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 66 Quick Quiz Sections 3.1 – 3.3

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.4 Velocity and Other Rates of Change

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 68 Quick Review

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

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 70 Quick Review Solutions

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 71 Quick Review Solutions

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 72 What you’ll learn about Instantaneous Rates of change Motion Along a Line Sensitivity to Change Derivatives in Economics … and why Derivatives give the rates at which things change in the world.

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 73 Instantaneous Rates of Change

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 74 Example Instantaneous Rates of Change

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 75 Motion Along a Line

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 76 Instantaneous Velocity

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

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

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 79 Free-fall Constants (Earth)

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 80 Example Finding Velocity

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 81 Sensitivity to Change When a small change in x produces a large change in the value of a function f(x), we say that the function is relatively sensitive to changes in x. The derivative f’(x) is a measure of this sensitivity.

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 82 Derivatives in Economics

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 83 Example Derivatives in Economics

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3.5 Derivatives of Trigonometric Functions

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 85 Quick Review

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 86 Quick Review

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 87 Quick Review Solutions

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 88 Quick Review Solutions

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 89 What you’ll learn about Derivative of the Sine Function Derivative of the Cosine Function Simple Harmonic Motion Jerk Derivatives of Other Basic Trigonometric Functions … and why The derivatives of sines and cosines play a key role in describing periodic change.

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 90 Derivative of the Sine Function

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 91 Derivative of the Cosine Function

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 92 Example Finding the Derivative of the Sine and Cosine Functions

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 93 Simple Harmonic Motion The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion.

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 94 Example Simple Harmonic Motion

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 95 Jerk

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 96 Derivative of the Other Basic Trigonometric Functions

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 97 Example Derivative of the Other Basic Trigonometric Functions

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 98 Example Derivative of the Other Basic Trigonometric Functions

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

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 100 Quick Review

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 101 Quick Review

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 102 Quick Review Solutions

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 103 Quick Review Solutions

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 104 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.

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 105 Rule 8 The Chain Rule

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 106 Example Derivatives of Composite Functions

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 107 “Outside-Inside” Rule

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 108 Example “Outside-Inside” Rule

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 109 Example Repeated Use of the Chain Rule

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 110 Slopes of Parametrized Curves

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 111 Finding dy/dx Parametrically

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 112 Power Chain Rule

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 113 Quick Quiz Sections 3.4 – 3.6

114. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 114 Quick Quiz Sections 3.4 – 3.6

115. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 115 Quick Quiz Sections 3.4 – 3.6

116. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 116 Quick Quiz Sections 3.4 – 3.6

117. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 117 Quick Quiz Sections 3.4 – 3.6

118. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 118 Quick Quiz Sections 3.4 – 3.6

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

120. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 120 Quick Review

121. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 121 Quick Review

122. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 122 Quick Review

123. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 123 Quick Review Solutions

124. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 124 Quick Review Solutions

125. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 125 Quick Review Solutions

126. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 126 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.

127. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 127 Implicitly Defined Functions

128. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 128 Implicitly Defined Functions

129. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 129 Example Implicitly Defined Functions

130. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 130 Implicit Differentiation Process

131. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 131 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.

132. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 132 Lenses, Tangents and Normal Lines

133. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 133 Example Lenses, Tangents and Normal Lines

134. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 134 Example Lenses, Tangents and Normal Lines

135. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 135 Example Derivatives of a Higher Order

136. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 136 Rule 9 Power Rule For Rational Powers of x

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

138. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 138 Quick Review

139. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 139 Quick Review

140. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 140 Quick Review Solutions

141. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 141 Quick Review Solutions

142. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 142 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.

143. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 143 Derivatives of Inverse Functions

144. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 144 Derivative of the Arcsine

145. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 145 Example Derivative of the Arcsine

146. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 146 Derivative of the Arctangent

147. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 147 Derivative of the Arcsecant

148. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 148 Example Derivative of the Arcsecant

149. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 149 Inverse Function – Inverse Cofunction Identities

150. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 150 Calculator Conversion Identities

151. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 151 Example Derivative of the Arccotangent

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

153. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 153 Quick Review

154. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 154 Quick Review

155. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 155 Quick Review Solutions

156. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 156 Quick Review Solutions

157. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 157 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.

158. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 158 Derivative of e x

159. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 159 Example Derivative of e x

160. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 160 Derivative of a x

161. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 161 Derivative of ln x

162. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 162 Example Derivative of ln x

163. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 163 Derivative of log a x

164. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 164 Rule 10 Power Rule For Arbitrary Real Powers

165. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 165 Example Power Rule For Arbitrary Real Powers

166. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 166 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.

167. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 167 Example Logarithmic Differentiation

168. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 168 Quick Quiz Sections 3.7 – 3.9

169. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 169 Quick Quiz Sections 3.7 – 3.9

170. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 170 Quick Quiz Sections 3.7 – 3.9

171. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 171 Quick Quiz Sections 3.7 – 3.9

172. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 172 Quick Quiz Sections 3.7 – 3.9

173. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 173 Quick Quiz Sections 3.7 – 3.9

174. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 174 Chapter Test

175. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 175 Chapter Test

176. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 176 Chapter Test Solutions

177. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 177 Chapter Test Solutions

178. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 178 Chapter Test Solutions