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Professor: I am Haynes
Miller, I am substituting
9
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.