1
00:00:07,107 --> 00:00:07,690
PROFESSOR: Hi.

2
00:00:07,690 --> 00:00:09,480
Welcome back to recitation.

3
00:00:09,480 --> 00:00:11,380
Last time in lecture
you started learning

4
00:00:11,380 --> 00:00:13,330
about implicit differentiation.

5
00:00:13,330 --> 00:00:16,450
And you saw some examples of
how implicit differentiation can

6
00:00:16,450 --> 00:00:19,130
be used to compute
derivatives of functions

7
00:00:19,130 --> 00:00:20,230
defined implicitly.

8
00:00:20,230 --> 00:00:23,390
So let's do another
example today.

9
00:00:23,390 --> 00:00:28,590
So here I have a curve that's
defined by the implicit

10
00:00:28,590 --> 00:00:32,220
equation y cubed plus
x cubed equals 3x*y.

11
00:00:32,220 --> 00:00:34,980
And I'd like to
know what the slope

12
00:00:34,980 --> 00:00:36,370
of the tangent
line to that curve

13
00:00:36,370 --> 00:00:38,800
is at the point (4/3, 2/3).

14
00:00:38,800 --> 00:00:41,140
So before we start
doing anything,

15
00:00:41,140 --> 00:00:43,450
let me just make a
couple of observations.

16
00:00:43,450 --> 00:00:47,290
If you don't believe me,
that the point (4/3, 2/3)

17
00:00:47,290 --> 00:00:51,950
is on this curve, you can always
check by plugging the values in

18
00:00:51,950 --> 00:00:55,680
and confirm that really, yes,
4/3 cubed plus 2/3 cubed is

19
00:00:55,680 --> 00:00:58,470
equal to 3 times 2/3 times 4/3.

20
00:00:58,470 --> 00:01:04,080
So it's-- how I found this
point is maybe a little magical.

21
00:01:04,080 --> 00:01:06,320
Because as you can
see, this equation

22
00:01:06,320 --> 00:01:08,420
is really a tough
one to solve for y.

23
00:01:08,420 --> 00:01:11,310
What you sort of--
natural thing to want

24
00:01:11,310 --> 00:01:14,040
to do when asked this
question is to solve for y

25
00:01:14,040 --> 00:01:16,950
and get an equation for
y in terms of x and then

26
00:01:16,950 --> 00:01:19,411
take the derivative using the
various differentiation rules

27
00:01:19,411 --> 00:01:20,244
that you've learned.

28
00:01:20,244 --> 00:01:24,320
But here, this is-- I'll
let you in a secret.

29
00:01:24,320 --> 00:01:25,990
There is a way to do this.

30
00:01:25,990 --> 00:01:27,060
But it's really hard.

31
00:01:27,060 --> 00:01:31,075
It's really ugly and it's
beyond the scope of this course.

32
00:01:33,800 --> 00:01:35,770
So really we're much
better off treating

33
00:01:35,770 --> 00:01:38,190
this as an implicit
differentiation problem

34
00:01:38,190 --> 00:01:41,400
than as an explicit
differentiation problem.

35
00:01:41,400 --> 00:01:45,650
So having said that, why don't
you take a minute or two.

36
00:01:45,650 --> 00:01:48,000
Try and have a go
at this yourself.

37
00:01:48,000 --> 00:01:52,240
And then we'll come back and
work through it together.

38
00:01:52,240 --> 00:01:54,265
All right, so welcome back.

39
00:01:54,265 --> 00:01:56,890
We were in the middle-- we were
just about to start, actually--

40
00:01:56,890 --> 00:01:59,990
solving this problem, computing
the slope of the tangent line

41
00:01:59,990 --> 00:02:03,060
to the curve y cubed plus
x cubed equals 3x*y at this

42
00:02:03,060 --> 00:02:04,680
point, (4/3, 2/3).

43
00:02:04,680 --> 00:02:08,610
So the slope of the tangent
line is the value y prime

44
00:02:08,610 --> 00:02:10,290
of x at that point.

45
00:02:10,290 --> 00:02:12,940
So we need to answer
this question.

46
00:02:12,940 --> 00:02:16,460
What we need to do is we need
to find the derivative of y.

47
00:02:16,460 --> 00:02:19,560
But as I said earlier, this
is tough to do explicitly,

48
00:02:19,560 --> 00:02:21,760
to find y in terms of
x, so we're going to use

49
00:02:21,760 --> 00:02:23,340
implicit differentiation.

50
00:02:23,340 --> 00:02:26,060
So, so we start
with this equation,

51
00:02:26,060 --> 00:02:30,330
y cubed plus x cubed equals 3x*y
and we can take a derivative

52
00:02:30,330 --> 00:02:31,530
with respect to x.

53
00:02:31,530 --> 00:02:33,580
So some parts--
all right, so let's

54
00:02:33,580 --> 00:02:35,990
start with it in the
order it's given.

55
00:02:35,990 --> 00:02:38,770
So you have y cubed.

56
00:02:38,770 --> 00:02:42,525
If you take a derivative of y
cubed with respect to x, what

57
00:02:42,525 --> 00:02:47,400
you need to use the chain
rule because y is implicitly

58
00:02:47,400 --> 00:02:51,320
a function of x and so y
cubed is the chain rule.

59
00:02:51,320 --> 00:02:54,330
It's the cubed function
applied to the y function.

60
00:02:54,330 --> 00:02:57,410
And this is true of implicit
differentiation in general.

61
00:02:57,410 --> 00:02:58,920
That the reason
that we can do this,

62
00:02:58,920 --> 00:03:01,210
the really fundamental
reason this works

63
00:03:01,210 --> 00:03:04,450
is that we have the chain rule
and that it lets us evaluate

64
00:03:04,450 --> 00:03:06,310
derivatives of compositions.

65
00:03:06,310 --> 00:03:10,178
So in our case we have,
so we take a derivative

66
00:03:10,178 --> 00:03:12,386
of the whole thing, of this
whole I'm going to write,

67
00:03:12,386 --> 00:03:12,518
this is a little
sloppy notation,

68
00:03:12,518 --> 00:03:13,851
but I hope you know what I mean.

69
00:03:20,080 --> 00:03:21,880
We have this
identity and so we're

70
00:03:21,880 --> 00:03:24,480
going to take a derivative
of the whole thing.

71
00:03:24,480 --> 00:03:26,910
And so the first
part on the left, we

72
00:03:26,910 --> 00:03:30,280
get the derivative of y cubed.

73
00:03:30,280 --> 00:03:32,520
So by the chain
rule, so we first

74
00:03:32,520 --> 00:03:34,740
take the derivative of
the cube function at y

75
00:03:34,740 --> 00:03:37,640
and then multiply by
the derivative of y.

76
00:03:37,640 --> 00:03:41,230
So this is the
derivative of y cubed.

77
00:03:41,230 --> 00:03:44,370
It just gives us 3 y squared.

78
00:03:44,370 --> 00:03:47,512
So that's what happens when you
just deal with the cubed part.

79
00:03:47,512 --> 00:03:49,720
But then we need to multiply
by-- in the chain rule--

80
00:03:49,720 --> 00:03:51,570
by the derivative of the inside.

81
00:03:51,570 --> 00:03:56,405
Which in this
context is dy by dx.

82
00:03:56,405 --> 00:03:58,490
OK.

83
00:03:58,490 --> 00:04:00,960
Plus the derivative of x cubed.

84
00:04:00,960 --> 00:04:02,470
That's more straightforward.

85
00:04:02,470 --> 00:04:04,520
Nothing really
complicated going on here.

86
00:04:04,520 --> 00:04:07,310
We've seen this for
a little while now.

87
00:04:07,310 --> 00:04:10,765
It's just 3x squared equal--

88
00:04:10,765 --> 00:04:11,265
OK.

89
00:04:11,265 --> 00:04:14,420
So on the right now, we don't
actually have a chain rule,

90
00:04:14,420 --> 00:04:16,470
we have a product
rule situation here.

91
00:04:16,470 --> 00:04:18,220
We have 3 times x times y.

92
00:04:18,220 --> 00:04:19,410
So 3 is just a constant.

93
00:04:19,410 --> 00:04:21,230
And so we could just
pull it out in front.

94
00:04:21,230 --> 00:04:23,600
So we take the
derivative of x*y.

95
00:04:23,600 --> 00:04:26,420
So we take the derivative of
the first times the second

96
00:04:26,420 --> 00:04:28,810
plus the derivative of the
second times the first.

97
00:04:28,810 --> 00:04:30,663
So the derivative of
the first is just--

98
00:04:30,663 --> 00:04:34,840
ah sorry, x is the first,
so its derivative is 1.

99
00:04:34,840 --> 00:04:38,720
So we got 3 times
the second is y.

100
00:04:38,720 --> 00:04:42,540
Plus-- OK, so we
take the first times

101
00:04:42,540 --> 00:04:49,230
the derivative of the
second, which is dy by dx.

102
00:04:49,230 --> 00:04:51,160
So because this is
an identity it holds

103
00:04:51,160 --> 00:04:53,690
for all values of x and y.

104
00:04:53,690 --> 00:04:56,900
This equality follows
just by taking

105
00:04:56,900 --> 00:04:59,051
the derivative of both sides.

106
00:04:59,051 --> 00:04:59,550
Good.

107
00:04:59,550 --> 00:05:01,460
So now the thing
we want is that we

108
00:05:01,460 --> 00:05:05,280
want the slope of the tangent
line at a particular point.

109
00:05:05,280 --> 00:05:08,190
So we want to isolate dy/dx.

110
00:05:08,190 --> 00:05:11,040
That's the thing
we're trying to find.

111
00:05:11,040 --> 00:05:15,530
So here, if you're only
interested in dy/dx,

112
00:05:15,530 --> 00:05:19,180
this is actually a linear
equation in some sense, right?

113
00:05:19,180 --> 00:05:21,470
We have dy/dx, a
constant, something--

114
00:05:21,470 --> 00:05:25,100
or, it's not a constant--
something times dy/dx

115
00:05:25,100 --> 00:05:28,860
plus something equals something
plus something times dy/dx.

116
00:05:28,860 --> 00:05:32,956
There are no squares of
dy/dx is what I really mean.

117
00:05:32,956 --> 00:05:33,456
So OK.

118
00:05:33,456 --> 00:05:34,430
So that that's nice.

119
00:05:34,430 --> 00:05:36,530
It makes it relatively
easier to solve,

120
00:05:36,530 --> 00:05:40,780
so we can just combine all
the terms with dy/dx in them.

121
00:05:40,780 --> 00:05:43,400
Let's say we'll combine maybe,
put them over here and put

122
00:05:43,400 --> 00:05:44,700
everything else over there.

123
00:05:44,700 --> 00:05:51,110
So over here we
get, so dy/dx, so we

124
00:05:51,110 --> 00:05:57,600
have a 3 y squared minus a 3x.

125
00:05:57,600 --> 00:06:06,580
And on the other side we have
a 3y minus a 3 x squared.

126
00:06:06,580 --> 00:06:09,590
And so this is times,
multiplication there.

127
00:06:09,590 --> 00:06:12,670
And so we want dy by
dx just by itself.

128
00:06:12,670 --> 00:06:16,370
So we can just divide through
by 3 y squared minus 3x.

129
00:06:16,370 --> 00:06:23,023
So then we have dy/dx
is equal to-- well,

130
00:06:23,023 --> 00:06:24,864
all right, so there
are a lot of 3's here.

131
00:06:24,864 --> 00:06:26,780
There's a constant
multiple of 3 on this side,

132
00:06:26,780 --> 00:06:27,930
a constant multiple
of 3 on this side.

133
00:06:27,930 --> 00:06:29,013
Those are going to cancel.

134
00:06:29,013 --> 00:06:39,210
So this is y minus x squared
over y squared minus x.

135
00:06:39,210 --> 00:06:41,850
OK, so at any point
(x, y) on this curve,

136
00:06:41,850 --> 00:06:48,190
the slope of the tangent line is
given by this expression here.

137
00:06:48,190 --> 00:06:49,950
And we're interested
in a particular point

138
00:06:49,950 --> 00:06:50,616
in this problem.

139
00:06:50,616 --> 00:06:54,220
We're interested in the
point 4/3 comma 2/3.

140
00:06:54,220 --> 00:07:06,910
So at let me take that back up
here so at the point 4/3 comma

141
00:07:06,910 --> 00:07:12,000
2/3 we have dy by dx.

142
00:07:12,000 --> 00:07:15,080
So OK, we just we just
plug that value of y

143
00:07:15,080 --> 00:07:17,550
and that value of x into
this formula that we've got.

144
00:07:17,550 --> 00:07:27,860
So that's 2/3 minus 4/3
squared is 16/9 over-- well,

145
00:07:27,860 --> 00:07:36,670
let's see, 2/3 squared
is 4/9 minus 4/3.

146
00:07:36,670 --> 00:07:39,600
All right, so we have a
little bit of rational number

147
00:07:39,600 --> 00:07:41,030
arithmetic here.

148
00:07:41,030 --> 00:07:45,380
Maybe I'll multiply top
and bottom through by 9

149
00:07:45,380 --> 00:07:51,860
to get 6 minus 16
over 4 minus 12.

150
00:07:51,860 --> 00:07:58,912
So this is negative 10 over
negative 8, which is 5 over 4.

151
00:08:01,570 --> 00:08:04,000
And if we go back to
the picture that I drew,

152
00:08:04,000 --> 00:08:06,640
it actually looks pretty
reasonable over here, right?

153
00:08:06,640 --> 00:08:09,930
This slope of this
tangent line is actually

154
00:08:09,930 --> 00:08:12,150
a little bit bigger than 1.

155
00:08:12,150 --> 00:08:13,220
Great.

156
00:08:13,220 --> 00:08:14,790
So that's that.