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Professor: I am Haynes
Miller, I am substituting

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00:00:25,050 --> 00:00:26,510
for David Jerison today.

10
00:00:26,510 --> 00:00:41,880
So you have a substitute
teacher today.

11
00:00:41,880 --> 00:00:44,400
So I haven't been here
in this class with you

12
00:00:44,400 --> 00:00:47,580
so I'm not completely
sure where you are.

13
00:00:47,580 --> 00:00:52,170
I think you've just been
talking about differentiation

14
00:00:52,170 --> 00:00:56,840
and you've got some examples
of differentiation like these

15
00:00:56,840 --> 00:00:59,860
basic examples: the
derivative of x^n is nx^(n-1).

16
00:01:03,197 --> 00:01:05,280
But I think maybe you've
spent some time computing

17
00:01:05,280 --> 00:01:11,230
the derivative of the sine
function as well, recently.

18
00:01:11,230 --> 00:01:16,290
And I think you have
some rules for extending

19
00:01:16,290 --> 00:01:18,950
these calculations as well.

20
00:01:18,950 --> 00:01:24,080
For instance, I think you
know that if you differentiate

21
00:01:24,080 --> 00:01:27,770
a constant times a
function, what do you get?

22
00:01:27,770 --> 00:01:32,590
Student: [INAUDIBLE].

23
00:01:32,590 --> 00:01:36,670
Professor: The constant
comes outside like this.

24
00:01:36,670 --> 00:01:40,030
Or I could write (cu)' = cu'.

25
00:01:42,550 --> 00:01:45,350
That's this rule,
multiplying by a constant,

26
00:01:45,350 --> 00:01:58,870
and I think you also know
about differentiating a sum.

27
00:01:58,870 --> 00:02:03,650
Or I could write this
as (u + v)' = u' + v'.

28
00:02:06,870 --> 00:02:09,850
So I'm going to be using
those but today I'll

29
00:02:09,850 --> 00:02:12,290
talk about a collection
of other rules

30
00:02:12,290 --> 00:02:15,080
about how to deal with
a product of functions,

31
00:02:15,080 --> 00:02:18,180
a quotient of functions,
and, best of all,

32
00:02:18,180 --> 00:02:20,024
composition of functions.

33
00:02:20,024 --> 00:02:21,690
And then at the end,
I'll have something

34
00:02:21,690 --> 00:02:23,480
to say about higher derivatives.

35
00:02:23,480 --> 00:02:26,670
So that's the story for today.

36
00:02:26,670 --> 00:02:29,120
That's the program.

37
00:02:29,120 --> 00:02:43,360
So let's begin by talking
about the product rule.

38
00:02:43,360 --> 00:02:44,750
So the product
rule tells you how

39
00:02:44,750 --> 00:02:46,760
to differentiate a
product of functions,

40
00:02:46,760 --> 00:02:49,270
and I'll just give you
the rule, first of all.

41
00:02:49,270 --> 00:02:51,600
The rule is it's u'v + uv'.

42
00:02:57,280 --> 00:02:58,500
It's a little bit funny.

43
00:02:58,500 --> 00:03:02,320
Differentiating a
product gives you a sum.

44
00:03:02,320 --> 00:03:06,730
But let's see how that works
out in a particular example.

45
00:03:06,730 --> 00:03:08,270
For example, suppose
that I wanted

46
00:03:08,270 --> 00:03:11,280
to differentiate the product.

47
00:03:11,280 --> 00:03:13,890
Well, the product
of these two basic

48
00:03:13,890 --> 00:03:15,584
examples that we
just talked about.

49
00:03:15,584 --> 00:03:17,000
I'm going to use
the same variable

50
00:03:17,000 --> 00:03:20,790
in both cases instead of
different ones like I did here.

51
00:03:20,790 --> 00:03:23,230
So the derivative
of x^n times sin x.

52
00:03:28,430 --> 00:03:30,300
So this is a new thing.

53
00:03:30,300 --> 00:03:36,120
We couldn't do this without
using the product rule.

54
00:03:36,120 --> 00:03:39,670
So the first function is x^n
and the second one is sin x.

55
00:03:39,670 --> 00:03:41,740
And we're going to
apply this rule.

56
00:03:41,740 --> 00:03:49,450
So u is x^n. u' is, according
to the rule, nx^(n-1).

57
00:03:49,450 --> 00:03:56,029
And then I take v and write it
down the way it is, sine of x.

58
00:03:56,029 --> 00:03:57,320
And then I do it the other way.

59
00:03:57,320 --> 00:04:00,690
I take u the way
it is, that's x^n,

60
00:04:00,690 --> 00:04:05,260
and multiply it by the
derivative of v, v'.

61
00:04:05,260 --> 00:04:09,234
We just saw v' is cosine of x.

62
00:04:09,234 --> 00:04:11,520
So that's it.

63
00:04:11,520 --> 00:04:14,980
Obviously, you can
differentiate longer products,

64
00:04:14,980 --> 00:04:20,560
products of more things
by doing it one at a time.

65
00:04:20,560 --> 00:04:22,700
Let's see why this is true.

66
00:04:22,700 --> 00:04:25,730
I want to try to show you
why the product rule holds.

67
00:04:25,730 --> 00:04:31,490
So you have a standard way
of trying to understand this,

68
00:04:31,490 --> 00:04:34,590
and it involves looking at
the change in the function

69
00:04:34,590 --> 00:04:37,350
that you're interested
in differentiating.

70
00:04:37,350 --> 00:04:41,390
So I should look at how
much the product uv changes

71
00:04:41,390 --> 00:04:44,630
when x changes a little bit.

72
00:04:44,630 --> 00:04:47,120
Well, so how do
compute the change?

73
00:04:47,120 --> 00:04:49,330
Well, I write down the
value of the function

74
00:04:49,330 --> 00:04:56,160
at some new value
of x, x + delta x.

75
00:04:56,160 --> 00:04:58,410
Well, I better write
down the whole new value

76
00:04:58,410 --> 00:05:01,480
of the function, and
the function is uv.

77
00:05:01,480 --> 00:05:05,230
So the whole new
value looks like this.

78
00:05:05,230 --> 00:05:09,450
It's u(x + delta x)
times v(x + delta x).

79
00:05:09,450 --> 00:05:10,960
That's the new value.

80
00:05:10,960 --> 00:05:13,200
But what's the change
in the product?

81
00:05:13,200 --> 00:05:15,410
Well, I better subtract
off what the old value

82
00:05:15,410 --> 00:05:20,920
was, which is u(x) v(x).

83
00:05:20,920 --> 00:05:24,950
Okay, according to the
rule we're trying to prove,

84
00:05:24,950 --> 00:05:27,750
I have to get u' involved.

85
00:05:27,750 --> 00:05:31,420
So I want to involve the
change in u alone, by itself.

86
00:05:31,420 --> 00:05:32,990
Let's just try that.

87
00:05:32,990 --> 00:05:36,620
I see part of the formula for
the change in u right there.

88
00:05:36,620 --> 00:05:40,260
Let's see if we can get
the rest of it in place.

89
00:05:40,260 --> 00:05:46,330
So the change in x is
(u(x + delta x) - u(x).

90
00:05:46,330 --> 00:05:50,080
That's the change in x
[Correction: change in u].

91
00:05:50,080 --> 00:05:54,760
This part of it occurs up here,
multiplied by v(x + delta x),

92
00:05:54,760 --> 00:05:57,890
so let's put that in too.

93
00:05:57,890 --> 00:06:00,440
Now this equality sign
isn't very good right now.

94
00:06:00,440 --> 00:06:04,770
I've got this
product here so far,

95
00:06:04,770 --> 00:06:06,690
but I've introduced
something I don't like.

96
00:06:06,690 --> 00:06:09,800
I've introduced u times
v(x delta x), right?

97
00:06:09,800 --> 00:06:12,010
Minus that.

98
00:06:12,010 --> 00:06:17,120
So the next thing I'm gonna do
is correct that little defect

99
00:06:17,120 --> 00:06:24,620
by adding in u(x)
v(x + delta x).

100
00:06:24,620 --> 00:06:28,834
Okay, now I cancelled off
what was wrong with this line.

101
00:06:28,834 --> 00:06:30,500
But I'm still not
quite there, because I

102
00:06:30,500 --> 00:06:32,670
haven't put this in yet.

103
00:06:32,670 --> 00:06:38,420
So I better subtract off
uv, and then I'll be home.

104
00:06:38,420 --> 00:06:40,500
But I'm going to do
that in a clever way,

105
00:06:40,500 --> 00:06:43,900
because I noticed that
I already have a u here.

106
00:06:43,900 --> 00:06:46,620
So I'm gonna take
this factor of u

107
00:06:46,620 --> 00:06:48,560
and make it the
same as this factor.

108
00:06:48,560 --> 00:06:52,510
So I get u(x) times this,
minus u(x) times that.

109
00:06:52,510 --> 00:06:57,180
That's the same thing as
u times the difference.

110
00:06:57,180 --> 00:06:59,406
So that was a
little bit strange,

111
00:06:59,406 --> 00:07:01,030
but when you stand
back and look at it,

112
00:07:01,030 --> 00:07:04,280
you can see multiplied out,
the middle terms cancel.

113
00:07:04,280 --> 00:07:07,340
And you get the right answer.

114
00:07:07,340 --> 00:07:10,420
Well I like that because
it's involved the change in u

115
00:07:10,420 --> 00:07:16,810
and the change in v. So this
is equal to delta u times v(x +

116
00:07:16,810 --> 00:07:26,320
delta x) minus u(x) times
the change in v. Well,

117
00:07:26,320 --> 00:07:27,510
I'm almost there.

118
00:07:27,510 --> 00:07:30,050
The next step in computing
the derivative is

119
00:07:30,050 --> 00:07:42,910
take difference quotient,
divide this by delta x.

120
00:07:42,910 --> 00:07:50,690
So, (delta (uv)) /
(delta x) is well,

121
00:07:50,690 --> 00:08:03,150
I'll say (delta u / delta
x) times v(x + delta x).

122
00:08:03,150 --> 00:08:10,000
Have I made a mistake here?

123
00:08:10,000 --> 00:08:13,340
This plus magically became a
minus on the way down here,

124
00:08:13,340 --> 00:08:18,810
so I better fix that.

125
00:08:18,810 --> 00:08:23,260
Plus u times (delta
v) / (delta x).

126
00:08:23,260 --> 00:08:27,950
This u is this u over here.

127
00:08:27,950 --> 00:08:30,310
So I've just divided
this formula by delta x,

128
00:08:30,310 --> 00:08:34,520
and now I can take the limit
as x goes to 0, so this

129
00:08:34,520 --> 00:08:42,070
is as delta x goes to 0.

130
00:08:42,070 --> 00:08:46,480
This becomes the definition
of the derivative,

131
00:08:46,480 --> 00:08:51,790
and on this side, I
get du/dx times...

132
00:08:51,790 --> 00:08:57,400
now what happens to this
quantity when delta x goes

133
00:08:57,400 --> 00:09:02,750
to 0?

134
00:09:02,750 --> 00:09:05,710
So this quantity is getting
closer and closer to x.

135
00:09:05,710 --> 00:09:09,000
So what happens
to the value of v?

136
00:09:09,000 --> 00:09:14,030
It becomes equal to x of v.
That uses continuity of v. So,

137
00:09:14,030 --> 00:09:22,590
v(x + delta x) goes
to v(x) by continuity.

138
00:09:22,590 --> 00:09:26,586
So this gives me times v,
and then I have u times,

139
00:09:26,586 --> 00:09:30,680
and delta v / delta
x gives me dv/dx.

140
00:09:30,680 --> 00:09:31,819
And that's the formula.

141
00:09:31,819 --> 00:09:33,360
That's the formula
as I wrote it down

142
00:09:33,360 --> 00:09:35,660
at the beginning over here.

143
00:09:35,660 --> 00:09:39,170
The derivative of a product
is given by this sum.

144
00:09:39,170 --> 00:09:46,223
Yeah?

145
00:09:46,223 --> 00:09:48,056
Student: How did you
get from the first line

146
00:09:48,056 --> 00:09:49,514
to the second of
the long equation?

147
00:09:49,514 --> 00:09:51,390
Professor: From here to here?

148
00:09:51,390 --> 00:09:53,460
Student: Yes.

149
00:09:53,460 --> 00:09:56,380
Professor: So maybe it's easiest
to work backwards and verify

150
00:09:56,380 --> 00:09:59,800
that what I wrote
down is correct here.

151
00:09:59,800 --> 00:10:05,130
So, if you look there's a u
times v(x + delta x) there.

152
00:10:05,130 --> 00:10:07,840
And there's also one here.

153
00:10:07,840 --> 00:10:09,940
And they occur with
opposite signs.

154
00:10:09,940 --> 00:10:11,490
So they cancel.

155
00:10:11,490 --> 00:10:20,530
What's left is u(x + delta
x) v(x + delta x) - uv.

156
00:10:20,530 --> 00:10:29,120
And that's just
what I started with.

157
00:10:29,120 --> 00:10:33,920
Student: [INAUDIBLE]
They cancel right?

158
00:10:33,920 --> 00:10:37,430
Professor: I cancelled out
this term and this term,

159
00:10:37,430 --> 00:10:39,700
and what's left is the ends.

160
00:10:39,700 --> 00:10:41,490
Any other questions?

161
00:10:41,490 --> 00:10:49,660
Student: [INAUDIBLE].

162
00:10:49,660 --> 00:10:55,640
Professor: Well, I just
calculated what delta uv is,

163
00:10:55,640 --> 00:10:57,980
and now I'm gonna divide
that by delta x on my way

164
00:10:57,980 --> 00:11:00,250
to computing the derivative.

165
00:11:00,250 --> 00:11:07,760
And so I copied down the right
hand side and divided delta x.

166
00:11:07,760 --> 00:11:11,550
I just decided to divide the
delta u by delta x and delta v

167
00:11:11,550 --> 00:11:16,230
by delta x.

168
00:11:16,230 --> 00:11:16,990
Good.

169
00:11:16,990 --> 00:11:22,490
Anything else?

170
00:11:22,490 --> 00:11:24,260
So we have the
product rule here.

171
00:11:24,260 --> 00:11:26,980
The rule for differentiating
a product of two functions.

172
00:11:26,980 --> 00:11:28,325
This is making us stronger.

173
00:11:28,325 --> 00:11:29,700
There are many
more functions you

174
00:11:29,700 --> 00:11:31,420
can find derivatives of now.

175
00:11:31,420 --> 00:11:33,580
How about quotients?

176
00:11:33,580 --> 00:11:35,370
Let's find out how
to differentiate

177
00:11:35,370 --> 00:11:47,669
a quotient of two functions.

178
00:11:47,669 --> 00:11:50,210
Well again, I'll write down what
the answer is and then we'll

179
00:11:50,210 --> 00:11:52,370
try to verify it.

180
00:11:52,370 --> 00:11:55,192
So there's a quotient.

181
00:11:55,192 --> 00:11:56,150
Let me write this down.

182
00:11:56,150 --> 00:11:58,970
There's a quotient
of two functions.

183
00:11:58,970 --> 00:12:00,340
And here's the rule for it.

184
00:12:00,340 --> 00:12:02,910
I always have to think about
this and hope that I get it

185
00:12:02,910 --> 00:12:09,140
right. (u'v - uv') / v^2.

186
00:12:09,140 --> 00:12:11,900
This may be the craziest rule
you'll see in this course,

187
00:12:11,900 --> 00:12:14,330
but there it is.

188
00:12:14,330 --> 00:12:18,014
And I'll try to show you why
that's true and see an example.

189
00:12:18,014 --> 00:12:18,930
Yeah there was a hand?

190
00:12:18,930 --> 00:12:27,300
Student: [INAUDIBLE]

191
00:12:27,300 --> 00:12:33,040
Professor: What letters look
the same? u and v look the same?

192
00:12:33,040 --> 00:12:37,040
I'll try to make them
look more different.

193
00:12:37,040 --> 00:12:38,520
The v's have points
on the bottom.

194
00:12:38,520 --> 00:12:41,160
u's have little round
things on the bottom.

195
00:12:41,160 --> 00:12:44,980
What's the new value of u?

196
00:12:44,980 --> 00:12:55,442
The value of u at x + delta
x is u + delta u, right?

197
00:12:55,442 --> 00:12:56,400
That's what delta u is.

198
00:12:56,400 --> 00:13:01,327
It's the change in u when
x gets replaced by delta x

199
00:13:01,327 --> 00:13:02,410
[Correction: x + delta x].

200
00:13:02,410 --> 00:13:09,700
And the change in v, the
new value v, is v + delta v.

201
00:13:09,700 --> 00:13:13,130
So this is the new value of u
divided by the new value of v.

202
00:13:13,130 --> 00:13:16,130
That's the beginning.

203
00:13:16,130 --> 00:13:18,705
And then I subtract off
the old values, which

204
00:13:18,705 --> 00:13:22,910
are u minus v. This'll
be easier to work out

205
00:13:22,910 --> 00:13:26,084
when I write it out this way.

206
00:13:26,084 --> 00:13:27,750
So now, we'll cross
multiply, as I said.

207
00:13:27,750 --> 00:13:38,890
So I get uv + (delta u)v minus,
now I cross multiply this way,

208
00:13:38,890 --> 00:13:46,330
you get uv - u(delta v).

209
00:13:46,330 --> 00:13:49,980
And I divide all this
by (v + delta v)u.

210
00:13:52,840 --> 00:13:55,970
Okay, now the reason
I like to do it

211
00:13:55,970 --> 00:13:59,490
this way is that you see the
cancellation happening here. uv

212
00:13:59,490 --> 00:14:02,187
and uv occur twice and
so I can cancel them.

213
00:14:02,187 --> 00:14:04,520
And I will, and I'll answer
these questions in a minute.

214
00:14:04,520 --> 00:14:06,260
Audience: [INAUDIBLE].

215
00:14:06,260 --> 00:14:14,030
Professor: Ooh,
that's a v. All right.

216
00:14:14,030 --> 00:14:15,570
Good, anything else?

217
00:14:15,570 --> 00:14:16,750
That's what all hands were.

218
00:14:16,750 --> 00:14:17,880
Good.

219
00:14:17,880 --> 00:14:20,870
All right, so I cancel these
and what I'm left with then

220
00:14:20,870 --> 00:14:24,900
is delta u times v
minus u times delta v

221
00:14:24,900 --> 00:14:32,250
and all this is over v
+ delta v times v. Okay,

222
00:14:32,250 --> 00:14:33,360
there's the difference.

223
00:14:33,360 --> 00:14:36,600
There's the change
in the quotient.

224
00:14:36,600 --> 00:14:39,580
The change in this function
is given by this formula.

225
00:14:39,580 --> 00:14:41,570
And now to compute
the derivative,

226
00:14:41,570 --> 00:14:45,000
I want to divide by delta
x, and take the limit.

227
00:14:45,000 --> 00:14:53,560
So let's write that down,
delta(u/v)/delta x is this

228
00:14:53,560 --> 00:14:56,820
formula here divided by delta x.

229
00:14:56,820 --> 00:15:00,986
And again, I'm going to put
the delta x under these delta u

230
00:15:00,986 --> 00:15:02,780
and delta v. Okay?

231
00:15:02,780 --> 00:15:05,080
I'm gonna put delta
x in the denominator,

232
00:15:05,080 --> 00:15:07,150
but I can think of
that as dividing

233
00:15:07,150 --> 00:15:09,920
into this factor
and this factor.

234
00:15:09,920 --> 00:15:16,980
So this is (delta u/ delta
x)v - u(delta v/delta x).

235
00:15:21,130 --> 00:15:23,130
And all that is divided
by the same denominator,

236
00:15:23,130 --> 00:15:28,970
(v + delta v)v. Right?

237
00:15:28,970 --> 00:15:33,010
Put the delta x up in
the numerator there.

238
00:15:33,010 --> 00:15:37,830
Next up, take the limit
as delta x goes to 0.

239
00:15:37,830 --> 00:15:43,470
I get, by definition,
the derivative of (u/v).

240
00:15:43,470 --> 00:15:46,210
And on the right
hand side, well, this

241
00:15:46,210 --> 00:15:51,300
is the derivative du/dx right?

242
00:15:51,300 --> 00:15:55,570
Times v. See and then
u times, and here it's

243
00:15:55,570 --> 00:15:56,480
the derivative dv/dx.

244
00:16:00,420 --> 00:16:04,250
Now what about the denominator?

245
00:16:04,250 --> 00:16:10,220
So when delta x goes to 0,
v stays the same, v stays

246
00:16:10,220 --> 00:16:10,720
the same.

247
00:16:10,720 --> 00:16:13,480
What happens to this delta v?

248
00:16:13,480 --> 00:16:17,970
It goes to 0, again,
because v is continuous.

249
00:16:17,970 --> 00:16:23,330
So again, delta v
goes to 0 with delta x

250
00:16:23,330 --> 00:16:28,180
because they're continuous
and you just get v times v.

251
00:16:28,180 --> 00:16:30,867
I think that's the formula
I wrote down over there.

252
00:16:30,867 --> 00:16:31,700
(du/dx)v - u(dv/dx).

253
00:16:35,510 --> 00:16:40,770
And all divided by the square
of the old denominator.

254
00:16:40,770 --> 00:16:42,160
Well, that's it.

255
00:16:42,160 --> 00:16:43,540
That's the quotient rule.

256
00:16:43,540 --> 00:16:44,520
Weird formula.

257
00:16:44,520 --> 00:16:46,160
Let's see an application.

258
00:16:46,160 --> 00:16:51,070
Let's see an example.

259
00:16:51,070 --> 00:16:54,680
So the example I'm going
to give is pretty simple.

260
00:16:54,680 --> 00:16:58,100
I'm going to take the
numerator to be just 1.

261
00:16:58,100 --> 00:17:02,790
So I'm gonna take u = 1.

262
00:17:02,790 --> 00:17:07,580
So now I'm
differentiating 1 / v,

263
00:17:07,580 --> 00:17:14,430
the reciprocal of a
function; 1 over a function.

264
00:17:14,430 --> 00:17:16,880
Here's a copy of my rule.

265
00:17:16,880 --> 00:17:22,790
What's du/ dx in that
case? u is a constant,

266
00:17:22,790 --> 00:17:27,050
so that term is 0 in this rule.

267
00:17:27,050 --> 00:17:28,700
I don't have to
worry about this.

268
00:17:28,700 --> 00:17:31,650
I get a minus.

269
00:17:31,650 --> 00:17:36,800
And then u is 1, and dv/dx.

270
00:17:36,800 --> 00:17:38,820
Well, v is whatever v is.

271
00:17:38,820 --> 00:17:40,790
I'll write dv/dx as v'.

272
00:17:43,854 --> 00:17:45,520
And then I get a v^2
in the denominator.

273
00:17:45,520 --> 00:17:50,070
So that's the rule.

274
00:17:50,070 --> 00:17:51,380
I could write it as v^(-2) v'.

275
00:17:56,840 --> 00:17:59,300
Minus v' divided by v^2.

276
00:17:59,300 --> 00:18:03,730
That's the derivative of 1 / v.

277
00:18:03,730 --> 00:18:12,110
How about sub-example of that?

278
00:18:12,110 --> 00:18:15,840
I'm going to take the special
case where u = 1 again.

279
00:18:15,840 --> 00:18:16,770
And v is a power of x.

280
00:18:21,000 --> 00:18:25,630
And I'm gonna use the rule
that we developed earlier about

281
00:18:25,630 --> 00:18:29,080
the derivative of x^n.

282
00:18:29,080 --> 00:18:33,230
So what do I get here?

283
00:18:33,230 --> 00:18:42,645
d/dx (1/x^n) is, I'm plugging
into this formula here with v =

284
00:18:42,645 --> 00:18:45,260
x^n.

285
00:18:45,260 --> 00:18:51,580
So I get minus, uh, v^-2.

286
00:18:51,580 --> 00:18:57,250
If v = x^n, v^-2 is, by the
rule of exponents, x^(-2n).

287
00:19:01,430 --> 00:19:05,550
And then v' is the derivative
of x^n, which is nx^(n-1).

288
00:19:10,150 --> 00:19:12,010
Okay, so let's put
these together.

289
00:19:12,010 --> 00:19:13,550
There's several
powers of x here.

290
00:19:13,550 --> 00:19:14,940
I can put them together.

291
00:19:14,940 --> 00:19:22,330
I get -n x to the -2n + n - 1.

292
00:19:22,330 --> 00:19:23,936
One of these n's cancels.

293
00:19:23,936 --> 00:19:25,310
And what I'm left
with is -n - 1.

294
00:19:29,260 --> 00:19:32,550
So we've computed the
derivative of 1 / x^n,

295
00:19:32,550 --> 00:19:39,210
which I could also
write as x^-n, right?

296
00:19:39,210 --> 00:19:42,640
So I've computed the derivative
of negative powers of x.

297
00:19:42,640 --> 00:19:46,560
And this is the
formula that I get.

298
00:19:46,560 --> 00:19:51,990
If you think of this -n as a
unit, as a thing to itself,

299
00:19:51,990 --> 00:19:54,310
it occurs here in the exponent.

300
00:19:54,310 --> 00:19:59,890
It occurs here,
and it occurs here.

301
00:19:59,890 --> 00:20:01,820
So how does that
compare with the formula

302
00:20:01,820 --> 00:20:04,120
that we had up here?

303
00:20:04,120 --> 00:20:06,860
The derivative of
a power of x is

304
00:20:06,860 --> 00:20:12,300
that power times x to
one less than that power.

305
00:20:12,300 --> 00:20:16,010
That's exactly the same as the
rule that I wrote down here.

306
00:20:16,010 --> 00:20:19,050
But the power here happens
to be a negative number,

307
00:20:19,050 --> 00:20:22,360
and the same negative number
shows up as a coefficient

308
00:20:22,360 --> 00:20:23,780
and there in the exponent.

309
00:20:23,780 --> 00:20:24,280
Yeah?

310
00:20:24,280 --> 00:20:30,440
Student: [INAUDIBLE].

311
00:20:30,440 --> 00:20:34,930
Professor: How did I do this?

312
00:20:34,930 --> 00:20:49,150
Student: [INAUDIBLE].

313
00:20:49,150 --> 00:20:55,990
Professor: Where did
that x^(-2n) come from?

314
00:20:55,990 --> 00:20:59,900
So I'm applying this rule.

315
00:20:59,900 --> 00:21:04,440
So the denominator in
the quotient rule is v^2.

316
00:21:04,440 --> 00:21:11,109
And v was x^n, so the
denominator is x^(2n).

317
00:21:11,109 --> 00:21:12,650
And I decided to
write it as x^(-2n).

318
00:21:19,010 --> 00:21:22,080
So the green comments there...

319
00:21:22,080 --> 00:21:26,270
What they say is that I
can enlarge this rule.

320
00:21:26,270 --> 00:21:31,230
This exact same rule is true
for negative values of n,

321
00:21:31,230 --> 00:21:36,310
as well as positive values of n.

322
00:21:36,310 --> 00:21:40,300
So there's something
new in your list

323
00:21:40,300 --> 00:21:46,670
of rules that you can apply,
of values of the derivative.

324
00:21:46,670 --> 00:21:49,550
That standard rule is true for
negative as well as positive

325
00:21:49,550 --> 00:21:51,120
exponents.

326
00:21:51,120 --> 00:21:57,290
And that comes out
of a quotient rule.

327
00:21:57,290 --> 00:21:59,020
Okay, so we've done two rules.

328
00:21:59,020 --> 00:22:04,650
I've talked about the product
rule and the quotient rule.

329
00:22:04,650 --> 00:22:05,670
What's next?

330
00:22:05,670 --> 00:22:07,150
Let's see the chain rule.

331
00:22:07,150 --> 00:22:22,220
So this is a composition rule.

332
00:22:22,220 --> 00:22:24,890
So the kind of thing that
I have in mind, composition

333
00:22:24,890 --> 00:22:28,210
of functions is
about substitution.

334
00:22:28,210 --> 00:22:31,106
So the kind of function that I
have in mind is, for instance,

335
00:22:31,106 --> 00:22:31,730
y = (sin t)^10.

336
00:22:39,700 --> 00:22:42,695
That's a new one.

337
00:22:42,695 --> 00:22:44,820
We haven't seen how to
differentiate that before, I

338
00:22:44,820 --> 00:22:46,590
think.

339
00:22:46,590 --> 00:22:50,600
This kind of power of a trig
function happens very often.

340
00:22:50,600 --> 00:22:53,540
You've seen them happen,
as well, I'm sure, already.

341
00:22:53,540 --> 00:22:58,020
And there's a little notational
switch that people use.

342
00:22:58,020 --> 00:22:59,320
They'll write sin^10(t).

343
00:23:02,910 --> 00:23:05,100
But remember that when
you write sin^10(t),

344
00:23:05,100 --> 00:23:08,030
what you mean is
take the sine of t,

345
00:23:08,030 --> 00:23:10,440
and then take the
10th power of that.

346
00:23:10,440 --> 00:23:13,590
It's the meaning of sin^10(t).

347
00:23:13,590 --> 00:23:20,950
So the method of dealing
with this kind of composition

348
00:23:20,950 --> 00:23:33,190
of functions is to use
new variable names.

349
00:23:33,190 --> 00:23:36,830
What I mean is, I can
think of this (sin t)^10.

350
00:23:39,710 --> 00:23:42,070
I can think of it it
as a two step process.

351
00:23:42,070 --> 00:23:44,160
First of all, I
compute the sine of t.

352
00:23:44,160 --> 00:23:47,450
And let's call the result x.

353
00:23:47,450 --> 00:23:50,150
There's the new variable name.

354
00:23:50,150 --> 00:23:53,340
And then, I express
y in terms of x.

355
00:23:53,340 --> 00:23:58,070
So y says take this and
raise it to the tenth power.

356
00:23:58,070 --> 00:23:59,360
In other words, y = x^10.

357
00:24:03,400 --> 00:24:06,420
And then you plug x
= sin(t) into that,

358
00:24:06,420 --> 00:24:10,590
and you get the formula for
what y is in terms of t.

359
00:24:10,590 --> 00:24:14,550
So it's good practice to
introduce new letters when

360
00:24:14,550 --> 00:24:17,060
they're convenient, and
this is one place where

361
00:24:17,060 --> 00:24:21,820
it's very convenient.

362
00:24:21,820 --> 00:24:24,260
So let's find a rule
for differentiating

363
00:24:24,260 --> 00:24:25,860
a composition, a
function that can

364
00:24:25,860 --> 00:24:27,770
be expressed by
doing one function

365
00:24:27,770 --> 00:24:30,270
and then applying
another function.

366
00:24:30,270 --> 00:24:32,880
And here's the rule.

367
00:24:32,880 --> 00:24:34,930
Well, maybe I'll actually
derive this rule first,

368
00:24:34,930 --> 00:24:37,420
and then you'll see what it is.

369
00:24:37,420 --> 00:24:40,600
In fact, the rule is
very simple to derive.

370
00:24:40,600 --> 00:24:43,890
So this is a proof first, and
then we'll write down the rule.

371
00:24:43,890 --> 00:24:51,950
I'm interested in delta y /
delta t. y is a function of x.

372
00:24:51,950 --> 00:24:53,760
x is a function of t.

373
00:24:53,760 --> 00:24:56,850
And I'm interested in how
y changes with respect

374
00:24:56,850 --> 00:25:00,850
to t, with respect to
the original variable t.

375
00:25:00,850 --> 00:25:05,160
Well, because of that
intermediate variable,

376
00:25:05,160 --> 00:25:12,670
I can write this as (delta y /
delta x) (delta x / delta t).

377
00:25:12,670 --> 00:25:15,330
It cancels, right?

378
00:25:15,330 --> 00:25:17,600
The delta x cancels.

379
00:25:17,600 --> 00:25:23,100
The change in that immediate
variable cancels out.

380
00:25:23,100 --> 00:25:26,120
This is just basic algebra.

381
00:25:26,120 --> 00:25:29,930
But what happens when I
let delta t get small?

382
00:25:29,930 --> 00:25:31,410
Well this give me dy/dt.

383
00:25:34,370 --> 00:25:42,220
On the right-hand side,
I get (dy/dx) (dx/dt).

384
00:25:42,220 --> 00:25:44,430
So students will often
remember this rule.

385
00:25:44,430 --> 00:25:47,130
This is the rule, by saying
that you can cancel out

386
00:25:47,130 --> 00:25:49,080
for the dx's.

387
00:25:49,080 --> 00:25:51,860
And that's not so
far from the truth.

388
00:25:51,860 --> 00:25:55,160
That's a good way
to think of it.

389
00:25:55,160 --> 00:26:01,410
In other words, this is
the so-called chain rule.

390
00:26:01,410 --> 00:26:26,690
And it says that differentiation
of a composition is a product.

391
00:26:26,690 --> 00:26:34,910
It's just the product
of the two derivatives.

392
00:26:34,910 --> 00:26:39,570
So that's how you differentiate
a composite of two functions.

393
00:26:39,570 --> 00:26:42,070
And let's just do an example.

394
00:26:42,070 --> 00:26:44,690
Let's do this example.

395
00:26:44,690 --> 00:26:48,820
Let's see how that comes out.

396
00:26:48,820 --> 00:26:55,250
So let's differentiate,
what did I say?

397
00:26:55,250 --> 00:26:56,530
(sin t)^10.

398
00:26:59,400 --> 00:27:03,130
Okay, there's an inside function
and an outside function.

399
00:27:03,130 --> 00:27:07,910
The inside function is
x as a function of t.

400
00:27:07,910 --> 00:27:19,170
This is the inside function, and
this is the outside function.

401
00:27:19,170 --> 00:27:22,590
So the rule says, first
of all let's differentiate

402
00:27:22,590 --> 00:27:23,550
the outside function.

403
00:27:23,550 --> 00:27:25,370
Take dy/dx.

404
00:27:25,370 --> 00:27:29,200
Differentiate it with
respect to that variable x.

405
00:27:29,200 --> 00:27:31,020
The outside function
is the 10th power.

406
00:27:31,020 --> 00:27:34,640
What's its derivative?

407
00:27:34,640 --> 00:27:37,530
So I get 10x^9.

408
00:27:42,440 --> 00:27:51,090
In this account, I'm using
this newly introduced variable

409
00:27:51,090 --> 00:27:53,990
named x.

410
00:27:53,990 --> 00:27:58,150
So the derivative of the
outside function is 10x^9.

411
00:27:58,150 --> 00:28:00,360
And then here's the
inside function,

412
00:28:00,360 --> 00:28:03,130
and the next thing I want
to do is differentiate it.

413
00:28:03,130 --> 00:28:07,730
So what's dx/dt, d/dt (sin
t), the derivative of sine t?

414
00:28:07,730 --> 00:28:11,619
All right, that's cosine t.

415
00:28:11,619 --> 00:28:13,160
That's what the
chain rule gives you.

416
00:28:13,160 --> 00:28:17,490
This is correct, but
since we were the ones

417
00:28:17,490 --> 00:28:20,730
to introduce this
notation x here,

418
00:28:20,730 --> 00:28:24,560
that wasn't given to us in
the original problem here.

419
00:28:24,560 --> 00:28:26,400
The last step in
this process should

420
00:28:26,400 --> 00:28:28,980
be to put back,
to substitute back

421
00:28:28,980 --> 00:28:32,440
in what x is in terms of t.

422
00:28:32,440 --> 00:28:35,320
So x = sin t.

423
00:28:35,320 --> 00:28:45,980
So that tells me that I get
10(sin(t))^9, that's x^9,

424
00:28:45,980 --> 00:28:47,860
times the cos(t).

425
00:28:47,860 --> 00:28:50,860
Or the same thing
is sin^9(t)cos(t).

426
00:28:56,040 --> 00:28:59,540
So there's an application
of the chain rule.

427
00:28:59,540 --> 00:29:02,504
You know, people often
wonder where the name chain

428
00:29:02,504 --> 00:29:03,170
rule comes from.

429
00:29:03,170 --> 00:29:06,340
I was just wondering
about that myself.

430
00:29:06,340 --> 00:29:15,230
So is it because
it chains you down?

431
00:29:15,230 --> 00:29:18,070
Is it like a chain fence?

432
00:29:18,070 --> 00:29:19,590
I decided what it is.

433
00:29:19,590 --> 00:29:21,910
It's because by
using it, you burst

434
00:29:21,910 --> 00:29:25,880
the chains of differentiation,
and you can differentiate

435
00:29:25,880 --> 00:29:28,040
many more functions using it.

436
00:29:28,040 --> 00:29:31,553
So when you want to
think of the chain rule,

437
00:29:31,553 --> 00:29:35,640
just think of that chain there.

438
00:29:35,640 --> 00:29:47,960
It lets you burst free.

439
00:29:47,960 --> 00:30:04,830
Let me give you another
application of the chain rule.

440
00:30:04,830 --> 00:30:16,220
Ready for this one?

441
00:30:16,220 --> 00:30:17,970
So I'd like to
differentiate the sin(10t).

442
00:30:25,524 --> 00:30:27,440
Again, this is the
composite of two functions.

443
00:30:27,440 --> 00:30:30,220
What's the inside function?

444
00:30:30,220 --> 00:30:35,640
Okay, so I think I'll introduce
this new notation. x = 10t,

445
00:30:35,640 --> 00:30:38,260
and the outside
function is the sine.

446
00:30:38,260 --> 00:30:41,320
So y = sin x.

447
00:30:41,320 --> 00:30:46,660
So now the chain
rule says dy/dt is...

448
00:30:46,660 --> 00:30:47,920
Okay, let's see.

449
00:30:47,920 --> 00:30:50,710
I take the derivative
of the outside function,

450
00:30:50,710 --> 00:30:54,240
and what's that?

451
00:30:54,240 --> 00:30:56,470
Sine prime and we can
substitute because we

452
00:30:56,470 --> 00:30:58,520
know what sine prime is.

453
00:30:58,520 --> 00:31:06,470
So I get cosine of whatever,
x, and then times what?

454
00:31:06,470 --> 00:31:11,400
Now I differentiate the inside
function, which is just 10.

455
00:31:11,400 --> 00:31:16,380
So I could write this
as 10cos of what?

456
00:31:16,380 --> 00:31:17,360
10t, x is 10t.

457
00:31:20,260 --> 00:31:26,170
Now, once you get used to
this, this middle variable,

458
00:31:26,170 --> 00:31:33,190
you don't have to
give a name for it.

459
00:31:33,190 --> 00:31:35,150
You can just to think
about it in your mind

460
00:31:35,150 --> 00:31:44,890
without actually writing
it down, d/dt (sin(10t)).

461
00:31:47,980 --> 00:31:49,860
I'll just do it again
without introducing

462
00:31:49,860 --> 00:31:52,240
this middle variable explicitly.

463
00:31:52,240 --> 00:31:54,530
Think about it.

464
00:31:54,530 --> 00:31:58,100
I first of all differentiate
the outside function,

465
00:31:58,100 --> 00:31:59,740
and I get cosine.

466
00:31:59,740 --> 00:32:03,170
But I don't change the thing
that I'm plugging into it.

467
00:32:03,170 --> 00:32:08,560
It's still x that I'm
plugging into it. x is 10t.

468
00:32:08,560 --> 00:32:11,470
So let's just write 10t and
not worry about the name

469
00:32:11,470 --> 00:32:12,720
of that extra variable.

470
00:32:12,720 --> 00:32:15,510
If it confuses you,
introduce the new variable.

471
00:32:15,510 --> 00:32:18,180
And do it carefully
and slowly like this.

472
00:32:18,180 --> 00:32:19,970
But, quite quickly,
I think you'll

473
00:32:19,970 --> 00:32:23,202
get to be able to keep
that step in your mind.

474
00:32:23,202 --> 00:32:24,160
I'm not quite done yet.

475
00:32:24,160 --> 00:32:26,900
I haven't differentiated
the inside function,

476
00:32:26,900 --> 00:32:29,190
the derivative of 10t = 10.

477
00:32:29,190 --> 00:32:33,250
So you get, again,
the same result.

478
00:32:33,250 --> 00:32:36,420
A little shortcut that
you'll get used to.

479
00:32:36,420 --> 00:32:38,680
Really and truly, once
you have the chain rule,

480
00:32:38,680 --> 00:32:41,110
the world is yours to conquer.

481
00:32:41,110 --> 00:32:46,730
It puts you in a very,
very powerful position.

482
00:32:46,730 --> 00:32:50,210
Okay, well let's see.

483
00:32:50,210 --> 00:32:51,310
What have I covered today?

484
00:32:51,310 --> 00:32:57,370
I've talked about product rule,
quotient rule, composition.

485
00:32:57,370 --> 00:32:59,580
I should tell you something
about higher derivatives,

486
00:32:59,580 --> 00:33:00,670
as well.

487
00:33:00,670 --> 00:33:10,440
So let's do that.

488
00:33:10,440 --> 00:33:12,150
This is a simple story.

489
00:33:12,150 --> 00:33:14,950
Higher is kind of
a strange word.

490
00:33:14,950 --> 00:33:32,950
It just means differentiate
over and over again.

491
00:33:32,950 --> 00:33:34,600
All right, so let's see.

492
00:33:34,600 --> 00:33:38,510
If we have a function
u or u(x), please

493
00:33:38,510 --> 00:33:45,010
allow me to just write
it as briefly as u.

494
00:33:45,010 --> 00:33:49,330
Well, this is a sort
of notational thing.

495
00:33:49,330 --> 00:33:51,780
I can differentiate
it and get u'.

496
00:33:54,790 --> 00:33:55,900
That's a new function.

497
00:33:55,900 --> 00:33:57,680
Like if you started
with the sine, that's

498
00:33:57,680 --> 00:34:00,760
gonna be the cosine.

499
00:34:00,760 --> 00:34:03,570
A new function, so I can
differentiate it again.

500
00:34:03,570 --> 00:34:05,780
And the notation for the
differentiating of it again,

501
00:34:05,780 --> 00:34:07,470
is u prime prime.

502
00:34:07,470 --> 00:34:12,930
So u'' is just u'
differentiated again.

503
00:34:12,930 --> 00:34:21,380
For example, if u is the sine
of x, so u' is the cosine of x.

504
00:34:21,380 --> 00:34:24,150
Has Professor Jerison
talked about what

505
00:34:24,150 --> 00:34:26,580
the derivative of cosine is?

506
00:34:26,580 --> 00:34:28,220
What is it?

507
00:34:28,220 --> 00:34:33,020
Ha, okay so u'' is -sin x.

508
00:34:36,810 --> 00:34:38,930
Let me go on.

509
00:34:38,930 --> 00:34:42,970
What do you suppose u''' means?

510
00:34:42,970 --> 00:34:46,420
I guess it's the
derivative of u''.

511
00:34:46,420 --> 00:34:53,050
It's called the
third derivative.

512
00:34:53,050 --> 00:34:56,210
And u'' is called the
second derivative.

513
00:34:56,210 --> 00:34:59,000
And it's u''
differentiated again.

514
00:34:59,000 --> 00:35:03,680
So to compute u''' in this
example, what do I do?

515
00:35:03,680 --> 00:35:05,340
I differentiate that again.

516
00:35:05,340 --> 00:35:08,460
There's a constant term,
-1, constant factor.

517
00:35:08,460 --> 00:35:09,950
That comes out.

518
00:35:09,950 --> 00:35:13,500
The derivative of sine is what?

519
00:35:13,500 --> 00:35:17,930
Okay, so u''' = -cos x.

520
00:35:17,930 --> 00:35:18,690
Let's do it again.

521
00:35:18,690 --> 00:35:21,890
Now after a while, you get
tired of writing these things.

522
00:35:21,890 --> 00:35:24,650
And so maybe I'll use
the notation u^(4).

523
00:35:24,650 --> 00:35:27,290
That's the fourth derivative.

524
00:35:27,290 --> 00:35:29,490
That's u''''.

525
00:35:29,490 --> 00:35:33,440
Or it's (u''') differentiated
again, the fourth derivative.

526
00:35:33,440 --> 00:35:37,970
And what is that
in this example?

527
00:35:37,970 --> 00:35:41,290
Okay, the cosine has
derivative minus the sine,

528
00:35:41,290 --> 00:35:42,010
like you told me.

529
00:35:42,010 --> 00:35:44,430
And that minus sign
cancels with that sign,

530
00:35:44,430 --> 00:35:47,640
and all together, I get sin x.

531
00:35:47,640 --> 00:35:48,940
That's pretty bizarre.

532
00:35:48,940 --> 00:35:51,720
When I differentiate the
function sine of x four times,

533
00:35:51,720 --> 00:35:56,920
I get back to the
sine of x again.

534
00:35:56,920 --> 00:36:00,290
That's the way it is.

535
00:36:00,290 --> 00:36:03,491
Now this notation, prime
prime prime prime, and things

536
00:36:03,491 --> 00:36:03,990
like that.

537
00:36:03,990 --> 00:36:13,650
There are different
variants of that notation.

538
00:36:13,650 --> 00:36:24,070
For example, that's
another notation.

539
00:36:24,070 --> 00:36:29,320
Well, you've used the
notation du/dx before. u'

540
00:36:29,320 --> 00:36:30,630
could also be denoted du/dx.

541
00:36:35,730 --> 00:36:38,460
I think we've
already here, today,

542
00:36:38,460 --> 00:36:43,230
used this way of
rewriting du/dx.

543
00:36:43,230 --> 00:36:48,150
I think when I was talking about
d/dt(uv) and so on, I pulled

544
00:36:48,150 --> 00:36:52,360
that d/dt outside and
put whatever function

545
00:36:52,360 --> 00:36:55,010
you're differentiating
over to the right.

546
00:36:55,010 --> 00:36:57,430
So that's just a
notational switch.

547
00:36:57,430 --> 00:36:58,110
It looks good.

548
00:36:58,110 --> 00:37:06,260
It looks like good
algebra doesn't it?

549
00:37:06,260 --> 00:37:12,410
But what it's doing is regarding
this notation as an operator.

550
00:37:12,410 --> 00:37:16,920
It's something you apply to a
function to get a new function.

551
00:37:16,920 --> 00:37:20,680
I apply it to the sine function,
and I get the cosine function.

552
00:37:20,680 --> 00:37:24,220
I apply it to x^2, and I get 2x.

553
00:37:24,220 --> 00:37:31,140
This thing here, that symbol,
represents an operator,

554
00:37:31,140 --> 00:37:40,340
which you apply to a function.

555
00:37:40,340 --> 00:37:44,860
And the operator says, take the
function and differentiate it.

556
00:37:44,860 --> 00:37:47,330
So further notation
that people often use,

557
00:37:47,330 --> 00:37:49,460
is they give a different
name to that operator.

558
00:37:49,460 --> 00:37:52,270
And they'll write
capital D for it.

559
00:37:52,270 --> 00:38:02,980
So this is just using capital
D for the symbol d/dx.

560
00:38:02,980 --> 00:38:05,050
So in terms of that
notation, let's see.

561
00:38:05,050 --> 00:38:20,440
Let's write down what higher
derivatives look like.

562
00:38:20,440 --> 00:38:21,870
So let's see.

563
00:38:21,870 --> 00:38:23,090
That's what u' is.

564
00:38:23,090 --> 00:38:24,360
How about u''?

565
00:38:24,360 --> 00:38:28,890
Let's write that in terms
of the d/dx notation.

566
00:38:28,890 --> 00:38:31,710
Well I'm supposed to
differentiate u' right?

567
00:38:31,710 --> 00:38:35,590
So that's d/dx applied
to the function du/dx.

568
00:38:40,920 --> 00:38:43,030
Differentiate the derivative.

569
00:38:43,030 --> 00:38:47,240
That's what I've done.

570
00:38:47,240 --> 00:38:54,350
Or I could write that as d/dx
applied to d/dx applied to u.

571
00:38:54,350 --> 00:38:57,850
Just pulling that u outside.

572
00:38:57,850 --> 00:38:59,570
So I'm doing d/dx twice.

573
00:38:59,570 --> 00:39:01,590
I'm doing that operator twice.

574
00:39:01,590 --> 00:39:08,030
I could write that as
(d/dx)^2 applied to u.

575
00:39:08,030 --> 00:39:15,170
Differentiate twice, and
do it to the function u.

576
00:39:15,170 --> 00:39:23,130
Or, I can write it as,
now this is a strange one.

577
00:39:23,130 --> 00:39:33,330
I could also write
as-- like that.

578
00:39:33,330 --> 00:39:36,630
It's getting stranger
and stranger, isn't it?

579
00:39:36,630 --> 00:39:40,770
This is definitely just a
kind of abuse of notation.

580
00:39:40,770 --> 00:39:45,030
But people will go even
further and write d squared

581
00:39:45,030 --> 00:39:46,030
u divided by dx squared.

582
00:39:50,500 --> 00:39:52,190
So this is the strangest one.

583
00:39:52,190 --> 00:39:56,190
This identity quality
is the strangest one,

584
00:39:56,190 --> 00:40:00,130
because you may think that
you're taking d of the quantity

585
00:40:00,130 --> 00:40:01,330
x^2.

586
00:40:01,330 --> 00:40:03,930
But that's not what's intended.

587
00:40:03,930 --> 00:40:08,240
This is not d(x^2).

588
00:40:08,240 --> 00:40:12,750
What's intended is the
quantity dx squared.

589
00:40:12,750 --> 00:40:14,630
In this notation,
which is very common,

590
00:40:14,630 --> 00:40:16,410
what's intended
by the denominator

591
00:40:16,410 --> 00:40:18,250
is the quantity dx squared.

592
00:40:18,250 --> 00:40:23,630
It's part of this second
differentiation operator.

593
00:40:23,630 --> 00:40:26,240
So I've written a bunch
of equalities down here,

594
00:40:26,240 --> 00:40:28,570
and the only content
to them is that these

595
00:40:28,570 --> 00:40:32,320
are all different notations
for the same thing.

596
00:40:32,320 --> 00:40:34,940
You'll see this
notation very commonly.

597
00:40:34,940 --> 00:40:37,050
So for instance the
third derivative

598
00:40:37,050 --> 00:40:47,330
is d cubed u divided
by dx cubed, and so on.

599
00:40:47,330 --> 00:40:47,830
Sorry?

600
00:40:47,830 --> 00:40:58,755
Student: [INAUDIBLE].

601
00:40:58,755 --> 00:40:59,880
Professor: Yes, absolutely.

602
00:40:59,880 --> 00:41:04,417
Or an equally good notation is
to write the operator capital

603
00:41:04,417 --> 00:41:05,500
D, done three times, to u.

604
00:41:09,400 --> 00:41:11,502
Absolutely.

605
00:41:11,502 --> 00:41:13,960
So I guess I should also write
over here when I was talking

606
00:41:13,960 --> 00:41:16,180
about d^2, the
second derivative,

607
00:41:16,180 --> 00:41:20,820
another notation is do the
operator capital D twice.

608
00:41:20,820 --> 00:41:22,820
Let's see an example of
how this can be applied.

609
00:41:22,820 --> 00:41:23,903
I'll answer this question.

610
00:41:23,903 --> 00:41:32,582
Student: [INAUDIBLE].

611
00:41:32,582 --> 00:41:34,040
Professor: Yeah,
so the question is

612
00:41:34,040 --> 00:41:36,230
whether the fourth
derivative always gives you

613
00:41:36,230 --> 00:41:38,880
the original function back,
like what happened here.

614
00:41:38,880 --> 00:41:39,580
No.

615
00:41:39,580 --> 00:41:43,470
That's very, very special
to sines and cosines.

616
00:41:43,470 --> 00:41:45,200
All right?

617
00:41:45,200 --> 00:41:47,850
And, in fact, let's
see an example of that.

618
00:41:47,850 --> 00:41:50,920
I'll do a calculation.

619
00:41:50,920 --> 00:42:06,130
Let's calculate the
nth derivative of x^n.

620
00:42:06,130 --> 00:42:13,190
Okay, n is a number,
like 1, 2, 3, 4.

621
00:42:13,190 --> 00:42:13,720
Here we go.

622
00:42:13,720 --> 00:42:15,360
Let's do this.

623
00:42:15,360 --> 00:42:17,650
So, let's do this bit by bit.

624
00:42:17,650 --> 00:42:22,500
What's the first
derivative of x^n?

625
00:42:22,500 --> 00:42:24,090
So everybody knows this.

626
00:42:24,090 --> 00:42:27,830
I'm just using a new notation,
this capital D notation.

627
00:42:27,830 --> 00:42:30,520
So it's nx^(n-1).

628
00:42:30,520 --> 00:42:34,380
Now know, by the way, n could
be a negative number for that,

629
00:42:34,380 --> 00:42:37,250
but for now, for
this application,

630
00:42:37,250 --> 00:42:41,280
I wanna take n to be
1, 2, 3, and so on;

631
00:42:41,280 --> 00:42:43,070
one of those numbers.

632
00:42:43,070 --> 00:42:44,550
Okay, we did one derivative.

633
00:42:44,550 --> 00:42:49,530
Let's compute the second
derivative of x^n.

634
00:42:49,530 --> 00:42:52,070
Well there's this n
constant that comes out,

635
00:42:52,070 --> 00:42:59,980
and then the exponent comes
down, and it gets reduced by 1.

636
00:42:59,980 --> 00:43:01,190
All right?

637
00:43:01,190 --> 00:43:03,780
Should I do one more?

638
00:43:03,780 --> 00:43:07,600
D^3 (x^n) is n(n-1).

639
00:43:07,600 --> 00:43:09,410
That's the constant from here.

640
00:43:09,410 --> 00:43:13,220
Times that exponent,
n - 2, times 1 less, n

641
00:43:13,220 --> 00:43:15,740
- 3 is the new exponent.

642
00:43:15,740 --> 00:43:26,430
Well, I keep on going until
I come to a new blackboard.

643
00:43:26,430 --> 00:43:28,100
Now, I think I'm
going to stop when

644
00:43:28,100 --> 00:43:29,980
I get to the n minus
first derivative,

645
00:43:29,980 --> 00:43:35,370
so we can see what's
likely to happen.

646
00:43:35,370 --> 00:43:38,970
So when I took the
third derivative,

647
00:43:38,970 --> 00:43:42,957
I had the n minus
third power of x.

648
00:43:42,957 --> 00:43:44,540
And when I took the
second derivative,

649
00:43:44,540 --> 00:43:45,760
I had the second power of x.

650
00:43:45,760 --> 00:43:48,310
So, I think what'll
happen when I

651
00:43:48,310 --> 00:43:49,730
have the n minus
first derivative

652
00:43:49,730 --> 00:43:53,510
is I'll have the first
power of x left over.

653
00:43:53,510 --> 00:43:55,390
The powers of x
keep coming down.

654
00:43:55,390 --> 00:43:59,350
And what I've done it n - 1
times, I get the first power.

655
00:43:59,350 --> 00:44:04,230
And then I get a big constant
out in front here times more

656
00:44:04,230 --> 00:44:07,450
and more and more of these
smaller and smaller integers

657
00:44:07,450 --> 00:44:08,500
that come down.

658
00:44:08,500 --> 00:44:12,310
What's the last integer that
came down before I got x^1

659
00:44:12,310 --> 00:44:17,460
here?

660
00:44:17,460 --> 00:44:19,390
Well, let's see.

661
00:44:19,390 --> 00:44:23,320
It's just 2, because this x^1
occurred as the derivative

662
00:44:23,320 --> 00:44:24,340
of x^2.

663
00:44:24,340 --> 00:44:27,800
And the coefficient
in front of that is 2.

664
00:44:27,800 --> 00:44:29,730
So that's what you get.

665
00:44:29,730 --> 00:44:35,140
The numbers n, n-1, and so
on down to 2, times x^1.

666
00:44:35,140 --> 00:44:41,560
And now we can differentiate
one more time and calculate what

667
00:44:41,560 --> 00:44:42,770
D^n x^n is.

668
00:44:42,770 --> 00:44:48,070
So I get the same number, n
times n-1 and so on and so on,

669
00:44:48,070 --> 00:44:49,680
times 2.

670
00:44:49,680 --> 00:44:52,500
And then I guess
I'll say times 1.

671
00:44:52,500 --> 00:44:58,640
Times, what's the derivative
of x^1= 1, so times 1.

672
00:44:58,640 --> 00:45:01,260
Time 1, times 1.

673
00:45:01,260 --> 00:45:10,490
Where this one means
the constant function 1.

674
00:45:10,490 --> 00:45:14,070
Does anyone know what
this number is called?

675
00:45:14,070 --> 00:45:15,110
That has a name.

676
00:45:15,110 --> 00:45:19,720
It's called n factorial.

677
00:45:19,720 --> 00:45:21,400
And it's written n
exclamation point.

678
00:45:24,240 --> 00:45:28,830
And we just used an example
of mathematical induction.

679
00:45:28,830 --> 00:45:37,750
So the end result is
D^n x^n is n!, constant.

680
00:45:37,750 --> 00:45:42,460
Okay that's a neat fact.

681
00:45:42,460 --> 00:45:47,570
Final question for the lecture
is what's D^(n + 1) applied

682
00:45:47,570 --> 00:45:49,730
to x^n?

683
00:45:49,730 --> 00:45:50,850
Ha.

684
00:45:50,850 --> 00:45:54,340
Excellent.

685
00:45:54,340 --> 00:45:56,620
It's the derivative
of a constant.

686
00:45:56,620 --> 00:45:57,940
So it's 0.

687
00:45:57,940 --> 00:45:58,440
Okay.

688
00:45:58,440 --> 00:45:59,980
Thank you.