Teaching Calculus

Site owners

  • Nikolay Brodskiy
  • Alexander Brodsky

Part 02 (Transcript)

1
00:00:01,200 --> 00:00:04,600
[...] sometimes ... Let's ...

2
00:00:05,900 --> 00:00:07,300
The chain rule.

3
00:00:07,400 --> 00:00:09,850
Right? So what's the chain rule about?

4
00:00:09,950 --> 00:00:13,000
It's about thinking of ...

5
00:00:14,200 --> 00:00:17,400
... a function f of independent variable x.

6
00:00:17,500 --> 00:00:20,900
And then all of a sudden you think of x ...

7
00:00:21,400 --> 00:00:23,700
... as a function of something else.

8
00:00:23,800 --> 00:00:28,900
And then you ask well if you know the rate of change of f with respect to x ...

9
00:00:29,000 --> 00:00:31,900
... and you also know the rate of change of with respect to t.

10
00:00:32,000 --> 00:00:36,300
Can you possibly find the rate of change of f with respect to t?

11
00:00:36,900 --> 00:00:40,200
Because now you can consider f as a function of t.

12
00:00:40,350 --> 00:00:43,900
Because now you can substitute that x into f, right?

13
00:00:44,000 --> 00:00:48,750
So being as a function of t you can ask for derivative.

14
00:00:48,850 --> 00:00:51,300
And that means ...

15
00:00:52,000 --> 00:00:54,100
So what is it that you can do?

16
00:00:54,200 --> 00:01:00,800
You can say df is f prime dx.

17
00:01:01,200 --> 00:01:05,300
Right? This is what you know from f being a function of x.

18
00:01:05,400 --> 00:01:09,400
And this f prime is f prime with respect to x.

19
00:01:09,500 --> 00:01:12,400
So let's put x there.

20
00:01:12,900 --> 00:01:16,400
And then from this formula you can say ...

21
00:01:16,500 --> 00:01:21,100
... dx is x prime ...

22
00:01:22,400 --> 00:01:24,500
Now it is a function of t.

23
00:01:24,800 --> 00:01:25,800
dt.

24
00:01:26,750 --> 00:01:31,700
Because in this setup you treat x as dependent on t.

25
00:01:31,800 --> 00:01:34,850
So dx is exactly that.

26
00:01:34,950 --> 00:01:39,150
And now ... dx was independent in this setting.

27
00:01:39,250 --> 00:01:44,200
But we switched to thinking of x being dependent.

28
00:01:44,300 --> 00:01:47,800
So you just substitute that dx into this ...

29
00:01:48,100 --> 00:01:53,600
... formula and you get df equals f prime ...

30
00:01:54,450 --> 00:01:57,700
... derivative with respect to x times ...

31
00:01:57,900 --> 00:01:59,500
... x prime.

32
00:02:00,300 --> 00:02:02,400
Derivative with respect to t.

33
00:02:02,900 --> 00:02:04,600
Times dt.

34
00:02:06,000 --> 00:02:08,000
And that's exactly the chain rule.

35
00:02:08,800 --> 00:02:11,200
Right? If you divide by dt ...

36
00:02:14,000 --> 00:02:16,800
... and then operating like this ...

37
00:02:17,750 --> 00:02:21,700
Those being quantities that I can divide.

38
00:02:22,200 --> 00:02:25,000
But essentially that derivative is ...

39
00:02:25,100 --> 00:02:27,500
... f prime of x times ...

40
00:02:28,200 --> 00:02:29,900
... x prime of t.

41
00:02:30,800 --> 00:02:33,500
So the chain rule follows from this basic idea ...

42
00:02:33,600 --> 00:02:36,000
... of what differentials are.

43
00:02:36,100 --> 00:02:40,600
And we will follow the same idea in function of two variables.

44
00:02:40,700 --> 00:02:43,850
Because the chain rule for function of many variables is ...

45
00:02:43,950 --> 00:02:46,500
... not one formula. It's much more complicated.

46
00:02:46,600 --> 00:02:49,700
But it will based on this very, very simple idea.

47
00:02:50,000 --> 00:02:53,400
[...] you operate at differentials and substitute.

48
00:02:54,700 --> 00:02:58,100
And essentially the chain rule ...

49
00:02:58,550 --> 00:03:04,000
Well in my mind the chain rule has no place in calculus because it doesn't belong to calculus.

50
00:03:04,750 --> 00:03:07,700
Because the chain rule basically tells you ...

51
00:03:08,200 --> 00:03:13,800
... something about combination of linear functions.

52
00:03:13,900 --> 00:03:15,900
So what is it about?

53
00:03:16,700 --> 00:03:19,000
What is it that we are saying here?

54
00:03:19,250 --> 00:03:23,600
We are saying that if you linearize this dependence ...

55
00:03:24,300 --> 00:03:26,600
And this is what we do here, right?

56
00:03:26,700 --> 00:03:30,800
We pretend dependence of f on x is linear.

57
00:03:30,900 --> 00:03:33,000
Which is simply multiplication by a number.

58
00:03:33,100 --> 00:03:36,700
So if you linearize that this is what you get.

59
00:03:37,000 --> 00:03:40,100
If you linearize this dependence this is what you get.

60
00:03:40,200 --> 00:03:43,200
And then you ask -- if you have ...

61
00:03:43,450 --> 00:03:45,400
... two linear dependencies ...

62
00:03:45,500 --> 00:03:49,200
f linearly depends on x and x linearly depends on t.

63
00:03:49,300 --> 00:03:52,000
How does f linearly depend on t?

64
00:03:53,000 --> 00:03:55,800
Well you just have to combine these two linear ...

65
00:03:55,900 --> 00:04:00,000
... dependencies so you have to multiply those two numbers.

66
00:04:00,550 --> 00:04:04,100
Right? So the idea is fundamentally very simple.

67
00:04:04,300 --> 00:04:08,400
If x is twice as fast as t, f is three times as fast as x ...

68
00:04:08,500 --> 00:04:11,200
... then f is six times as fast as t.

69
00:04:11,300 --> 00:04:13,100
So that's all it is about it.

70
00:04:13,200 --> 00:04:18,300
So that idea is about linear algebra.

71
00:04:18,600 --> 00:04:20,700
And for many variables ...

72
00:04:22,350 --> 00:04:24,500
... what we will have is ...

73
00:04:26,100 --> 00:04:29,450
... f being multivariable object.

74
00:04:29,550 --> 00:04:32,300
x being multivariable object.

75
00:04:32,400 --> 00:04:36,800
And how do you describe linear dependence of one object on the other?

76
00:04:36,900 --> 00:04:38,600
Well you use a matrix ...

77
00:04:38,700 --> 00:04:40,100
... instead of a number.

78
00:04:40,200 --> 00:04:44,200
So you multiply a vector by a matrix to get another vector.

79
00:04:44,300 --> 00:04:47,200
And then similar thing here.

80
00:04:47,300 --> 00:04:52,600
Multivariable vector, multivariable vector, multiplication by a matrix.

81
00:04:52,700 --> 00:04:57,950
And then you ask 'well, how do you do you combine these two linear dependences?'

82
00:04:58,050 --> 00:05:01,550
Well that's exactly the topic of linear algebra saying you ...

83
00:05:01,650 --> 00:05:04,300
You have to multiply these two matrices.

84
00:05:06,200 --> 00:05:09,550
Right? So the chain rule is saying effectively that ...

85
00:05:09,650 --> 00:05:14,800
... all you have to do is you have to multiply those two multivariable objects.

86
00:05:14,900 --> 00:05:19,400
And actually the concept of multiplication of matrices ...

87
00:05:19,500 --> 00:05:22,700
... is designed by looking at this rule.

88
00:05:23,200 --> 00:05:25,000
With the way you ...

89
00:05:25,750 --> 00:05:26,750
... [...] did it.

90
00:05:26,850 --> 00:05:29,100
The way you multiply matrices together ...

91
00:05:29,200 --> 00:05:32,200
Well it something like an artificial rule ... well somebody tells you ...

92
00:05:32,300 --> 00:05:35,600
Well you have to take this, multiply by that plus take that, multiply by this.

93
00:05:35,700 --> 00:05:43,200
So essentially that rule is designed to make this simple rule work.

94
00:05:43,550 --> 00:05:46,000
If you want this to happen ...

95
00:05:46,100 --> 00:05:51,300
... that combination is determined by multiplication of these coefficients.

96
00:05:51,400 --> 00:05:57,000
As you make those coefficients matrices you have to design multiplication and ...

97
00:05:57,300 --> 00:05:59,500
... linear algebra tells you what it is.

98
00:06:02,900 --> 00:06:07,000
So unfortunately as linear algebra is not [...] ...

99
00:06:07,100 --> 00:06:10,300
... this chain rule is not something very easy for you ...

100
00:06:10,400 --> 00:06:15,100
... that is well understood at the level of linear dependences.

101
00:06:15,200 --> 00:06:17,100
So I will go to that quickly.

102
00:06:17,200 --> 00:06:23,000
But there is nothing mysterious and it's really not a part of calculus from my point of view.

103
00:06:24,050 --> 00:06:27,200
It's something that happens after we linearize.

104
00:06:27,300 --> 00:06:31,500
And calculus to me is about this linearization.

105
00:06:33,500 --> 00:06:36,400
All right. So that's kind of preview ...

106
00:06:37,200 --> 00:06:39,500
... of multivariable stuff.

107
00:06:39,700 --> 00:06:42,600
So any questions about differentials?

108
00:06:44,650 --> 00:06:47,250
Because that's basically the ...

109
00:06:47,950 --> 00:06:53,400
... the different way of wording what Taylor series says after degree one.

110
00:06:53,500 --> 00:06:55,600
Nothing more, nothing less.

111
00:06:55,700 --> 00:06:57,800
Just different notation for the same stuff.

Comments