1. 3.1 – Derivative of a Function

2. Slope of the Tangent Line
If f is defined on an open interval containing c and the limit exists, then
and the line through (c, f (c)) with slope m is the line tangent to the graph of f at the point (c, f (c)).

3. The Slope of the Graph of a Linear Function
Find the slope of the graph of
at the point (2, 1).

4. The Slope of the Graph of a Nonlinear Function
Find the slope of the graph of
at the point (0, 1) and (-1, 2).
Find the equation of the tangent line at each point.

5. Definition Derivative – The derivative of f at x is given by
provided the limit exists. For all x for which this limit exists, is a function of x.

6. Find the Derivative of the Function

7. Alternate Definition
The derivative of the function f at the point x = a is the limit
provided the limit exists.

8. Find the Derivative of the Functions

9. Homework
p.105 ~ 1-9 (O), 13-16, 17, 19

10. Reflection
p.105 ~ 1-9 (O), 13-16, 17, 19

11. Differentiation Rules
3.3.1 (also 3.2)

12. When Derivatives Do Not Exist

13. When Derivatives Do Not Exist

14. When Derivatives Do Not Exist

15. When Derivatives Do Not Exist

16. The Constant Rule
The derivative of a constant function is 0. That is, if c is a real number, then .

17. The Power Rule
If n is a rational number, then the function f (x) = xn is differentiable and
For f to be differentiable at x = 0, n must be a number such that xn–1 is defined on an interval containing 0.

18. Find the Derivative of the Function

19. The Constant Multiple Rule
If f is a differentiable function and c is a real number, then cf is also differentiable and .

20. Find the Derivative of the Function

21. The Sum and Difference Rules
The sum (or difference) of two differentiable functions f and g is itself differentiable. Moreover, the derivative of f + g (or f – g) is the sum (or difference) of the derivatives of f and g.

22. Find the Derivative of the Function

23. The Slope of a Graph
Find the slope of the graph of when , , and

24. The Tangent Line Find an equation of the tangent line to the graph
of when

25. The Product Rule
The product of two differentiable functions f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function time the derivative of the first.

26. Find the Derivative

27. Find the Derivative

28. The Quotient Rule
The quotient of two differentiable functions f and g is itself differentiable for all values of x for which g(x) ≠ 0. Moreover, the derivative of f / g is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

29. Find the Derivative

30. Find the Derivative

31. Homework
p. 114 ~ 1-10
p. 124 ~ 1-5, 7-29 (O), 30, 32

32. Higher Order Derivatives
3.3.2

33. Which Rule Do I Use?
Find the Derivative

34. Higher-order Derivative Notation
First Derivative:
Second Derivative:
Third Derivative:
Fourth Derivative:
nth Derivative:

35. Find the Second Derivative

36. Instantaneous Rate of Change
A population of 500 bacteria is introduced into a culture and grows in number according to the equation
where t is measured in hours. Find the rate at which the population is growing when t = 2

37. Velocity and Other Rates of Change
3.4

38. Position and Velocity
If a billiard ball is dropped from a height of 100 feet, its height s at time t is given by the position function
where s is measured in feet and t is measured in seconds. Find the average velocity over each of the following time intervals.
[1, 2] [1, 1.5] [1, 1.1]

39. Position Function

40. Instantaneous Velocity
At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. the position of the diver is given by
where s is measured in feet and t is measured in seconds.
When does the diver hit the water?
What is the diver’s velocity at impact?

41. Higher-order Derivatives
Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by
where s(t) is the height in meters and t is the time in seconds. What is the ratio of Earth’s gravitational force to the moon’s?

42. Free Fall
A silver dollar is dropped from the top of a building that is 1362 feet tall.
Determine the position, velocity, and acceleration functions for the coin.
Determine the average velocity on the interval [1, 2].
Find the instantaneous velocities when t = 1 and t = 2.
Find the time required for the coin to reach ground level.
Find the velocity of the coin at impact.

43. Free-Fall
A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds?
What is the maximum height of the projectile?

44. Homework
p.135/1, 3, 7, 9-15 (O), 19, 21

45. Derivatives of Trigonometric Functions
3.5

46. The Derivative of Sine

47. Derivatives of Sine and Cosine Functions

48. Find the Derivative of the Function

49. Find the Derivative

50. Simple Harmonic Motion
A weight hanging from a spring is stretched 5 units beyond its rest position (x = 0) and released at time t = 0 to bob up and down. Its position at any later time t is
What are its velocity and acceleration at time t? Describe its motion.

51. Derivative of Tangent

52. Derivatives of Trigonometric Functions

53. Find the Derivative

54. Find the Second Derivative

55. Homework
p. 146/ 1-9 (O), (O), (O), 43

56. The Chain Rule
4.1

57. The Chain Rule
If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f (g(x)) is a differentiable function of x and
or, equivalently,

58. Identify the inner and outer functions
Composite y = f (g(x)) Inner u = g(x) Outer y = f (u)

59. The General Power Rule
If , where u is a differentiable function of x and n is a rational number, then
or, equivalently,

60. Find the Derivative

61. Find the Derivative

62. Homework
Chain Rule Worksheet

63. Factoring Out the Least Powers
Find the Derivative

64. Factoring Out the Least Powers
Find the Derivative

65. Factoring Out the Least Powers
Find the Derivative

66. Find the Derivative

67. Find the Derivative

68. Trig Tangent Line Find an equation of the tangent line to the graph of
at the point (π, 1). Then determine all values of x in the interval (0, 2π) at which the graph of f has a horizontal tangent.

69. Homework
p.153/ 1-11odd, 21-39odd, 59

70. Implicit Differentiation
4.2

71. Find dy/dx

72. Guidelines for Implicit Differentiation
Differentiate both sides of the equation with respect to x.
Collect all terms involving dy / dx on the left side of the equation and move all other terms to the right side of the equation.
Factor dy / dx out of the left side of the equation.
Solve for dy / dx.

73. Find the derivative

74. Homework
p.162/ 1-19odd, 49, 51

75. Example Determine the slope of the tangent line to the graph of
at the point

76. Example Determine the slope of the tangent line to the graph of
at the point

77. Finding the Second Derivative Implicitly

78. Finding the Second Derivative Implicitly

79. Example Find the tangent and normal line to the graph given by
at the point

80. Homework
p.162/ 21-25odd, 27-30, 31-43odd

81. Inverse Functions
4.3

82. Definition of Inverse Function
A function g is the inverse function of the function f if
for each x in the domain of g.
and
for each x in the domain of f.

83. Verifying Inverse Functions
Show that the functions are inverse functions of each other.
and

84. The Existence of an Inverse Function
A function has an inverse function if and only if it is one-to-one.
If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.

85. Existence of an Inverse Function
Which of the functions has an inverse function?

86. Finding an Inverse
Find the inverse function of .

87. The Derivative of an Inverse Function
Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which Moreover,

88. Example Let . What is the value of when x = 3?

89. Homework
p. 44/ 1-6, 7-23odd, 43
p. 170/ 28, 29bc

90. Inverse Trigonometric Functions
3.8

91. The Inverse Trigonometric Functions

92. Evaluating Inverse Trigonometric Functions
Evaluate each function.

93. Solving an Equation

94. Using Right Triangles a) Given y = arcsin x, where , find cos y.
b) Given , find tan y.

95. Homework
3.8 Inverse Trig Review worksheet

96. Derivatives of Inverse Trigonometric Functions
Find

97. Derivatives of Inverse Trigonometric Functions

98. Differentiating Inverse Trigonometric Functions

99. Differentiate and Simplify

100. Homework
p. 170/ 1-27odd, 31ab

101. Derivatives of Exponential and Logarithmic Functions
4.4

102. Properties of Logarithms
If a and b are positive numbers and n is rational, then 1. 2. 3. 4.

103. Expand the following logarithms

104. Solving Equations
Solve 7 = ex + 1
Solve ln(2x  3) = 5

105. Derivative of the Natural Exponential Function

106. Examples

107. Derivatives for Bases Other than e

108. Find the derivative of each function.

109. Homework
p.44/ p.178/ 1-13odd, 29, 30

110. Derivative of the Natural Logarithmic Function

111. Example
Find the derivative of ln (2x).

112. Find the Derivative 1. 2. 3.

113. Derivatives for Bases Other than e

114. Example
Differentiate

115. Example
Differentiate

116. Example
Differentiate

117. Find an equation of the tangent line to the graph at the given point.

118. Homework
p. 178/15-27odd, 37-41odd

119. Comparing Variable and Constants

120. Derivative Involving Absolute Value

121. Find the derivative

122. Use implicit differentiation to find dy/dx.
ln xy + 5x = 30

123. Logarithmic Differentiation
Differentiate .

124. Logarithmic Differentiation

125. More Logarithmic Differentiation

126. Homework
p. 179/ 31, 33-36, 43-55odd