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 Derivatives

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 Determine the derivative of the function at the given point. f(x) = x 2 + 4, at a =1. Slide 3- 12

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Determine the derivative of the function at the given point. Slide 3- 13

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 14 Derivative at a Point (alternate)

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use the alternate definition of derivative to determine the derivative at the indicated point. f(x) = x 2 + 4 at x = 1 Slide 3- 15

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use the alternate definition of derivative to determine the derivative at the indicated point. Slide 3- 16

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 17 Notation

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 18 The derivative of a linear function is the slope of the line, so dy/dx here is 7.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Find the derivative of the function y = 2x 2 -13x+5 and use it to find an equation of the line tangent to the curve at x = 3. Slide 3- 19

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Find the lines that are a) tangent and b) normal to the curve below. Slide 3- 20

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

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Match the graph of the function to one of the derivative graphs shown below. Slide 3- 22 y=f(x)

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Match the graph of the function to one of the derivative graphs shown below. Slide 3- 23 y=f(x)

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Below is the graph of f(x) = xlnx -x Which of the functions below is the derivative of f(x)? i. y = sin x ii. y = ln x iii. y = x.5 iv. y = x 2 v. y= 3x - 1 Slide 3- 24

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Sketch the graph of a continuous function f with f(0) = 1 and Slide 3- 25

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

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Sketch a graph of the elevation (y) as a function of distance downriver. Slide 3- 27

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Use your graph from the previous problem to get an approximate graph of the derivative, dy/dx. Slide 3- 28

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall The average change in elevation over a given distance is called a gradient. What units of measure would be appropriate for a gradient in this problem? What units of measure would be appropriate for the derivative? Slide 3- 29

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Using your function graph, how would you determine the most dangerous section of the river (ignoring rocks)? Slide 3- 30

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Using your derivative graph, how would you determine the most dangerous section of the river (ignoring rocks)? Slide 3- 31

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 32 One-sided Derivatives

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

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

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Assignment pages 105 – 107, # 1 – 23 odds, 24 – 27 all, 29, 31, 33, 34 Slide 3- 35

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

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 37 Quick Review

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 38 Quick Review

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 39 Quick Review Solutions

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

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

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 42 How f’(a) Might Fail to Exist

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 43 How f’(a) Might Fail to Exist

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 44 How f’(a) Might Fail to Exist

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 45 How f’(a) Might Fail to Exist

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 46 How f’(a) Might Fail to Exist

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 47 Example How f’(a) Might Fail to Exist

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 48 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.

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

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 50 Differentiability Implies Local Linearity

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 51 Derivatives on a Calculator

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 52 Example Derivatives on a Calculator

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

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 54 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.

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 55 Intermediate Value Theorem for Derivatives Not every function can be a derivative.

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

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

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

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

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 61 Quick Review Solutions

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 62 Quick Review Solutions

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

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 64 Rule 1 Derivative of a Constant Function

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 65 Rule 2 Power Rule for Positive Integer Powers of x.

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 66 Rule 3 The Constant Multiple Rule

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 67 Rule 4 The Sum and Difference Rule

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 68 Example Positive Integer Powers, Multiples, Sums, and Differences

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 69 Example Positive Integer Powers, Multiples, Sums, and Differences

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 70 Rule 5 The Product Rule

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 71 Example Using the Product Rule

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 72 Rule 6 The Quotient Rule

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 73 Example Using the Quotient Rule

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 74 Rule 7 Power Rule for Negative Integer Powers of x

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 75 Example Negative Integer Powers of x

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 76 Second and Higher Order Derivatives

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 77 Second and Higher Order Derivatives

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 78 Quick Quiz Sections 3.1 – 3.3

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 79 Quick Quiz Sections 3.1 – 3.3

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 80 Quick Quiz Sections 3.1 – 3.3

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 81 Quick Quiz Sections 3.1 – 3.3

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 82 Quick Quiz Sections 3.1 – 3.3

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 83 Quick Quiz Sections 3.1 – 3.3

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

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 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.

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 90 Instantaneous Rates of Change

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 91 Example Instantaneous Rates of Change

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 92 Motion Along a Line

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 93 Instantaneous Velocity

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 94 Speed

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

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 96 Free-fall Constants (Earth)

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 97 Example Finding Velocity

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 98 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.

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 99 Derivatives in Economics

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 100 Example Derivatives in Economics

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

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

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

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

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 105 Quick Review Solutions

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 106 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.

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 107 Derivative of the Sine Function

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 108 Derivative of the Cosine Function

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 109 Example Finding the Derivative of the Sine and Cosine Functions

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

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 111 Example Simple Harmonic Motion

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

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 113 Derivative of the Other Basic Trigonometric Functions

114. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 114 Example Derivative of the Other Basic Trigonometric Functions

115. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 115 Example Derivative of the Other Basic Trigonometric Functions

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

117. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 117 Quick Review

118. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 118 Quick Review

119. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 119 Quick Review Solutions

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

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

122. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 122 Rule 8 The Chain Rule

123. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 123 Example Derivatives of Composite Functions

124. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 124 “Outside-Inside” Rule

125. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 125 Example “Outside-Inside” Rule

126. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 126 Example Repeated Use of the Chain Rule

127. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 127 Slopes of Parametrized Curves

128. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 128 Finding dy/dx Parametrically

129. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 129 Power Chain Rule

130. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 130 Quick Quiz Sections 3.4 – 3.6

131. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 131 Quick Quiz Sections 3.4 – 3.6

132. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 132 Quick Quiz Sections 3.4 – 3.6

133. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 133 Quick Quiz Sections 3.4 – 3.6

134. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 134 Quick Quiz Sections 3.4 – 3.6

135. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 135 Quick Quiz Sections 3.4 – 3.6

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

137. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 137 Quick Review

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 Quick Review Solutions

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

144. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 144 Implicitly Defined Functions

145. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 145 Implicitly Defined Functions

146. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 146 Example Implicitly Defined Functions

147. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 147 Implicit Differentiation Process

148. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 148 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.

149. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 149 Lenses, Tangents and Normal Lines

150. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 150 Example Lenses, Tangents and Normal Lines

151. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 151 Example Lenses, Tangents and Normal Lines

152. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 152 Example Derivatives of a Higher Order

153. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 153 Rule 9 Power Rule For Rational Powers of x

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

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

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

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

158. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 158 Quick Review Solutions

159. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 159 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.

160. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 160 Derivatives of Inverse Functions

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

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

163. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 163 Derivative of the Arctangent

164. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 164 Derivative of the Arcsecant

165. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 165 Example Derivative of the Arcsecant

166. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 166 Inverse Function – Inverse Cofunction Identities

167. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 167 Calculator Conversion Identities

168. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 168 Example Derivative of the Arccotangent

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

170. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 170 Quick Review

171. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 171 Quick Review

172. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 172 Quick Review Solutions

173. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 173 Quick Review Solutions

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

175. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 175 Derivative of e x

176. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 176 Example Derivative of e x

177. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 177 Derivative of a x

178. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 178 Derivative of ln x

179. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 179 Example Derivative of ln x

180. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 180 Derivative of log a x

181. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 181 Rule 10 Power Rule For Arbitrary Real Powers

182. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 182 Example Power Rule For Arbitrary Real Powers

183. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 183 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.

184. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 184 Example Logarithmic Differentiation

185. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 185 Quick Quiz Sections 3.7 – 3.9

186. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 186 Quick Quiz Sections 3.7 – 3.9

187. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 187 Quick Quiz Sections 3.7 – 3.9

188. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 188 Quick Quiz Sections 3.7 – 3.9

189. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 189 Quick Quiz Sections 3.7 – 3.9

190. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 190 Quick Quiz Sections 3.7 – 3.9

191. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 191 Chapter Test

192. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 192 Chapter Test

193. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 193 Chapter Test Solutions

194. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 194 Chapter Test Solutions

195. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 195 Chapter Test Solutions