1
00:00:07,170 --> 00:00:08,906
PROFESSOR: Welcome
back to recitation.

2
00:00:08,906 --> 00:00:10,280
In this video
segment we're going

3
00:00:10,280 --> 00:00:12,020
to look at the chain rule.

4
00:00:12,020 --> 00:00:14,250
And specifically we're
going to answer a question

5
00:00:14,250 --> 00:00:16,420
that I've placed on the board.

6
00:00:16,420 --> 00:00:18,710
I want you to find
two values for theta

7
00:00:18,710 --> 00:00:21,620
so that d d theta of
the function cosine

8
00:00:21,620 --> 00:00:23,870
squared theta to
the fourth equals 0.

9
00:00:23,870 --> 00:00:26,430
At this point I should also
point out a few things.

10
00:00:26,430 --> 00:00:30,080
One, theta is a
variable in this case.

11
00:00:30,080 --> 00:00:31,730
And if you haven't
seen it before,

12
00:00:31,730 --> 00:00:33,960
frequently we use it
when we're dealing

13
00:00:33,960 --> 00:00:35,470
with trigonometric functions.

14
00:00:35,470 --> 00:00:39,490
Theta often represents the
variable that measures angle.

15
00:00:39,490 --> 00:00:41,430
So if you haven't
seen theta before,

16
00:00:41,430 --> 00:00:42,950
this is what it looks like.

17
00:00:42,950 --> 00:00:44,080
That's how you say it.

18
00:00:44,080 --> 00:00:46,910
And I'm actually doing the
same things we've done before.

19
00:00:46,910 --> 00:00:50,140
I'm taking the derivative with
respect to the variable theta,

20
00:00:50,140 --> 00:00:52,481
of a function of theta.

21
00:00:52,481 --> 00:00:53,980
The other thing I
want to point out,

22
00:00:53,980 --> 00:00:56,980
which I think we know
already, but just to be sure,

23
00:00:56,980 --> 00:00:59,360
is it that this squared
here, cosine squared theta

24
00:00:59,360 --> 00:01:03,150
to the fourth, means I'm taking
cosine of theta to the fourth

25
00:01:03,150 --> 00:01:04,820
and I'm actually squaring it.

26
00:01:04,820 --> 00:01:07,510
So with that
knowledge I would like

27
00:01:07,510 --> 00:01:10,520
us to use the chain rule to
find two values for theta

28
00:01:10,520 --> 00:01:13,340
where this derivative
is equal to 0.

29
00:01:13,340 --> 00:01:15,900
So I'll give you a
moment to take a stab

30
00:01:15,900 --> 00:01:18,690
at finding the derivative,
setting it equal to 0

31
00:01:18,690 --> 00:01:23,590
and finding some values for
theta that make this equation,

32
00:01:23,590 --> 00:01:25,190
the derivative equal 0.

33
00:01:25,190 --> 00:01:29,990
And we'll come back and then
I'll work it out as well.

34
00:01:29,990 --> 00:01:32,540
OK, so the first thing
we obviously need to do

35
00:01:32,540 --> 00:01:35,240
is be able to take the
derivative of this function

36
00:01:35,240 --> 00:01:36,360
on the left-hand side.

37
00:01:36,360 --> 00:01:39,480
And I should say it's a
little more complicated

38
00:01:39,480 --> 00:01:42,690
than an example you saw in the
lecture, because in the lecture

39
00:01:42,690 --> 00:01:44,950
you were given an example
with just two functions.

40
00:01:44,950 --> 00:01:47,400
So I want to write it a
little differently just

41
00:01:47,400 --> 00:01:49,780
to show very obviously what
the three functions are

42
00:01:49,780 --> 00:01:50,830
we're composing.

43
00:01:50,830 --> 00:01:53,400
So this function
I can rewrite it

44
00:01:53,400 --> 00:01:55,963
as cosine theta to the fourth
does that look like a 4,

45
00:01:55,963 --> 00:02:04,362
there we go and then I
square that whole thing.

46
00:02:04,362 --> 00:02:05,820
That's really what
the function is.

47
00:02:05,820 --> 00:02:07,040
OK?

48
00:02:07,040 --> 00:02:10,690
So we can see what is the
outermost function here?

49
00:02:10,690 --> 00:02:14,070
The outermost function
is actually the quantity

50
00:02:14,070 --> 00:02:15,150
of something squared.

51
00:02:15,150 --> 00:02:17,660
So the outermost
function is x squared.

52
00:02:17,660 --> 00:02:20,520
What's the next function
in, in this composition?

53
00:02:20,520 --> 00:02:23,450
The next function in
is the cosine function.

54
00:02:23,450 --> 00:02:27,120
And then the last function in is
the function taking something,

55
00:02:27,120 --> 00:02:28,537
raising it to the fourth.

56
00:02:28,537 --> 00:02:30,078
So it's very important
you understand

57
00:02:30,078 --> 00:02:32,040
sort of the
composition, which is

58
00:02:32,040 --> 00:02:35,430
the outermost function, which
is the innermost function?

59
00:02:35,430 --> 00:02:37,770
In order to do this chain rule.

60
00:02:37,770 --> 00:02:40,220
Now as you saw in
recitation, you

61
00:02:40,220 --> 00:02:44,850
were given the example dy-- you
had y as as a function of t--

62
00:02:44,850 --> 00:02:46,130
sorry, not in recitation.

63
00:02:46,130 --> 00:02:50,352
In the lecture, you were
given y as a function of t

64
00:02:50,352 --> 00:02:51,810
or a function of
x and then you had

65
00:02:51,810 --> 00:02:54,330
to put one other
variable in the middle.

66
00:02:54,330 --> 00:02:57,310
Here we're going to have a
composition of three functions.

67
00:02:57,310 --> 00:03:00,650
So we need to have two other
things sort of in the middle.

68
00:03:00,650 --> 00:03:02,340
So let's write this out.

69
00:03:02,340 --> 00:03:06,130
First, the outermost function,
we'll call the whole thing y.

70
00:03:06,130 --> 00:03:11,440
So y is equal to x squared
is the outermost function.

71
00:03:11,440 --> 00:03:14,690
So then this whole
thing is x now.

72
00:03:14,690 --> 00:03:19,305
So then we'll write x
is equal to cosine of w.

73
00:03:21,980 --> 00:03:27,602
And then w is equal to
theta to the fourth.

74
00:03:27,602 --> 00:03:28,600
OK.

75
00:03:28,600 --> 00:03:33,440
Again this is what you
saw before in the lecture.

76
00:03:33,440 --> 00:03:36,050
So you write the
outermost function.

77
00:03:36,050 --> 00:03:38,510
And then that
function is a function

78
00:03:38,510 --> 00:03:41,129
of cosine of another
one, cosine of w,

79
00:03:41,129 --> 00:03:42,920
and w is a function of
theta to the fourth.

80
00:03:42,920 --> 00:03:45,290
So if I put all
these back together,

81
00:03:45,290 --> 00:03:49,100
I have theta to the fourth in
here, and then I square that,

82
00:03:49,100 --> 00:03:51,420
and I come back to
the function I wanted.

83
00:03:51,420 --> 00:03:56,650
Now we know from the lecture
what we need to do to find,

84
00:03:56,650 --> 00:03:59,061
essentially, dy/d theta.

85
00:03:59,061 --> 00:04:00,310
That's what we're looking for.

86
00:04:03,750 --> 00:04:08,380
So dy/d theta, you'll see,
you remember from the lecture,

87
00:04:08,380 --> 00:04:18,149
should be dy/dx times
dx/dw times dw/d theta.

88
00:04:18,149 --> 00:04:19,524
So it's slightly
more complicated

89
00:04:19,524 --> 00:04:21,565
than we saw before because
there's one more term.

90
00:04:21,565 --> 00:04:22,970
OK.

91
00:04:22,970 --> 00:04:24,780
So now let's work
out what these are.

92
00:04:24,780 --> 00:04:27,820
Well dy/dx is fairly
straightforward.

93
00:04:27,820 --> 00:04:29,900
dy/dx is just 2x.

94
00:04:33,010 --> 00:04:35,330
And then dx/dw, well
what's the derivative

95
00:04:35,330 --> 00:04:36,300
of the cosine function?

96
00:04:36,300 --> 00:04:39,090
The derivative of
cosine is negative sine.

97
00:04:39,090 --> 00:04:45,310
So this is times negative
sine of w times--

98
00:04:45,310 --> 00:04:47,330
and what's dw/d theta?

99
00:04:47,330 --> 00:04:49,800
So w is the function
theta to the fourth.

100
00:04:49,800 --> 00:04:54,870
So dw/d theta is 4
theta to the third.

101
00:04:54,870 --> 00:04:58,670
Now when you look at
this you should remember,

102
00:04:58,670 --> 00:05:00,830
x we sort of inserted
into the problem

103
00:05:00,830 --> 00:05:03,527
to make the problem easier, and
w we inserted into the problem

104
00:05:03,527 --> 00:05:05,360
to make the problem
easier for us to follow.

105
00:05:05,360 --> 00:05:07,520
So we don't want all
these x's and w's.

106
00:05:07,520 --> 00:05:11,260
We want everything in terms
of, in terms of theta.

107
00:05:11,260 --> 00:05:12,390
But what is w?

108
00:05:12,390 --> 00:05:14,110
Well w is theta to the fourth.

109
00:05:14,110 --> 00:05:16,600
So I can replace that
by theta to the fourth.

110
00:05:16,600 --> 00:05:18,650
And what is x?

111
00:05:18,650 --> 00:05:20,590
x is cosine w.

112
00:05:20,590 --> 00:05:22,160
But w is theta to the fourth.

113
00:05:22,160 --> 00:05:24,355
So x is actually cosine
of theta to the fourth.

114
00:05:24,355 --> 00:05:26,000
OK?

115
00:05:26,000 --> 00:05:33,990
So I get 2 cosine of theta
to the fourth times negative

116
00:05:33,990 --> 00:05:38,060
sine-- w again is theta
to the fourth-- of theta

117
00:05:38,060 --> 00:05:43,710
to the fourth times
4 theta to the third.

118
00:05:43,710 --> 00:05:45,910
Let's make this nicer.

119
00:05:45,910 --> 00:05:47,590
I'll bring the
coefficient and the theta

120
00:05:47,590 --> 00:05:50,150
to the third in front and
this minus sign in front.

121
00:05:50,150 --> 00:05:55,700
I get negative 8 theta to
the third cosine of theta

122
00:05:55,700 --> 00:06:00,340
to fourth sine of
theta to the fourth.

123
00:06:00,340 --> 00:06:02,610
OK, and then the problem
asks to find where

124
00:06:02,610 --> 00:06:04,370
the derivative is equal to 0.

125
00:06:04,370 --> 00:06:05,502
Find two values.

126
00:06:05,502 --> 00:06:07,210
Now why did I ask you
to find two values?

127
00:06:07,210 --> 00:06:08,770
Because one is very easy.

128
00:06:08,770 --> 00:06:11,640
If this thing is set
equal to 0, one value

129
00:06:11,640 --> 00:06:13,190
should stand out
right away for theta

130
00:06:13,190 --> 00:06:15,750
that makes this product 0.

131
00:06:15,750 --> 00:06:17,840
And that is theta equals 0.

132
00:06:17,840 --> 00:06:20,320
So theta equals 0 is
our easiest answer.

133
00:06:26,690 --> 00:06:28,174
So if you didn't
do that, then you

134
00:06:28,174 --> 00:06:29,590
wanted the more
challenging stuff,

135
00:06:29,590 --> 00:06:32,310
you could have done
the other things also.

136
00:06:32,310 --> 00:06:34,570
So what about cosine
theta to the fourth?

137
00:06:34,570 --> 00:06:36,990
If we want a product of
three things to be 0,

138
00:06:36,990 --> 00:06:39,430
then at least one
of them has to be 0.

139
00:06:39,430 --> 00:06:42,410
So I could have set cosine
theta to the fourth equal to 0.

140
00:06:48,560 --> 00:06:51,870
And for what values
of-- an angle

141
00:06:51,870 --> 00:06:54,840
I should just say-- for what
values of theta to the fourth

142
00:06:54,840 --> 00:06:56,340
is this going to be 0?

143
00:06:56,340 --> 00:07:01,550
Well this means that
theta to the fourth

144
00:07:01,550 --> 00:07:06,682
is equal to either maybe pi over
2, or you could add another pi.

145
00:07:06,682 --> 00:07:07,182
OK?

146
00:07:07,182 --> 00:07:11,690
So we could say, well let's
just say that's one example.

147
00:07:11,690 --> 00:07:14,830
Theta to the fourth equals
pi over 2 would work right?

148
00:07:14,830 --> 00:07:18,230
Because cosine of pi
over 2 is equal to 0.

149
00:07:18,230 --> 00:07:20,200
OK?

150
00:07:20,200 --> 00:07:22,740
And so then we could
have said theta is

151
00:07:22,740 --> 00:07:29,390
equal to pi over 2 to the 1/4.

152
00:07:29,390 --> 00:07:31,470
That's another example there.

153
00:07:31,470 --> 00:07:34,797
We could have also done, we
could have added pi to this

154
00:07:34,797 --> 00:07:35,880
and gotten another answer.

155
00:07:35,880 --> 00:07:37,360
But I only asked for two.

156
00:07:37,360 --> 00:07:40,294
So I guess that I'm
allowed to stop there.