1. Find the derivative of the function f(x) = x 2 – 2x

2. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions. Objective – To be able to find the derivative of a function.

3. RULE 1 Derivative of a Constant Function If f has the constant value f(x) = c, then If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

4. Example 1a Find the derivative of f(x) = 8 Example 1b Find the derivative of

5. RULE 2 Power Rule for Positive Integers If n is a positive integer, then

6. In the Warm-Up we saw that if,. This is part of a pattern. examples: power rule

8. RULE 3 Constant Mulitple Rule If u is a differentiable function of x, and c is a constant, then

9. examples: constant multiple rule:

10. RULE 4 The Sum Rule If f and g are both differentiable, then

11. Example Find the derivative of y = x 4 + 12x

12. RULE 5 The Difference Rule If f and g are both differntiable, then

13. Example Find the derivative of y = x 3 – 3x

14. RULE 6 The Derivative of the Natural Exponential Function

15. Pg. 191 3 – 31 odd

16. Find the derivative of the function f(x) = 3x 2 – 5x + 1

17. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions. Objective – To be able to find the derivative of a function.

18. Derivative of Sine and Cosine Functions:

19. THE PRODUCT RULE:

20. Example 1 Find the derivative of f(x) = (2x+5)(3+4x) f(x)g(x)

21. Example 2 Find the derivative of f(x) = (4x 3 )(sin x) f(x)g(x)

22. Example 3 Find the derivative of y = (5x 2 )(cos x) + (3x)(sinx)

23. 3-25 odd

24. Find the derivative of the function f(x) = (x – 4)(x + 3)

25. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions. Objective – To be able to find the derivative of a function.

26. THE QUOTIENT RULE:

27. Example 1 Find the derivative of f(x) g(x)

28. Example 2 Find the derivative of f(x) g(x)

29. Example 3 Find the derivative of f(x) g(x)

30. 3 – 25 odd

31. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions. Objective – To be able to find the derivative of a function.

32. Derivatives of the remaining trig functions:

33. Example 1 Find the derivative of y = (sec x)(tan x)

34. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for. 

36. WS

37. Find the Derivative:

38. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions. Objective – To be able to find the derivative of a function.

39. So far we have been memorizing the derivatives of the trig functions. And today we will be investigating this further. This will help us as we go into the next section of using the Chain Rule.

40. 2) Find the derivative of f(x) = (x)(sin x)

41. ON WHITE BOARD a)Find the derivative of f(x) = (x)(cos x) b)Find the derivative of f(x) = (x)(tan x)

42. 4) Find the derivative of y = 2 csc x + 5 cos x

43. ON WHITE BOARD a)Find the derivative of y = 4 sec x + 3 sin x b)Find the derivative of y = 7 cot x + 2 tan x

44. 10) Find the derivative of 1

45. ON WHITE BOARD 12)

46. Pg. 216 1 – 15 odd

47. Find y´´

49. State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions. Objective – To be able to use the Chain Rule to solve applications.

50. We now have a pretty good list of “shortcuts” to find derivatives of simple functions. Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.

51. Consider a simple composite function:

52. one more: This pattern is called the chain rule.

53. The Chain Rule can be written either in Leibniz notation: Or in Prime Notation:

54. Example 1 Find of

55. On White Board Find of

56. On White Board Find of

57. 9 – 42 mult of 3

58. Find y´´

60. State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions. Objective – To be able to use the Chain Rule to solve applications.

61. The Chain Rule can be written either in Leibniz notation: Or in Prime Notation:

62. Example 1 Find of

63. On White Board Find of

64. On White Board Find of

65. 9 – 42 mult of 3

66. Find the Derivative

67. Find y´´

69. State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions. Objective – To be able to use the Chain Rule to solve applications.

70. Repeated Use of the Chain Rule We sometimes have to use the Chain Rule two or more times to find a derivative.

71. Example 1 Find the derivative of g(t) = tan (5 – sin 2t)

72. Example 2 Find the derivative of f(x) = cos ( sin 3x)

73. Find the Derivative

74. 9 – 42 mult of 3

75. Find the Derivative

76. Find y´´ Find the Derivative:

77. State Standard – 6.0 Students use implicit differentiation in a wide variety of problems. Objective – To be able to use implicit differentiation

78. Implicit Differentiation (Takes Four Steps) 1) Differentiate both sides of the equation with respect to ‘x’, treating ‘y’ as a differentiable function of ‘x’. 2) Collect the terms with the dy / dx on one side of the equation. 3) Factor out the dy / dx. 4) Solve for dy / dx.

79. This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. Note use of chain rule.

80. This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for.

81. Example 1 Find of

82. Example 2 Find of

83. Example 3 Find of

84. Example 4 Find of

85. Pg. 233 1a, 2a, 3a, 5 – 9, 11, 13, 14, and 20

86. From www.Dictionary.com Explicit- 6.Mathematics. (of a function) having the dependent variable expressed directly in terms of the independent variables, as y = 3x + 4. Implicit- 4.Mathematics. (of a function) having the dependent variable not explicitly expressed in terms of the independent variables, as x 2 + y 2 = 1.

87. Find of

88. State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of inverse trig functions. Objective – To be able to take derivatives of Inverse Trig Functions.

90. 1 y x b

91. Example 1 Find the derivative of

92. Example 2 Find the derivative of

93. Find the Derivative.

94. In Sec. 3.2 we learned the basics of Higher Order Derivatives. So Find the first four derivatives of First Derivative: Second Derivative: Third Derivative: Fourth Derivative:

95. Velocity and Acceleration

96. State Standard – 7.0 Students compute derivatives of higher order. Objective – To be able to solve problems involving multiple derivative steps.

97. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for.

98. Example 1 Find the second derivative of

99. Example 2 Find the second derivative of

100. Acceleration is the first derivative of Velocity. Velocity is the first derivative of position. Acceleration is the second derivative of position.

101. Example 3 The position of a particle is given by the equation: Where t is measured in seconds and s in meters. a) Find the acceleration at time t. What is the acceleration after 4 seconds.

102. Pg. 240 5 – 15 odd, 29, 31, 43 and 44

103. The position of a particle is given by the equation: Where t is measured in seconds and s in meters. a) Find the acceleration at time t. What is the acceleration after 5 seconds.

104. A dynamite blast blows a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of: Where t is measured in seconds and s in feet. a)How high does the rock go? What is the velocity at the rock’s height?

105. State Standard – 4.4 Students find the derivatives of logarithmic functions. Objective – Students will be able to solve problems involving logarithmic functions.

106. Derivative of Log Functions Example 1

107. Derivative of Log Functions Example 2 Product RuleChain Rule

108. More Derivatives Example 3

109. Derivative of Log Functions (Base other than e) Example 4

110. Derivative of Log Functions (Base other than e) Example 5

111. Pg. 249 2 – 5, 7, 9, 21 – 23, and 30

112. Find the First Derivative: 1) Find the second Derivative: 2)

113. State Standard – 4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Objective – Students will be able to solve problems involving rate of change.

114. If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. We will find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.

115. Example 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm 3 /s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

116. Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

117. Example 2 How rapidly will the fluid level inside a vertical cylindrical tank drop if we pump the fluid out at the rate of 3000 L/min? But since there are 1000L in a cubic meter. The fluid level will drop at the rate of

118. Example 3 A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2 m 3 /min, find the rate of rising water when the water is 3 m deep.

119. Example 4 Water runs into a conical tank at the rate of 9 ft 3 /min. The tank stands point down and has a height of 10 ft. and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep?

120. Pg. 260 1, 3 – 5, 7, 10, and 19