1. Day 1: Derivatives in mathematics come from something else, just like derivative prose comes from something else. What do you think derivatives come from?
DERIVATIVES

2. Right! Functions!! Linear Quadratic Cubic Exponentials Logarithms Sine
Cosine
(and maybe even a few more)

3. Derivative: Slope of a curve at a point
Remember this:
Derivative: Slope of a curve at a point

4. Recall or Learn Point-Slope Equation of a Line:
y-y1=m(x-x1)
TI84 this is nDeriv under Math
Which is math 8: Or Calc from the graph.
nDeriv (function, variable, value)

5. Zooooooooooooooooom!
Let’s graph a function in our calculators. Let’s graph
Now go to where x=2.5 by hitting “trace” and then Zoom in over and over until the graph looks like a straight line

6. And….
Let’s try doing a derivative of it in the calculator. Do the derivative of the function at 2.5.
To do this, go to Math 8: nDeriv ( 9− 𝑥 2 , 𝑥,2.5)
Also try doing calc 6 from the graph

7. Discussion
When you zoomed in until it looked like a straight line, that is the beginning of a linearization of the function. Why do you think it is called linearization?
What is a derivative? (All together now!)
How does a derivative relate to a linearization?

8. Analytic Definition of Derivative
The derivative of the function f with respect to the variable x is the function f’ whose value at x is
𝑓 ′ 𝑥 = lim 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ

9. Two Definitions of Derivative
Let’s look at a four-step process for computing derivatives from the analytic definition

10. Four Step Process for Derivative from Definition
Find f(x+h)
Determine f(x+h)-f(x) note: all terms must end up with “h” as a factor
Simplify f(x+h)-f(x) h
Take its limit as h 0

11. Alternate definition of derivative
Now, let’s look at the alternate definition of derivative at a point:
𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓(𝑎) 𝑥−𝑎
Is the derivative of the function f at the point x=a, provided the limit exists
How does this relate to the other definition?

12. Examples
Use each definition (and the four steps process if you need it)to find the derivative of:
𝑓 𝑥 =5𝑥−2

13. Homework Page 101 Quick Review: 1-4 Exercises: 1-6,12
Read page 99 example 5

14. Day 2: Homework – Work Day in Class
Page 101
7-25 (odd)

15. Day 3: Differentiability
Big idea: The ability for a function to have a derivative

16. Differentiability All of the following are notations for derivative:
y’ “y prime”
𝑑𝑦 𝑑𝑥 “ dy dx” or “the derivative of y with respect to x”
𝑑𝑓 𝑑𝑥 “df dx” or “the derivative of f with respect to x”
𝑑 𝑑𝑥 𝑓(𝑥) “d dx of f at x” or “the derivative of f at x”
Where do you think derivatives might not exist?

17. Derivatives fail to exist at:
Cusps
Corners
Vertical Tangents
Discontinuity
Infinite Roughness
Sheesh! How can you find these out?

18. Continuity?
Do you think continuity is necessary in order to have a derivative at a point?
It is. In order to have a derivative at a point, the function must be continuous there.

19. Intermediate Value Theorem for Derivatives
If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b)
Let’s discuss

20. Homework 111(1-15,17,21,23) Finish in class tomorrow
Read Example 3 carefully (page 108)
Try Example 4 in your calculator (page 109)

21. Day 4: Work in Class
Finish from Day (1-15,17,21,23)

22. Day 5: Linearization Recall the point slope equation 𝑦−𝑦 1 =𝑚 𝑥− 𝑥 1
Imagine that we use, instead of y, the expression L(x) for linearization of x and f’(x) instead of m for the slope. Make sense?
Then
𝐿 𝑥 − 𝑦 1 = 𝑓 ′ 𝑥 𝑥− 𝑥 1 OR
𝐿 𝑥 = 𝑓 ′ 𝑥 𝑥− 𝑥 1 + 𝑦 1

23. Local Linearity handout Do 229(1,3,7,11)
Linearization
Local Linearity handout
Do 229(1,3,7,11)
Here are the derivatives for the book work:
1. 3 𝑥 2 − − 1 𝑥 2
7. 𝑘 (1+𝑥) 𝑘− 𝑥+4

24. Relationships between graphs of f, f’ and f’’
Day 6
Relationships between graphs of f, f’ and f’’

25. Facts about f’ (First Derivative)
-When f has a positive slope, f’ is
positive (above the x-axis)
-When f has a slope of 0 (horizontal
tangent line), f’ has an x-intercept
-When f has a negative slope, f is
negative (below x-axis)

26. Facts about f” (Second Derivative)
-When f changes from concave to convex, f” has an x-intercept (inflection point)
-When f is concave up (like a cup), f” is positive (above x – axis)
-When f is concave down (like a frown), f” is negative (below x – axis)

27. Work
Complete Relationship between graphs of f and f’ worksheet (see video for help)
Graphing f and f’ worksheet for homework
Go to classroom website for other resources

28. Day 7: by tomorrow, you will know
Power Rule
Derivative of a Constant
Derivatives of sin, cos, and tan
Derivative of ex
Derivative of lnx

29. Derivative of a Constant Function
What does a constant function mean?
What does a constant function look like?
What is the slope of a constant function?
What is the derivative of a constant function?

30. Power Rule lim ℎ→0 (𝑥+ℎ) 𝑛 − 𝑥 𝑛 ℎ Let’s develop this together
If f(x) = xn , then we need to find
I will now walk through the proof of this power rule with you from page 113 in your text
lim ℎ→0 (𝑥+ℎ) 𝑛 − 𝑥 𝑛 ℎ

31. Proof to know for AP Test

32. Power Rule and Constant Multiple Rule
If n is a positive integer, then
If u is a differentiable function of x and c is a constant, then
𝑑 𝑑𝑥 𝑥 𝑛 = 𝑛𝑥 𝑛−1

33. Let’s practice the power rule and the constant multiple rule
Try finding the derivative for each of the following. You may do more than one term. Just do them term by term (that is the sum and difference rule)
𝑓 𝑥 =3 𝑥 7
g(x)= 𝑥
f(c)=6 𝑐 2 − 3 𝑐 +10

36. Derivative of lnx
Although we will have to revisit the derivation, you need to know the derivative of ln⁡(𝑥)

37. The second derivative is the derivative of the derivative
It is denoted f’’(x) or y’’ or

38. What do you call a third derivative, d3y/dx3?
Calculus Joke?
What do you call a third derivative, d3y/dx3?
A Jerk (Now you can call people third derivatives if you need to)

39. With the derivative, we can find…
Antiderivatives!!
*Always add a +C to the end of your answer.

40. What if I know the derivative?
Tell me the original function if you know the derivative is as follows: (work together)
𝑓 ′ 𝑥 = 6𝑥 5
𝑔 ′ 𝑟 = 4𝑟 3 − 𝑟
ℎ ′ 𝑛 = 𝑛 2

41. Homework
Pg. 120(1-10,25,26,27) Pg. 192(25-27,31,32) Pg. 140(1-3)

42. Day 8: Integral Notation (What is definite? Indefinite?)

43. Separable Differential Equations
If you know that y’=2x and are asked, what is y? How would you approach that?
This is a simple differential equation and it is separable. Let’s examine why.

44. Process to Solve – AP Test
1. Separate the variables (get all x’s to one side and all the y’s to the other)
2. Take the antiderivative of each side (being sure to include the variable of integration). On the y side you’ll have a dy and on the x side you’ll have a dx.
3. Make sure you put a “+ C” on the end!

45. Initial Condition??
Let’s examine the same problem, only this time we know that (1,5) is on the final function.
How does this help? What can we find out?

46. Homework on Integrals
Pg. 312
#1-3,8-13,31,32,34,35, Omit 9

47. Day 9
Recall how a differential equation results in an indefinite integral for a solution. Since it is indefinite, there is a family of functions that are possible solutions. Work together on the slope field worksheets Note: On the AP test, this entire section is with no calculator

48. Day 10: Work in Class on Slope Fields
Do page 313(27, 28, 30, 43, 44)
And AP problem 2005#6

49. Day 11: Product Rule and Quotient Rule
The Product Rule:
The product of two differentiable functions u and v is differentiable and
The Quotient Rule:
At a point where v≠0, the quotient y=u/v of two differentiable functions is differentiable, and

50. Day 11: Product Rule and Quotient Rule
The Product Rule:
𝑢 𝑣 ′ +𝑣𝑢′
The Quotient Rule:
𝑣 𝑢 ′ −𝑢 𝑣 ′ 𝑣 2

51. Pg. 120(11-20) Begin the product and quotient rule handout
Assignment
Pg. 120(11-20) Begin the product and quotient rule handout

52. Continue practice on Product and Quotient Rule
Day 12: More Practice
Continue practice on Product and Quotient Rule

53. Day 13: More Product/Quotient
120(21-33odd, 34)

54. Day 14: Displacement, Velocity and Acceleration
What do you know about displacement? Distance? Velocity? Speed? Acceleration?
What does instantaneous mean?
How about average?
Free-fall
Sensitivity to change
How could derivatives be involved in economics?

55. Distinctions between
Speed is the magnitude (or absolute value) of velocity
Distance is the magnitude (or absolute value) of displacement

56. Rates of change
What is the rate of change of displacement with respect to time?
What is the rate of change of velocity with respect to time?
What accumulates as one travels at a velocity over a period of time?

57. Example
S(t)=1/2 at2 + vot + so
If a(t)=9.8 and v0=25 and so=12
Remember: s’(t)=v(t) and s’’(t)=v’(t)=a(t)
Find v(t) and a(t)
Marginal cost is the derivative of cost and represents the instantaneous change in cost

58. Example Average velocity = displacement/time
Instantaneous velocity: Derivative
Acceleration: Second Derivative of position or first derivative of velocity
Maximum height: Find derivative then set equation = 0. Solve

59. Example
3. A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec reaches a height of 𝑠=24𝑡−0.8 𝑡 2 meters in t seconds.
a. Find the rock’s velocity and acceleration as functions of time.
Velocity 𝑣 𝑡 = 𝑑𝑠 𝑑𝑡 = 𝑑 𝑑𝑡 24𝑡− 0.8𝑡 2 =𝟐𝟒− 𝟏.𝟔𝒕 m/sec
Acc 𝑎 𝑡 = 𝑑𝑣 𝑑𝑡 = 𝑑 𝑑𝑡 24−1.6𝑡 =−𝟏.𝟔 𝐦/ 𝐬𝐞𝐜 𝟐

60. Example
b. How long did it take the rock to to reach its highest point?
Look for when the derivative is 0.
24−1.6𝑡=0
t= 15. So it took 15 seconds

61. Example 24−1.6𝑡=0 24−1.6 15 =𝟏𝟖𝟎 meters c. How high did the rock go?
Plug in previous answer into original equation.
24−1.6𝑡=0
24− =𝟏𝟖𝟎 meters

62. Example
d. When did the rock reach half its maximum height? (If the max height was 180, then we want to set original equation = 90)
24𝑡−0.8 𝑡 2 =90
0=0.8 𝑡 2 −24𝑡+90
If you cannot factor, then you need to use Quadratic Formula.
So, it took about seconds to reach half its maximum height.

63. Example
e. How long was the rock aloft?
To find how long the rock was in the air, we can find the x-intercepts of the original equation.
−0.8 𝑡 2 +24𝑡=0
−0.8𝑡 𝑡−30 =0
t = 0 and 30. The rock was in the air for 30 seconds.

64. Assignment
Pg. 129(1, 2, 5, 9, 10, 14, 15) For extra problems- check website for video called “Pg. 115,117,118 – Video”

65. Day 15: Work work work
130(12, 21, 23, 25, 26, 28, 31)

66. Day 16: Initial Value Problem
When you do an integral you always have to “C” and you do not know what specific function is the antiderivative.
In an initial value problem, a value is given so that the “C” may be computed and the function determined.

67. Initial Value Problems
Problem # AP Test

68. Initial Value Problems

69. Initial Value Problems
Plug in -1 for x to find the slope
Use point-slope form to make eq.
𝑦−2=2(𝑥+1)

70. Initial Value Problems
Separate the variables
Find antiderivative
Plug in numbers to find C
Plug C back in and solve for y

71. Homework
Assignment: (39,41,42)

72. Day 17: More Derivatives
If you were asked to find the derivative of the sine function, how could you do that?
Right! Use the definition of derivative and trig identities. “Take it to the Limit!”
See and know page 135 derivation

73. Derivation of Sine Derivative

74. Trig Derivatives (Use Radians)

75. More Trig Derivatives

76. Homework
140(1-10)
192(28,29,30,32,33,3 4,42)

77. Day 18
140(11-21, odd)

78. Give me some examples of composite functions
Day 19: Chain Rule
Give me some examples of composite functions

79. How can we differentiate composite functions?
The chain rule!!
If f is differentiable at the point u=g(x) and g is differentiable at x, the composite function (f◦g)’(x)=f’(g(x))·g’(x)

80. Or in Leibniz notation

81. Still not making sense?
𝑑𝑦 𝑑𝑥 = 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝐨𝐮𝐭𝐬𝐢𝐝𝐞, 𝑙𝑒𝑎𝑣𝑖𝑛𝑔 𝑖𝑛𝑠𝑖𝑑𝑒 𝑎𝑙𝑜𝑛𝑒 𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝒊𝒏𝒔𝒊𝒅𝒆

82. 𝑓 𝑥 = sin ( 3𝑥 2 −4𝑥) 𝑔 𝑥 =16 ( 3𝑥 4 − 7𝑥 6 +9) 3
Let’s try some….
𝑓 𝑥 = sin ( 3𝑥 2 −4𝑥)
𝑔 𝑥 =16 ( 3𝑥 4 − 7𝑥 6 +9) 3
ℎ 𝑥 = (2𝑥−3) 4 ( 𝑥 2 +𝑥+1) 5

83. Work Space

84. Work Space

85. Work Space

86. What was the chain rule again?

87. Let me show you “u” on those same problems
𝑓 𝑥 = sin ( 3𝑥 2 −4𝑥)
𝑓 𝑢 = sin 𝑢 let 𝑢= 3𝑥 2 −4𝑥
𝑓 ′ 𝑢 = cos 𝑢 𝑑𝑦 𝑑𝑥 =6𝑥−4
= cos 𝑢 ∙ 𝑢 ′
= cos 3𝑥 2 −4𝑥 (6𝑥−4)
= 𝟔𝒙−𝟒 𝐜𝐨𝐬 𝟑𝒙 𝟐 −𝟒𝒙

88. 𝑔 𝑥 = cos 2 ( ln ( 5𝑥 3 +2𝑥)) Multiple Chain Rules
𝟐 𝐜𝐨𝐬 (𝐥𝐧⁡( 𝟓𝒙 𝟑 +𝟐𝒙)(− 𝐬𝐢𝐧 𝐥𝐧 𝟓𝒙 𝟑 +𝟐𝒙 𝟏 𝟓𝒙 𝟑 +𝟐𝒙 ∙(𝟏𝟓 𝒙 𝟐 +𝟐)
−𝟐(𝟏 𝟓𝒙 𝟐 +𝟐) 𝟓𝒙 𝟑 +𝟐𝒙 𝐜𝐨𝐬 ( 𝐥𝐧 ( 𝟓𝒙 𝟑 +𝟐𝒙))( 𝐬𝐢𝐧 ( 𝐥𝐧 ( 𝟓𝒙 𝟑 +𝟐𝒙)))

89. 𝑦= tan ( 2𝑥−𝑥 3 ) 𝑑𝑦 𝑑𝑥 = sec 2 (2𝑥− 𝑥 3 )(2− 3𝑥 2 )
Another example pg. 146 #4
𝑦= tan ( 2𝑥−𝑥 3 )
𝑑𝑦 𝑑𝑥 = sec 2 (2𝑥− 𝑥 3 )(2− 3𝑥 2 )
=(𝟐− 𝟑𝒙 𝟐 )( 𝐬𝐞𝐜 𝟐 (𝟐𝒙− 𝒙 𝟑 )

90. What about integrals?
Determine what u will be and take the derivative. Solve for dx
Replace dx with your substitution found in step 1
Find the antiderivative of the equation (+C)
Replace u with the original information

91. What about integrals? Let u = 4x -1 𝑑𝑢 𝑑𝑥 =4 𝑑𝑢=4𝑑𝑥 𝒅𝒙= 𝟏 𝟒 𝒅𝒖 4𝑥−1 𝑑𝑥
𝑢 𝑑𝑥
𝑢 𝑑𝑢
Let u = 4x -1
𝑑𝑢 𝑑𝑥 =4
𝑑𝑢=4𝑑𝑥
𝒅𝒙= 𝟏 𝟒 𝒅𝒖

92. What about integrals? 𝟏 𝟔 (𝟒𝒙−𝟏) 𝟑 𝟐 +𝑪 1 4 𝑢 1 2 𝑑𝑢
𝑢 𝑑𝑢
2 3 ∙ 1 4 𝑢 𝐶
1 6 𝑢 𝐶
𝟏 𝟔 (𝟒𝒙−𝟏) 𝟑 𝟐 +𝑪

93. 146(1-23, odd) 321(1-11, odd) omit 5 312(4) 140(27)
Now you can do homework
146(1-23, odd)
321(1-11, odd) omit 5
312(4) 140(27)

94. Day 20
147(25-31, odd)
322(15-21, odd)
312 (7)

95. Day 21: Work Day
312(9,14,16,18, ,29,32,35,37)

96. Day 22
146(2, 10, 16, 24, 32, 54, 61, 62)
321(2, 8, 16)

97. Day 23
Pg. 147(33,36,39, 51,54,57,63,69)

98. Day 24: Implicit Differentiation
What is the meaning of implicit?
implied, rather than expressly stated: implicit agreement.
Mathematics. (of a function) having the dependent variable not explicitly expressed in terms of the independent variables, as x2 + y2 = 1.
Implicit Differentiation is just an application of the chain rule

99. Implicit Differentiation Procedure
Differentiate both sides of the equation with respect to x.
Collect the terms with dy/dx on one side of the equation.
Factor out dy/dx
Solve for dy/dx

100. What’s different here?
Since you are differentiating with respect to x, any time you differentiate terms containing y you will need to do the chain rule and multiply dy/dx on the end. (Same thing for other variables)

101. Example

102. Cautions
Remember that you need dy/dx after any y terms you differentiate
Remember that the derivative of a constant is zero
Sometimes you will go back and substitute the original y into the final answer

103. Example: 1. Take derivative of eq 2. Put a dy/dx when taking
2𝑥 2 +2𝑥𝑦=16
1. Take derivative of eq
2. Put a dy/dx when taking
deriv of y
3. Solve equation for dy/dx
4𝑥+2 1∙𝑦+𝑥 𝑑𝑦 𝑑𝑥 =0
4𝑥+2 𝑦+𝑥 𝑑𝑦 𝑑𝑥 =0
4𝑥+2𝑦+2𝑥 𝑑𝑦 𝑑𝑥 =0
2𝑥 𝑑𝑦 𝑑𝑥 =−4𝑥−2𝑦
𝑑𝑦 𝑑𝑥 = −4𝑥−2𝑦 2𝑥
𝑑𝑦 𝑑𝑥 = −2𝑥−𝑦 𝑥

104. Assignment
155( 3-18 mult 3, 21, (y’ only), 28, 30, 34, 42, 45)

105. Day 25: Derivatives of Inverse Functions
If f is differentiable at every point of an interval I and df/dx is never zero on I, then f has an inverse and f -1 is differentiable at every point in the interval f(I)

106. Important Point
If f(x) and g(x) are inverses, then g’(x)=1/f’(g(x)) Proof: Because f(x) and g(x) are inverses, f(g(x))=x Differentiate both sides and f’(g(x))g’(x)=1 Divide and g’(x)=1/f’(g(x))

107. Example
Find (𝑓 −1 ) ′ (𝑎) for the function f and the given real number a. 𝑓 𝑥 = 𝑥 3 +1, 𝑎=9 1. Plug a into y. (Hint: use graph/table in calc) 9= 𝑥 3 +1 →𝑥=2 (this is our g(x) or 𝒇 −𝟏 𝒙 ) 2. Find 𝑓 ′ 𝑥 𝑓 ′ 𝑥 =3 𝑥 2

108. Example
3. Use equation to fill in what you know: 1 𝑓′( 𝑓 −1 𝑥 ) = 1 𝑓′(2) = 1 3 (2) 2 = 𝟏 𝟏𝟐 **The equation outlined in the calc book is: 1 𝑓′(𝑔 𝑥 ) but since 𝑔 𝑥 = 𝑓 −1 (𝑥) they can be interchanged**

109. Example 2
Find (𝑓 −1 ) ′ (𝑎) for the function f and the given real number a. 𝑓 𝑥 = 3𝑥 3 −2𝑥+3, 𝑎=4 1. Plug a into y. (Hint: use graph/table in calc) 4= 3𝑥 3 −2𝑥+3 →𝑥=1 (this is our g(x) or 𝒇 −𝟏 𝒙 ) 2. Find 𝑓 ′ 𝑥 𝑓 ′ 𝑥 =9 𝑥 2 −2

110. Example 2
3. Use equation to fill in what you know: 1 𝑓′( 𝑓 −1 𝑥 ) = 1 𝑓′(1) = 1 9 (1) 2 −2 = 𝟏 𝟕 **The equation outlined in the calc book is: 1 𝑓′(𝑔 𝑥 ) but since 𝑔 𝑥 = 𝑓 −1 (𝑥) they can be interchanged**

111. Homework
Do circled problems from homework handout
For additional help, read the information on Page 157 in your text

112. Day 26: Derivatives of Inverse Trig Functions
For starters, let’s read on page 158 “Derivative of Arcsine”

113. Similarly….

114. Keep in mind
Sin -1 is also called arcsin and the same is true for all the other inverse trig functions. They can be named either way.

115. Also notice
The cofunction inverses have derivatives that are just the negatives of the other ones. Let’s quickly look at the middle of page to see why (Plus you can jot down the calculator conversion identities if you like)

116. Assignment
162(1-17odd,20,22)
321(5)

117. Day 27
What was the derivative of lnx again?

118. Day 27: Derivatives of Exponential and Logarithmic Functions
Let’s revisit

119. e
You may recall from precalc that

120. More on e
When you combine that with the definition of derivative, you can find that:

121. Also, since ax=exlna
You can do the derivative of both sides to find: for a>0 and a≠1

122. Derivative of logax

123. So…..
For a>0 and a≠1

124. Logarithmic Differentiation (page 169)
Use when the base and the exponent are both variable expressions.
Take the log of both sides
Differentiate implicitly
May have to substitute y back in at the end

125. 𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 𝑒 2𝑥 = 𝑒 2𝑥 𝑑 𝑑𝑥 2𝑥 =𝟐 𝒆 𝟐𝒙
Examples
𝑦= 𝑒 2𝑥
𝑑𝑦 𝑑𝑥 = 𝑑 𝑑𝑥 𝑒 2𝑥 = 𝑒 2𝑥 𝑑 𝑑𝑥 2𝑥 =𝟐 𝒆 𝟐𝒙

126. 3 csc 𝑥 ( ln 3) 𝑑 𝑑𝑥 csc 𝑥 = 3 csc 𝑥 ( ln 3) (− csc 𝑥 cot 𝑥)
Examples
𝑦= 3 csc 𝑥
3 csc 𝑥 ( ln 3) 𝑑 𝑑𝑥 csc 𝑥 = csc 𝑥 ( ln 3) (− csc 𝑥 cot 𝑥)
− 𝟑 𝒄𝒔𝒄 𝒙 (𝒍𝒏 𝟑)(𝒄𝒔𝒄𝒙 𝒄𝒐𝒕𝒙)

127. Examples 𝑦= ln ( 𝑥 2 +1) 1 ( 𝑥 2 +1) 𝑑 𝑑𝑥 𝑥 2 +1 = 1 𝑥 2 +1 2𝑥 =
1 ( 𝑥 2 +1) 𝑑 𝑑𝑥 𝑥 2 +1 = 𝑥 𝑥 =
𝟐𝒙 ( 𝒙 𝟐 +𝟏)

128. 1 𝑥 2 ( ln 4) 𝑑 𝑑𝑥 𝑥 2 = 1 𝑥 2 ( ln 4) 2𝑥 = 𝟐 𝒙( 𝒍𝒏 𝟒)
Examples
𝑦= log 4 𝑥 2
1 𝑥 2 ( ln 4) 𝑑 𝑑𝑥 𝑥 2 = 1 𝑥 2 ( ln 4) 2𝑥 = 𝟐 𝒙( 𝒍𝒏 𝟒)

129. Homework
170 (3-39, multiples of 3)

130. Day 28: Homework
170(26,27,43,44,47)

131. Day 29: Work Day
172(1-57, odd)omit 49,51

132. Day 30: Work Day
172(59-67, 70, 78)