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The directions are to differentiate.

2
00:00:09,780 --> 00:00:12,740
What does that mean in calculus, differentiate?

3
00:00:15,620 --> 00:00:18,020
All that means is to find the derivative.

4
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Find the derivative.

5
00:00:30,580 --> 00:00:34,980
And the answer is...

6
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8 u to the 7th.

7
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And what do I put on the left side?

8
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Y prime.

9
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However, I want to use Leibniz notation.

10
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That would be dy over du.

11
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Why is it dy over du?

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as the representation with respect to u.

13
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But what I didn't tell you is u is some function of x.

14
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And I really want dy dx, not dy du.

15
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Okay, so dy dx, I don't see any x's here.

16
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Okay, so we assume this variable is some function of x.

17
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Right now y is a function of u.

18
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So if I want dy dx, I go 8u to the 7th, and then by the chain rule, we multiply by

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the

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derivative of u with respect to x.

21
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This is 8u to the seventh times du over dx.

22
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The derivative of u with respect to x is cosine x.

23
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In terms of x, the result is 8 sine to the seventh x times cosine x.

24
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cosine of x.

25
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What if the directions asked us to find dy over dt?

26
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Okay. This implies, right? We only see y's and u's here. But if we're trying to find

27
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this, this

28
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implies that every variable is some function of t, but you just don't even know what

29
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it is. And

30
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we don't even care. So all we do here is

31
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the derivative of this function is dy with respect to t. We differentiate this 8u to

32
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the seventh and then by the chain rule we multiply by the derivative of this

33
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function.

34
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du over dt.

35
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Okay. You're done.

36
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Because we don't even know what U is in terms of T.

37
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We don't...

38
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Who cares?

39
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So we can take the derivative with respect to any variable.

40
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Even if we don't see the variable.

41
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you take the derivative of whatever this function is and then by the chain rule

42
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we take the derivative of this function u, since it's just u it's times derivative

43
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of u with

44
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This process is called implicit differentiation.

45
00:05:23,020 --> 00:05:24,900
What's this in that equation?

46
00:05:31,760 --> 00:05:35,780
A circle of radius?

47
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3.

48
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How do I know it's 3?

49
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It's r squared.

50
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squared. Okay. And where's the center of the circle?

51
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Oh, it's, um, zero, zero.

52
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Zero, zero. Okay. Do you remember this from a couple years ago?

53
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No. No? Okay.

54
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The equation of a circle is (x-h) squared plus (y-k) squared equals r squared.

55
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squared plus y minus k

56
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squared equals r squared. Is the equation

57
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have a circle of radius r centered at the e, okay. Okay, anyways, this is a circle

58
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centered at the origin of radius n. Is this a function? No. Why is this not a

59
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function? It

60
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fails the vertical line test. Okay, but still we call it in calculus called a

61
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curve. Okay, we don't care if it's not a function. Calculus doesn't really care

62
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about functions, whether it's a function or not. Okay, this is some curve. It has to

63
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be a

64
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We could solve for y: y squared is 9 minus x squared.

65
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Then y is plus or minus the square root of 9 minus x squared.

66
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which is two separate functions now this is a function this is a function okay

67
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The top semicircle is y equals the square root of 9 minus x squared.

68
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The bottom semicircle is y equals negative the square root of 9 minus x squared.

69
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Okay.

70
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Okay, is y a function of x here?

71
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It's not. For it to be a function of x, you have to have y equals something. This

72
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here, y is a function of x.

73
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Okay, Y is explicit. Y is explicit.

74
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A function of X.

75
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Here,

76
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We can imply it's a function of x, but we don't know.

77
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Okay, but here it's explicitly it's written y is explicitly some function of x here

78
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it might be might not we don't really care actually but.

79
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But we can apply it is even though it's not written it want as long.

80
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So anytime we have y equals some function, this is like explicit, and then here we

81
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can

82
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imply it is or it's not, so on. So we're gonna differentiate this even though y is

83
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not explicitly

84
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a function. Okay, and here's how we do it. The x squared plus y squared equals 9.

85
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We're

86
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to differentiate both sides of the equation not just the right side although we have

87
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been doing

88
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that. We just haven't really been talking about that. Okay. What's it like here?

89
00:09:34,840 --> 00:09:42,037
We just said this equals this before we went to you. I told you that. We actually

90
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are

91
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The derivative of y with respect to x is dy over dx.

92
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The derivative of 8u to the eighth is 8u to the seventh times du over dx.

93
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this kind of makes sense so we're gonna do the same thing except now we have

94
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variables on both sides so we just use all the same rules on the left hand side

95
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we have a sum. So how do I take the derivative of a sum? It's just the sum of

96
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each derivative. Okay so we start with this one. Derivative of x squared is 2x,

97
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except x is, could be a function of any variable we want. So we do need, I should

98
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We are differentiating with respect to x.

99
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with respect to x so it's 2x times the derivative of x with respect to x dx dx

100
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The derivative of y squared is 2y times dy over dx.

101
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dy dx

102
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we're done with the left hand side

103
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and the derivative of 9 is

104
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0 we don't need any chain rule because there's no variables

105
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okay and what is dx dx

106
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1 the rate of change of x

107
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with respect to the change of x is one. The rate of change is the same. So that's

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one.

109
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You can take it as a fraction as well. We get this.

110
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Okay, and if I want to find dy dx, we now just solve for this.

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Two y times dy over dx equals negative 2x, so divide by 2y.

112
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So what does this tell us?

113
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Okay.

114
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That's not a very good circle, but if we want to find the slope of the tangent line,

115
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which

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is our dy dx, it is our x and y.

117
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Okay.

118
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Okay.

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At x equals 2.

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So find the slope.

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So I need to plug in x equals 2 into here.

122
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At x equals 2, the derivative is negative 2 over y; we still need the y-coordinate.

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What's that?

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So this is a little this is a little weird because we normally don't do this very

125
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often. We need a y coordinate just like we plugged in an x coordinate. So how do I

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find

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this y coordinate right here? The original function right? That's this. Okay, so x

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squared

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At x equals 2, y squared equals 5.

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Thus y equals plus or minus the square root of 5.

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Yeah, when x is 2, we got one value here and we got one value here. So this will

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The two points are (2, square root of 5) and (2, negative square root of 5).

133
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The two slopes are negative 2 over square root of 5 and positive 2 over square root

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of 5.

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is going to be dy dx of 2, square root of 5 is negative 2 over square root of 5 and

136
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the other one

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So the slope of this line is exactly negative 2 over square root of 5 and the slope

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of this line is

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positive 2 over square root of 5.

140
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The harder method is to solve for y first and then differentiate.

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So dy dx would be what?

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That's 9 minus X squared.

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The 1 hat.

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And then 1 half times the derivative of the inside, which is negative 2x.

145
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And then if we wanted at x equals 2, we get 1 half 9 minus 2 squared 1 half times

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negative

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two times two

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And if you do a bunch of work comes out to be

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negative two over

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Five or something like that

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And we would only gotten one of the tangent lines slope to the tangent lines instead

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of two of them

153
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Then we have to do the same thing with the other one

154
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Okay, a lot of functions. There's no way to

155
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Change it to make it explicitly

156
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Okay

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Now take the same curve and find dy over dz.

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Now let's find d y d z I

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Don't even see these

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And we don't care. This implies that every variable is a function and it's a

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function of z.

162
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Okay. So we just start on the left hand side, differentiate the whole left hand

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side, which again is a sum.

164
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So we take the derivative of each piece separately.

165
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We get 2x times dx over dz.

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chain rule we multiply by the derivative of that function with respect to whatever

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variable.

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Plus 2y times dy over dz.

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to the variable z.

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And the derivative of constant is 0.

171
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And then we just solve for

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dy dz, which is this one right here.

173
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So we have two while.

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And then what those simplify and that's it.

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You will still have a, you have this rate of change in our answer and that's okay.

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We can differentiate with respect to any variable we want.

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If we are asked to find the slope of a tangent line, that's always dy ds.

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But if they just say find dy dz, that's all we do.

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Yeah, these are actually.

180
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Okay, this is definitely not a function.

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Who cares? Some curve.

182
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Let's see what it looks like.

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In fact, just for fun.

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The left side is sine x times e to the y.

185
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What did I say? Y cubed? That's kind of cool. Definitely not a function, but who

186
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cares?

187
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It's occurred. Okay, if we wanted to find the two slopes of the tangent lines at x

188
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equals 2,

189
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we're gonna have to differentiate. We'd have to plug it, figure out those y-values.

190
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It's not the point.

191
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Okay, let's find

192
00:21:27,580 --> 00:21:28,540
dy dx.

193
00:21:32,200 --> 00:21:35,900
Okay, we just differentiate the left hand side separately from the right hand side.

194
00:21:36,600 --> 00:21:39,400
So on the left hand side, what do we have here?

195
00:21:42,220 --> 00:21:43,120
A product.

196
00:21:43,980 --> 00:21:44,060
A product.

197
00:21:44,060 --> 00:21:45,060
So what rule do we got each?

198
00:21:45,960 --> 00:21:46,860
A product rule.

199
00:21:47,460 --> 00:21:47,580
Okay.

200
00:21:47,580 --> 00:21:48,520
Okay.

201
00:21:58,180 --> 00:22:01,233
Use the product rule: the derivative of sine x times e to the y, plus sine x times

202
00:22:01,233 --> 00:22:02,420
the derivative of e to the y.

203
00:22:03,880 --> 00:22:04,780
t to the y,

204
00:22:06,200 --> 00:22:08,700
sine t to the y.

205
00:22:17,940 --> 00:22:24,811
The derivative of sine x is cosine x, and the derivative of e to the y is e to the y

206
00:22:24,811 --> 00:22:26,120
times dy over dx.

207
00:22:26,120 --> 00:22:40,659
derivative of x with respect x times e to the y. The derivative of e to the y is e

208
00:22:40,659 --> 00:22:42,840
to the y

209
00:22:42,840 --> 00:22:47,780
times the derivative of y, which is dy dx.

210
00:22:51,380 --> 00:22:58,020
The derivative of y cubed is 3y squared times dy over dx.

211
00:22:59,700 --> 00:23:04,960
3y squared times the derivative of y with respect x.

212
00:23:20,260 --> 00:23:25,480
And now if we want to find dy, dx, we have to solve for these.

213
00:23:25,660 --> 00:23:26,560
How are we going to do that?

214
00:23:29,040 --> 00:23:30,820
We got to bring both to one side.

215
00:23:32,280 --> 00:23:34,660
Okay. And dx, dx is?

216
00:23:36,180 --> 00:23:37,040
One.

217
00:23:37,040 --> 00:23:47,760
2 months, bud.

218
00:24:09,380 --> 00:24:12,940
Okay, try them both to one side.

219
00:24:12,940 --> 00:24:14,113
There is the implicit derivative. It is not pretty, but it gives a tangent slope at

220
00:24:14,113 --> 00:24:14,863
any point on the curve.

221
00:24:16,060 --> 00:24:19,820
But if we note any ordered pair that's on the curve,

222
00:24:21,400 --> 00:24:25,900
we plug in the x-coordinates into the x's, the y-coordinates into the y's,

223
00:24:26,040 --> 00:24:28,500
and we're going to get some slope of the tangent line.

224
00:24:29,740 --> 00:24:32,320
Okay, up that crazy looking curve.

225
00:24:34,860 --> 00:24:36,960
Okay, what questions we got on this?

226
00:24:38,780 --> 00:24:43,233
Differentiate as usual, and every time a variable depends on the differentiation

227
00:24:43,233 --> 00:24:45,460
variable, include the chain-rule derivative factor.

228
00:24:45,460 --> 00:24:50,935
by the chain will multiply by the derivative of that inside function with respect to

229
00:24:50,935 --> 00:24:51,300
any

230
00:24:51,300 --> 00:24:52,200
variable we want.

231
00:24:53,540 --> 00:24:55,680
Okay, if at the very beginning...

232
00:25:02,460 --> 00:25:04,060
If I said find

233
00:25:06,660 --> 00:25:07,980
dy, dt,

234
00:25:10,040 --> 00:25:11,880
instead of dy, dx,

235
00:25:12,180 --> 00:25:14,200
it would be the only thing we do different.

236
00:25:18,520 --> 00:25:20,760
The only thing different is

237
00:25:20,760 --> 00:25:24,180
And this would be DX DT.

238
00:25:25,860 --> 00:25:27,500
This would be DY DT.

239
00:25:28,600 --> 00:25:30,560
This would be DY DT.

240
00:25:33,660 --> 00:25:39,020
And then everything, this would not simplify to one.

241
00:25:40,060 --> 00:25:41,780
It just stay in there part of that product.

242
00:25:43,120 --> 00:25:44,020
Makes sense?

243
00:25:44,420 --> 00:25:45,320
Kind of?

244
00:25:46,000 --> 00:25:46,900
OK.

245
00:25:46,960 --> 00:25:48,680
Let us do another problem.

246
00:26:04,160 --> 00:26:07,100
What is this formula for?

247
00:26:08,080 --> 00:26:11,760
The volume of the classroom here?

248
00:26:12,860 --> 00:26:15,420
The volume of a cylinder is the area of the circular base times its height.

249
00:26:16,760 --> 00:26:20,480
This is the area of the base times our hives.

250
00:26:23,940 --> 00:26:26,640
Here volume is a function of what?

251
00:26:32,280 --> 00:26:36,100
Volume is a function of both radius and height.

252
00:26:37,500 --> 00:26:38,880
Do we see any X's?

253
00:26:38,880 --> 00:26:44,152
What is the variable? Radius and height? Yeah, here the volume is a function of both

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the radius and the height.

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00:26:53,800 --> 00:27:04,840
We could find dV over dr, dV over dh, or dV over dt.

256
00:27:08,680 --> 00:27:23,420
We will find dV over dt, so volume, radius, and height are all functions of time.

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and A's are functions of T's. Okay, so let's start the left-hand side. We

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differentiate this with

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00:27:36,080 --> 00:27:48,449
respect to time and it is or with respect to T. That was the left side easy. Now the

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right-hand

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00:27:49,100 --> 00:27:50,000
hand side. What do we have?

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00:27:53,460 --> 00:27:57,220
A product. We do have two products, but it's

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just a constant. So, let's keep it with this guy right here.

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Okay,

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that's that product right there.

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00:28:09,760 --> 00:28:10,720
So, we need what role?

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00:28:16,120 --> 00:28:17,280
Okay, so.

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00:28:19,020 --> 00:28:21,700
The derivative is dV over dt.

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00:28:26,020 --> 00:28:34,780
The derivative of pi r squared is 2 pi r times dr over dt.

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Then apply the product rule and include dh over dt on the second term.

271
00:29:01,000 --> 00:29:07,060
And that's it. Okay. Why would we ever want to do this?

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So soon we're gonna have cylinders pretend this is a cylinder.

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00:29:29,920 --> 00:29:30,820
It does not.

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00:29:33,200 --> 00:29:35,000
Water is not coming out of it.

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Water is going in. Water is going in.

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Oh!

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But it does have a leak.

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00:29:40,380 --> 00:29:41,300
So it's going out as well.

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00:29:42,380 --> 00:29:43,840
Guys, why are you freaking out?

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00:29:45,180 --> 00:29:46,260
This is it right here.

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00:29:49,780 --> 00:29:51,300
This is how...

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This is the rate of change of volume with respect to time.

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00:29:56,020 --> 00:30:04,020
These factors describe how radius and height change with time.

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00:30:05,140 --> 00:30:08,920
And they're all related. Why are they all related? Or how are they all related?

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00:30:11,940 --> 00:30:14,460
They're related by exactly this equation.

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00:30:17,420 --> 00:30:19,660
Okay, this is the relation right here.

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00:30:20,420 --> 00:30:26,443
So if I know how fast I'm pouring water in, that would be the rate of change of the

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volume.

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00:30:27,040 --> 00:30:33,720
Let's say I'm putting water in at three gallons an hour.

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It's pretty slow, but who cares?

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This would be our dv dt.

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Okay.

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00:30:46,500 --> 00:30:54,971
And so for example if the radius is not changing right, fix cop the radius doesn't

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change.

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00:30:56,280 --> 00:31:04,939
So would be the rate of change of the radius would be DR. That'd be zero. Okay the

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00:31:04,939 --> 00:31:05,420
height

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00:31:05,420 --> 00:31:12,864
would be changing as we add water and it's changing exactly this whatever that is.

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00:31:12,864 --> 00:31:13,360
So

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00:31:13,360 --> 00:31:15,580
If we plug in three actually right here,

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and we knew the radius of the top, whatever,

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00:31:20,180 --> 00:31:22,880
and we could find exactly how fast the height is changing.

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00:31:26,020 --> 00:31:29,140
This topic is called related rates.

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00:31:31,600 --> 00:31:33,260
It is super, it was easy.

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00:31:33,800 --> 00:31:34,700
We just did it.

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00:31:38,180 --> 00:31:41,560
This is exactly how the different rates are related.

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00:31:41,560 --> 00:31:44,393
the rate of change of the volume the rate of change the radius and the rate of

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00:31:44,393 --> 00:31:45,143
change of the height.

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00:31:45,540 --> 00:31:49,040
It's not complicated. It's that easy. Okay.

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00:31:54,740 --> 00:32:01,620
What if, instead of dV over dt, we find dV over dh?

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00:32:04,400 --> 00:32:05,960
What would be the only difference?

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00:32:09,240 --> 00:32:14,140
We have an 8 here, an 8 here, and an 8 here.

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00:32:15,060 --> 00:32:16,420
And then this would simplify to 1.

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00:32:17,700 --> 00:32:19,260
Well, it's not really simplifying to d1.

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00:32:19,820 --> 00:32:21,420
That's easy peasy.

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00:32:25,120 --> 00:32:26,820
And then we're done.

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00:32:30,000 --> 00:32:31,920
This is why I like the fractions.

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00:32:33,760 --> 00:32:41,781
Okay, what does R prime even mean? What does H prime mean? The ray of change, but

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00:32:41,781 --> 00:32:44,140
with respect to what variables?

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00:32:47,040 --> 00:32:58,140
But how do we know that? How do I know that H prime is equal to 1?

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00:33:00,220 --> 00:33:06,230
Why is x prime equal to 1? Only when we're differentiating with respect to x. Only

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when we're differentiating with respect to h.

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00:33:09,360 --> 00:33:22,394
Leibniz notation records the variable with respect to which each derivative is

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00:33:22,394 --> 00:33:23,480
taken.

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00:33:26,540 --> 00:33:32,832
Implicit differentiation uses the same derivative rules, with a chain-rule factor

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00:33:32,832 --> 00:33:35,120
for each dependent variable.

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00:33:35,120 --> 00:33:39,426
derivative of the function you multiply by the derivative of the variable with

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00:33:39,426 --> 00:33:40,420
respect to whatever

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00:33:40,420 --> 00:33:41,320
favorite.

329
00:33:42,520 --> 00:33:43,900
Can you try one more?

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00:33:44,660 --> 00:33:45,560
Where should we start at home?

331
00:33:50,020 --> 00:33:50,920
One more.

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00:33:51,200 --> 00:33:52,100
Let's do one more.

333
00:33:52,220 --> 00:33:53,120
I like it.

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00:33:54,860 --> 00:33:57,740
We will find dx over dn.

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00:34:00,680 --> 00:34:01,860
Whatever that means.

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00:34:04,320 --> 00:34:09,060
It's the rate of change of X with respect to the change in n.

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00:34:11,140 --> 00:34:12,400
So we started the left hand side.

338
00:34:13,940 --> 00:34:14,880
The left-hand side requires the quotient rule.

339
00:34:16,180 --> 00:34:17,220
Quotient rule.

340
00:34:17,580 --> 00:34:18,560
We've got to use the quotient rule.

341
00:34:30,600 --> 00:34:31,560
So, the

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00:34:31,560 --> 00:34:33,640
The derivative of x to the fourth is 4x cubed times dx over dn.

343
00:34:36,340 --> 00:34:37,700
4x cubed.

344
00:34:37,700 --> 00:34:40,240
Times the derivative of our inside,

345
00:34:40,760 --> 00:34:41,880
which is

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00:34:43,960 --> 00:34:47,160
dx with respect to n.

347
00:34:47,900 --> 00:34:49,140
We don't care what goes.

348
00:34:52,820 --> 00:34:54,960
And then the second function

349
00:34:57,660 --> 00:34:59,680
minus the first function.

350
00:35:00,780 --> 00:35:04,320
Differentiate square root of y as one over 2 square root of y times dy over dn.

351
00:35:08,560 --> 00:35:22,037
y to the negative one half times the derivative of y which is dy dn divided by the

352
00:35:22,037 --> 00:35:25,780
square root of y squared.

353
00:35:30,440 --> 00:35:39,840
The right-hand side is a product.

354
00:35:39,840 --> 00:35:41,780
So we got to use the product pool.

355
00:35:50,240 --> 00:35:57,760
The derivative of x cubed is 3x squared times dx over dn.

356
00:35:59,420 --> 00:36:00,460
dx dn.

357
00:36:02,600 --> 00:36:05,980
Then keep x cubed and differentiate y as dy over dn.

358
00:36:08,100 --> 00:36:13,860
So, dy over dn.

359
00:36:14,360 --> 00:36:18,149
You could think of it as the derivative of y as 1, but then we still have to

360
00:36:18,149 --> 00:36:18,360
multiply

361
00:36:18,360 --> 00:36:22,920
by the derivative of that variable, which is dy over dn.

362
00:36:26,300 --> 00:36:27,200
Okay.

363
00:36:28,980 --> 00:36:30,440
If I said find

364
00:36:30,440 --> 00:36:34,220
dy dx, what would be the only change?

365
00:36:37,580 --> 00:36:39,140
All the dn's would be

366
00:36:39,140 --> 00:36:42,920
dx's and then the dx over dx

367
00:36:42,920 --> 00:36:43,980
simplifies to one.

368
00:36:46,460 --> 00:36:50,020
What was it? I asked for dx dn.

369
00:36:54,100 --> 00:37:00,942
We could collect all terms containing dx over dn and solve, but we will stop after

370
00:37:00,942 --> 00:37:03,080
setting up the derivative correctly.

371
00:37:03,740 --> 00:37:09,290
We'd have to clear the denominator here, bring all everything with the DX on one

372
00:37:09,290 --> 00:37:12,620
side, everything without DX, XDN on the other side.

373
00:37:13,100 --> 00:37:17,160
But we're going to stop there. That wasn't bad, right?