1
00:00:00,000 --> 00:00:07,170

2
00:00:07,170 --> 00:00:08,980
PROFESSOR: Welcome back
to recitation.

3
00:00:08,980 --> 00:00:10,630
In this video segment
we're going to look

4
00:00:10,630 --> 00:00:12,020
at the chain rule.

5
00:00:12,020 --> 00:00:14,690
And specifically we're going to
answer a question that I've

6
00:00:14,690 --> 00:00:16,420
placed on the board.

7
00:00:16,420 --> 00:00:18,710
I want you to find two
values for theta.

8
00:00:18,710 --> 00:00:22,080
So that the d d theta of the
function cosine squared theta

9
00:00:22,080 --> 00:00:23,870
to the fourth equals 0.

10
00:00:23,870 --> 00:00:26,430
At this point I should also
point out a few things.

11
00:00:26,430 --> 00:00:30,080
One, theta is a variable
in this case.

12
00:00:30,080 --> 00:00:33,230
And if you haven't seen it
before, frequently we use it

13
00:00:33,230 --> 00:00:35,470
when we're dealing 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 --> 00:00:42,130
So if you haven't seen
theta before, this is

16
00:00:42,130 --> 00:00:42,950
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 --> 00:00:52,690
of a function of theta.

21
00:00:52,690 --> 00:00:54,740
The other thing I want to point
out, which I think we

22
00:00:54,740 --> 00:00:57,880
know already, but just to be
sure, is it that this squared

23
00:00:57,880 --> 00:01:01,370
here, cosine squared theta to
the fourth, means I'm taking

24
00:01:01,370 --> 00:01:04,820
cosine of theta to the fourth
and I'm actually squaring it.

25
00:01:04,820 --> 00:01:08,870
So with that knowledge I would
like us to use the chain rule

26
00:01:08,870 --> 00:01:11,060
to find two values for
theta where this

27
00:01:11,060 --> 00:01:13,340
derivative is equal to 0.

28
00:01:13,340 --> 00:01:16,420
So I'll give you a moment to
take a stab at finding the

29
00:01:16,420 --> 00:01:19,620
derivative, setting it equal to
0 and finding some values

30
00:01:19,620 --> 00:01:23,750
for theta that make
this equation, the

31
00:01:23,750 --> 00:01:25,190
derivative equal 0.

32
00:01:25,190 --> 00:01:29,990
And we'll come back and then
I'll work it out as well.

33
00:01:29,990 --> 00:01:32,790
OK, so the first thing we
obviously need to do is be

34
00:01:32,790 --> 00:01:35,360
able to take the derivative
of this function on

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

36
00:01:36,360 --> 00:01:40,270
And I should say it's a little
more complicated than an

37
00:01:40,270 --> 00:01:41,960
example you saw in the lecture,
because in the

38
00:01:41,960 --> 00:01:43,570
lecture you were given
an example

39
00:01:43,570 --> 00:01:44,950
with just two functions.

40
00:01:44,950 --> 00:01:48,030
So I want to write it a little
differently just to show very

41
00:01:48,030 --> 00:01:50,830
obviously what the three
functions are we're composing.

42
00:01:50,830 --> 00:01:55,760
So this function I can rewrite
it as cosine theta to the

43
00:01:55,760 --> 00:01:57,010
fourth-- does that look like
a 4, there we go--

44
00:01:57,010 --> 00:02:02,160

45
00:02:02,160 --> 00:02:04,600
and then I square that
whole thing.

46
00:02:04,600 --> 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 --> 00:02:14,170
The outermost function is
actually the quantity of

50
00:02:14,170 --> 00:02:15,150
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 --> 00:02:26,790
And then the last function
in is the function taking

55
00:02:26,790 --> 00:02:28,570
something, raising
it to the fourth.

56
00:02:28,570 --> 00:02:31,090
So it's very important you
understand sort of the

57
00:02:31,090 --> 00:02:33,760
composition, which is the
outermost function, which is

58
00:02:33,760 --> 00:02:35,430
the innermost function?

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

60
00:02:37,770 --> 00:02:47,750
Now as you saw in the
recitation, you were given the

61
00:02:47,750 --> 00:02:47,990
example, you had y as a function
of t-- sorry, not in

62
00:02:47,990 --> 00:02:49,570
recitation--

63
00:02:49,570 --> 00:02:50,220
in the lecture you were given,
you were given y as a function

64
00:02:50,220 --> 00:02:53,090
of t or a function of x and then
you had to put one other

65
00:02:53,090 --> 00:02:54,330
variable in the middle.

66
00:02:54,330 --> 00:02:56,090
Here we're going to have a

67
00:02:56,090 --> 00:02:57,310
composition of three functions.

68
00:02:57,310 --> 00:03:00,650
So we need to have two other
things sort of in the middle.

69
00:03:00,650 --> 00:03:02,340
So let's write this out.

70
00:03:02,340 --> 00:03:06,130
First, the outermost function,
we'll call the whole thing y.

71
00:03:06,130 --> 00:03:11,440
So y is equal to x squared is
the outermost function.

72
00:03:11,440 --> 00:03:14,690
So then this whole
thing is x now.

73
00:03:14,690 --> 00:03:19,305
So then we'll write x is
equal to cosine of w.

74
00:03:19,305 --> 00:03:21,980

75
00:03:21,980 --> 00:03:27,602
And then w is equal to
theta to the fourth.

76
00:03:27,602 --> 00:03:28,600
OK.

77
00:03:28,600 --> 00:03:33,440
Again this is what you saw
before in the lecture.

78
00:03:33,440 --> 00:03:36,050
So you write the outermost
function.

79
00:03:36,050 --> 00:03:38,600
And then that function
is a function of

80
00:03:38,600 --> 00:03:40,590
cosine of another one.

81
00:03:40,590 --> 00:03:42,400
Cosine of w and w
is a function of

82
00:03:42,400 --> 00:03:42,920
theta to the fourth.

83
00:03:42,920 --> 00:03:45,990
So if I put all these back
together, I have theta to the

84
00:03:45,990 --> 00:03:49,830
fourth in here, and then I
square that, and I come back

85
00:03:49,830 --> 00:03:51,420
to the function I wanted.

86
00:03:51,420 --> 00:03:56,650
Now we know from the lecture
what we need to do to find,

87
00:03:56,650 --> 00:03:59,290
essentially, dy d theta.

88
00:03:59,290 --> 00:04:00,540
That's what we're looking for.

89
00:04:00,540 --> 00:04:03,750

90
00:04:03,750 --> 00:04:08,380
So dy d theta, you'll see, you
remember from the lecture,

91
00:04:08,380 --> 00:04:18,500
should be dy dx times dx
dw times dw d theta.

92
00:04:18,500 --> 00:04:20,490
So it's slightly more
complicated than we saw before

93
00:04:20,490 --> 00:04:21,350
because there's one more term.

94
00:04:21,350 --> 00:04:22,970
OK.

95
00:04:22,970 --> 00:04:24,780
So now let's work out
what these are.

96
00:04:24,780 --> 00:04:27,820
Well dy dx is fairly
straightforward.

97
00:04:27,820 --> 00:04:29,900
dy dx is just 2x.

98
00:04:29,900 --> 00:04:33,010

99
00:04:33,010 --> 00:04:35,330
And then dx dw, well what's
the derivative

100
00:04:35,330 --> 00:04:36,300
of the cosine function?

101
00:04:36,300 --> 00:04:39,090
The derivative of cosine
is negative sine.

102
00:04:39,090 --> 00:04:45,310
So this is times negative
sine of w times--

103
00:04:45,310 --> 00:04:47,330
and what's dw d theta?

104
00:04:47,330 --> 00:04:49,800
So w is the function theta
to the fourth.

105
00:04:49,800 --> 00:04:54,870
So dw d theta is 4 theta
to the third.

106
00:04:54,870 --> 00:04:59,570
Now when you look at this you
should remember, x we sort of

107
00:04:59,570 --> 00:05:02,510
inserted into the problem to
make the problem easier, and w

108
00:05:02,510 --> 00:05:04,730
we inserted into the problem to
make the problem easier for

109
00:05:04,730 --> 00:05:05,360
us to follow.

110
00:05:05,360 --> 00:05:07,520
So we don't want all
these x's and w's.

111
00:05:07,520 --> 00:05:11,260
We want everything in terms
of, in terms of theta.

112
00:05:11,260 --> 00:05:12,390
But what is w?

113
00:05:12,390 --> 00:05:14,110
Well w is theta to the fourth.

114
00:05:14,110 --> 00:05:16,600
So I can replace that by
theta to the fourth.

115
00:05:16,600 --> 00:05:18,650
And what is x?

116
00:05:18,650 --> 00:05:20,590
x is cosine w.

117
00:05:20,590 --> 00:05:22,160
But w is theta to the fourth.

118
00:05:22,160 --> 00:05:24,355
So x is actually cosine of
theta to the fourth.

119
00:05:24,355 --> 00:05:26,000
OK?

120
00:05:26,000 --> 00:05:33,260
So I get 2 cosine of
theta to the fourth

121
00:05:33,260 --> 00:05:35,260
times negative sine--

122
00:05:35,260 --> 00:05:37,570
w again is theta
to the fourth--

123
00:05:37,570 --> 00:05:43,710
of theta to the fourth times
4 theta to the third.

124
00:05:43,710 --> 00:05:45,910
Let's make this nicer.

125
00:05:45,910 --> 00:05:48,100
I'll bring the coefficient and
the theta to the third in

126
00:05:48,100 --> 00:05:50,150
front and this minus
sign in front.

127
00:05:50,150 --> 00:05:55,850
I get negative 8 theta to the
third cosine of theta to

128
00:05:55,850 --> 00:06:00,340
fourth sine of theta
to the fourth.

129
00:06:00,340 --> 00:06:03,190
OK, and then the problem asks
to find where the derivative

130
00:06:03,190 --> 00:06:04,370
is equal to 0.

131
00:06:04,370 --> 00:06:05,640
Find two values.

132
00:06:05,640 --> 00:06:07,210
Now why did I ask you
to find two values?

133
00:06:07,210 --> 00:06:08,770
Because one is very easy.

134
00:06:08,770 --> 00:06:12,310
If this thing is set equal to 0,
one value should stand out

135
00:06:12,310 --> 00:06:15,750
right away for theta that
makes this product 0.

136
00:06:15,750 --> 00:06:17,840
And that is theta equals 0.

137
00:06:17,840 --> 00:06:20,320
So theta equals 0 is
our easiest answer.

138
00:06:20,320 --> 00:06:26,690

139
00:06:26,690 --> 00:06:28,760
So if you didn't do that,
then you wanted the more

140
00:06:28,760 --> 00:06:30,810
challenging stuff, you
could have done the

141
00:06:30,810 --> 00:06:32,310
other things also.

142
00:06:32,310 --> 00:06:34,570
So what about cosine theta
to the fourth?

143
00:06:34,570 --> 00:06:37,830
If we want a product of three
things to be 0, then at least

144
00:06:37,830 --> 00:06:39,430
one of them has to be 0.

145
00:06:39,430 --> 00:06:41,660
So I could have set cosine
theta to the

146
00:06:41,660 --> 00:06:42,910
fourth equal to 0.

147
00:06:42,910 --> 00:06:48,560

148
00:06:48,560 --> 00:06:51,170
And for what values of--

149
00:06:51,170 --> 00:06:54,230
an angle I should just say--
for what values of theta to

150
00:06:54,230 --> 00:06:56,340
the fourth is this
going to be zero?

151
00:06:56,340 --> 00:07:04,070
Well this means that theta to
the fourth is equal to either

152
00:07:04,070 --> 00:07:07,121
maybe pi over 2, or you
could add another pi.

153
00:07:07,121 --> 00:07:07,182
OK?

154
00:07:07,182 --> 00:07:11,690
So we could say, well let's just
say that's one example.

155
00:07:11,690 --> 00:07:14,830
Theta to the fourth equals pi
over 2 would work right?

156
00:07:14,830 --> 00:07:18,230
Because it cosine of pi
over 2 is equal to 0.

157
00:07:18,230 --> 00:07:20,200
OK?

158
00:07:20,200 --> 00:07:24,520
And so then we could have said
theta is equal to pi

159
00:07:24,520 --> 00:07:29,390
over 2 to the 1/4.

160
00:07:29,390 --> 00:07:31,470
That's another example there.

161
00:07:31,470 --> 00:07:34,900
We could have also done, we
could have added pi to this

162
00:07:34,900 --> 00:07:35,880
and gotten another answer.

163
00:07:35,880 --> 00:07:37,360
But I only asked for two.

164
00:07:37,360 --> 00:07:40,320
So I guess that I'm allowed
to stop there.

165
00:07:40,320 --> 00:07:40,794