1
00:00:00 --> 00:00:00,11
2
00:00:00,11 --> 00:00:02,32
The following content is
provided under a Creative
3
00:00:02,32 --> 00:00:03,15
Commons license.
4
00:00:03,15 --> 00:00:06,57
Your support will help MIT
OpenCourseWare continue to
5
00:00:06,57 --> 00:00:09,96
offer high quality educational
resources for free.
6
00:00:09,96 --> 00:00:13,17
To make a donation, or to view
additional materials from
7
00:00:13,17 --> 00:00:15,91
hundreds of MIT courses visit
MIT OpenCourseWare
8
00:00:15,91 --> 00:00:22,24
at ocw.mit.edu.
9
00:00:22,24 --> 00:00:25,23
Professor: I am Haynes Miller,
I am substituting for
10
00:00:25,23 --> 00:00:26,51
David Jerison today.
11
00:00:26,51 --> 00:00:41,88
So you have a substitute
teacher today.
12
00:00:41,88 --> 00:00:45,08
So I haven't been here in this
class with you so I'm not
13
00:00:45,08 --> 00:00:47,58
completely sure where you are.
14
00:00:47,58 --> 00:00:53,59
I think just been talking about
differentiation and you've
15
00:00:53,59 --> 00:00:56,84
got some examples of
differentiation like these
16
00:00:56,84 --> 00:00:59,86
basic examples: the derivative
of x^n is nx^(x-1).
17
00:00:59,86 --> 00:01:03,35
18
00:01:03,35 --> 00:01:05,42
But I think maybe you've spent
some time computing the
19
00:01:05,42 --> 00:01:11,23
derivative of the sine
function as well, recently.
20
00:01:11,23 --> 00:01:16,51
And I think you have some
rules for extending these
21
00:01:16,51 --> 00:01:18,95
calculations as well.
22
00:01:18,95 --> 00:01:24,19
For instance, I think you know
that if you differentiate a
23
00:01:24,19 --> 00:01:27,77
constant times a function,
what do you get?
24
00:01:27,77 --> 00:01:32,59
Student: [INAUDIBLE].
25
00:01:32,59 --> 00:01:36,67
Professor: The constant
comes outside like this.
26
00:01:36,67 --> 00:01:40,03
Or I could write (cu)' = cu'.
27
00:01:40,03 --> 00:01:42,55
28
00:01:42,55 --> 00:01:45,51
That's this rule, multiplying
by a constant, and I
29
00:01:45,51 --> 00:01:58,87
think you also know about
differentiating a sum.
30
00:01:58,87 --> 00:02:00,3
Or I could write this as (u
31
00:02:00,3 --> 00:02:03,65
v)' = u' + v'.
32
00:02:03,65 --> 00:02:06,87
33
00:02:06,87 --> 00:02:10,83
So I'm going to be using those
but today I'll talk about a
34
00:02:10,83 --> 00:02:14,14
collection of other rules about
how to deal with a product of
35
00:02:14,14 --> 00:02:18,18
functions, a quotient of
functions, and, best of all,
36
00:02:18,18 --> 00:02:20,34
composition of functions.
37
00:02:20,34 --> 00:02:22,03
And then at the end, I'll
have something to say
38
00:02:22,03 --> 00:02:23,48
about higher derivatives.
39
00:02:23,48 --> 00:02:26,67
So that's the story for today.
40
00:02:26,67 --> 00:02:29,12
That's the program.
41
00:02:29,12 --> 00:02:43,36
So let's begin by talking
about the product rule.
42
00:02:43,36 --> 00:02:45,855
So the product rule tells you
how to differentiate a product
43
00:02:45,855 --> 00:02:48,04
of functions, and I'll just
give you the rule,
44
00:02:48,04 --> 00:02:49,27
first of all.
45
00:02:49,27 --> 00:02:51,6
The rule is it's u'v + uv'.
46
00:02:51,6 --> 00:02:57,28
47
00:02:57,28 --> 00:02:58,5
It's a little bit funny.
48
00:02:58,5 --> 00:03:02,32
Differentiating a product
gives you a sum.
49
00:03:02,32 --> 00:03:06,73
But let's see how that works
out in a particular example.
50
00:03:06,73 --> 00:03:10,3
For example, suppose that
I wanted to differentiate
51
00:03:10,3 --> 00:03:11,28
the product.
52
00:03:11,28 --> 00:03:14,67
Well, the product of these
two basic examples that
53
00:03:14,67 --> 00:03:15,61
we just talked about.
54
00:03:15,61 --> 00:03:18,06
I'm going to use the same
variable in both cases
55
00:03:18,06 --> 00:03:20,79
instead of different
ones like I did here.
56
00:03:20,79 --> 00:03:23,23
So the derivative
of (x^n)sin x.
57
00:03:23,23 --> 00:03:28,43
58
00:03:28,43 --> 00:03:30,3
So this is a new thing.
59
00:03:30,3 --> 00:03:36,12
We couldn't do this without
using the product rule.
60
00:03:36,12 --> 00:03:39,67
So the first function is x^n
and the second one is sin x.
61
00:03:39,67 --> 00:03:41,74
And we're going to
apply this rule.
62
00:03:41,74 --> 00:03:49,45
So u is x^n. u' is, according
to the rule, nx^(n - 1).
63
00:03:49,45 --> 00:03:56,05
And then I take v and write it
down the way it is, sine of x.
64
00:03:56,05 --> 00:03:57,32
And then I do it the other way.
65
00:03:57,32 --> 00:04:02,45
I take u the way it is, that's
x^n, and multiply it by
66
00:04:02,45 --> 00:04:05,26
the derivative of v, v'.
67
00:04:05,26 --> 00:04:09,235
We just saw v' is cosine of x.
68
00:04:09,235 --> 00:04:11,52
So that's it.
69
00:04:11,52 --> 00:04:14,98
Obviously, you can
differentiate longer products,
70
00:04:14,98 --> 00:04:20,56
products of more things by
doing it one at a time.
71
00:04:20,56 --> 00:04:22,7
Let's see why this is true.
72
00:04:22,7 --> 00:04:25,73
I want to try to show you
why the product rule holds.
73
00:04:25,73 --> 00:04:31,63
So you have a standard way of
trying to understand this, and
74
00:04:31,63 --> 00:04:34,71
it involves looking at the
change in the function that
75
00:04:34,71 --> 00:04:37,35
you're interested in
differentiating.
76
00:04:37,35 --> 00:04:41,64
So I should look at how much
the product uv changes when
77
00:04:41,64 --> 00:04:44,63
x changes a little bit.
78
00:04:44,63 --> 00:04:47,12
Well, so how do
compute the change?
79
00:04:47,12 --> 00:04:51,48
Well, I write down the value
of the function at some
80
00:04:51,48 --> 00:04:53,48
new value of x, (x
81
00:04:53,48 --> 00:04:56,16
delta x).
82
00:04:56,16 --> 00:04:58,41
Well, I better write down
the whole new value
83
00:04:58,41 --> 00:05:01,48
of the function, and
the function is uv.
84
00:05:01,48 --> 00:05:05,23
So the whole new value
looks like this.
85
00:05:05,23 --> 00:05:06,19
It's u (x
86
00:05:06,19 --> 00:05:09,45
delta x)v(x + delta x).
87
00:05:09,45 --> 00:05:10,96
That's the new value.
88
00:05:10,96 --> 00:05:13,2
But what's the change
in the product?
89
00:05:13,2 --> 00:05:18,27
Well, I better subtract off
what the old value was,
90
00:05:18,27 --> 00:05:20,92
which is u(x) v(x).
91
00:05:20,92 --> 00:05:25,05
Okay, according to the rule
we're trying to prove, I
92
00:05:25,05 --> 00:05:27,75
have to get u' involved.
93
00:05:27,75 --> 00:05:31,42
So I want to involve the
change in u alone, by itself.
94
00:05:31,42 --> 00:05:32,99
Let's just try that.
95
00:05:32,99 --> 00:05:36,62
I see part of the formula for
the change in u right there.
96
00:05:36,62 --> 00:05:40,26
Let's see if we can get
the rest of it in place.
97
00:05:40,26 --> 00:05:42,66
So the change in x = ((u(x)
98
00:05:42,66 --> 00:05:46,33
+ delta x) - u(x)).
99
00:05:46,33 --> 00:05:49,312
That's the change in
x [Correction:___c
100
00:05:49,312 --> 00:05:50,08
hange___in___u].
101
00:05:50,08 --> 00:05:54,1
This part of it occurs up
here, multiplied by v (x
102
00:05:54,1 --> 00:05:57,89
delta x), so let's
put that in too.
103
00:05:57,89 --> 00:06:00,44
Now this equality sign
isn't very good right now.
104
00:06:00,44 --> 00:06:05,65
I've got this product here
so far, but I've introduced
105
00:06:05,65 --> 00:06:06,69
something I don't like.
106
00:06:06,69 --> 00:06:09,05
I've introduced u (v ( x
107
00:06:09,05 --> 00:06:09,8
delta x)), right?
108
00:06:09,8 --> 00:06:12,01
Minus that.
109
00:06:12,01 --> 00:06:16,37
So the next thing I'm gonna
do is correct that little
110
00:06:16,37 --> 00:06:24,62
defect by adding in
u(x) (v (x + delta x)).
111
00:06:24,62 --> 00:06:28,88
Okay, now I cancelled off what
was wrong with this line.
112
00:06:28,88 --> 00:06:30,81
But I'm still not quite
there, because I haven't
113
00:06:30,81 --> 00:06:32,67
put this in yet.
114
00:06:32,67 --> 00:06:38,42
So I better subtract off
uv, and then I'll be home.
115
00:06:38,42 --> 00:06:41,2
But I'm going to do that in a
clever way, because I noticed
116
00:06:41,2 --> 00:06:43,9
that I already have a u here.
117
00:06:43,9 --> 00:06:47,15
So I'm gonna take this
factor of u and make it
118
00:06:47,15 --> 00:06:48,56
the same as this factor.
119
00:06:48,56 --> 00:06:52,51
So I get u(x) times this,
minus u(x) times that.
120
00:06:52,51 --> 00:06:57,18
That's the same thing as
u times the difference.
121
00:06:57,18 --> 00:07:00,18
So that was a little bit
strange, but when you stand
122
00:07:00,18 --> 00:07:02,86
back and look at it, you can
see multiplied out, the
123
00:07:02,86 --> 00:07:04,28
middle terms cancel.
124
00:07:04,28 --> 00:07:07,34
And you get the right answer.
125
00:07:07,34 --> 00:07:09,615
Well I like that because
it's involved the change
126
00:07:09,615 --> 00:07:12,24
in u and the change in v.
127
00:07:12,24 --> 00:07:16,81
So this is equal to
(delta u) v (x
128
00:07:16,81 --> 00:07:26,07
delta x) - u(x) times
the change in v.
129
00:07:26,07 --> 00:07:27,51
Well, I'm almost there.
130
00:07:27,51 --> 00:07:30,66
The next step in computing the
derivative is take difference
131
00:07:30,66 --> 00:07:42,91
quotient, divide
this by delta x.
132
00:07:42,91 --> 00:07:51,89
So, (delta (uv)) / (delta x)
is well, I'll say (delta
133
00:07:51,89 --> 00:07:55,58
u / delta x) v ( x
134
00:07:55,58 --> 00:08:03,15
delta x).
135
00:08:03,15 --> 00:08:10
Have I made a mistake here?
136
00:08:10 --> 00:08:13,34
This plus magically became a
minus on the way down here,
137
00:08:13,34 --> 00:08:18,81
so I better fix that.
138
00:08:18,81 --> 00:08:23,26
Plus u(delta v / (delta x).
139
00:08:23,26 --> 00:08:27,95
This u is this u over here.
140
00:08:27,95 --> 00:08:30,55
So I've just divided this
formula by delta x, and now I
141
00:08:30,55 --> 00:08:35,72
can take the limit as goes
to 0, so this is as
142
00:08:35,72 --> 00:08:42,07
delta x goes to 0.
143
00:08:42,07 --> 00:08:46,86
This becomes the definition of
the derivative, and on this
144
00:08:46,86 --> 00:08:54,62
side, I get du/dx times... now
what happens to this quantity
145
00:08:54,62 --> 00:09:02,75
when delta x goes to 0?
146
00:09:02,75 --> 00:09:05,71
So this quantity is getting
closer and closer to x.
147
00:09:05,71 --> 00:09:09
So what happens to
the value of v?
148
00:09:09 --> 00:09:10,75
It becomes equal to x(v).
149
00:09:10,75 --> 00:09:13,02
That uses continuity of v.
150
00:09:13,02 --> 00:09:15,35
So, v (x
151
00:09:15,35 --> 00:09:22,59
delta x) goes to
v(x) by continuity.
152
00:09:22,59 --> 00:09:27,07
So this gives me times v, and
then I have u(n delta v/ v
153
00:09:27,07 --> 00:09:30,68
delta x) gives me dv/dx.
154
00:09:30,68 --> 00:09:31,93
And that's the formula.
155
00:09:31,93 --> 00:09:33,53
That's the formula as I
wrote it down at the
156
00:09:33,53 --> 00:09:35,66
beginning over here.
157
00:09:35,66 --> 00:09:39,17
The derivative of a product
is given by this sum.
158
00:09:39,17 --> 00:09:46,29
Yeah?
159
00:09:46,29 --> 00:09:46,871
Student: How did you get from
the first line to the second
160
00:09:46,871 --> 00:09:49,3
of the long equation?
161
00:09:49,3 --> 00:09:51,39
Professor: From here to here?
162
00:09:51,39 --> 00:09:53,46
Student: Yes.
163
00:09:53,46 --> 00:09:55,89
Professor: So maybe it's
easiest to work backwards and
164
00:09:55,89 --> 00:09:59,8
verify that what I wrote
down is correct here.
165
00:09:59,8 --> 00:10:03,48
So, if you look
there's a u(v (x
166
00:10:03,48 --> 00:10:05,13
delta x)) there.
167
00:10:05,13 --> 00:10:07,84
And there's also one here.
168
00:10:07,84 --> 00:10:09,94
And they occur with
opposite signs.
169
00:10:09,94 --> 00:10:11,49
So they cancel.
170
00:10:11,49 --> 00:10:13,71
What's left is u ( x
171
00:10:13,71 --> 00:10:15,09
delta x) v (x
172
00:10:15,09 --> 00:10:20,53
delta x) - (uv).
173
00:10:20,53 --> 00:10:29,12
And that's just what
I started with.
174
00:10:29,12 --> 00:10:33,92
Student: They cancel right?
175
00:10:33,92 --> 00:10:37,55
Professor: I cancelled out
this term and this term, and
176
00:10:37,55 --> 00:10:39,7
what's left is the ends.
177
00:10:39,7 --> 00:10:41,49
Any other questions?
178
00:10:41,49 --> 00:10:49,66
Student: [INAUDIBLE].
179
00:10:49,66 --> 00:10:55,64
Professor: Well, I just
calculated what delta uv is,
180
00:10:55,64 --> 00:10:58,08
and now I'm gonna divide that
by delta x on my way to
181
00:10:58,08 --> 00:11:00,25
computing the derivative.
182
00:11:00,25 --> 00:11:07,76
And so I copied down the right
hand side and divided delta x.
183
00:11:07,76 --> 00:11:10,7
I just decided to divide
the delta u by delta x
184
00:11:10,7 --> 00:11:16,23
and delta v by delta x.
185
00:11:16,23 --> 00:11:16,99
Good.
186
00:11:16,99 --> 00:11:22,49
Anything else?
187
00:11:22,49 --> 00:11:24,26
So we have the
product rule here.
188
00:11:24,26 --> 00:11:26,98
The rule for differentiating
a product of two functions.
189
00:11:26,98 --> 00:11:28,55
This is making us stronger.
190
00:11:28,55 --> 00:11:30,24
There are many more
functions you can find
191
00:11:30,24 --> 00:11:31,42
derivatives of now.
192
00:11:31,42 --> 00:11:33,58
How about quotients?
193
00:11:33,58 --> 00:11:36,19
Let's find out how to
differentiate a quotient
194
00:11:36,19 --> 00:11:47,83
of two functions.
195
00:11:47,83 --> 00:11:49,97
Well again, I'll write down
what the answer is and then
196
00:11:49,97 --> 00:11:52,37
we'll try to verify it.
197
00:11:52,37 --> 00:11:55,38
So there's a quotient.
198
00:11:55,38 --> 00:11:56,15
Let me write this down.
199
00:11:56,15 --> 00:11:58,97
There's a quotient
of two functions.
200
00:11:58,97 --> 00:12:00,34
And here's the rule for it.
201
00:12:00,34 --> 00:12:02,82
I always have to think about
this and hope that I get
202
00:12:02,82 --> 00:12:09,14
it right. u'v - uv' / v^2.
203
00:12:09,14 --> 00:12:11,26
This may be the craziest
rule you'll see in this
204
00:12:11,26 --> 00:12:14,33
course, but there it is.
205
00:12:14,33 --> 00:12:18,21
And I'll try to show you why
that's true and see an example.
206
00:12:18,21 --> 00:12:18,93
Yeah there was a hand?
207
00:12:18,93 --> 00:12:27,3
Student: [INAUDIBLE]
208
00:12:27,3 --> 00:12:31,45
Professor: What letters
look the same? u and
209
00:12:31,45 --> 00:12:33,04
v look the same?
210
00:12:33,04 --> 00:12:37,04
I'll try to make them
look more different.
211
00:12:37,04 --> 00:12:39,34
The v's have points on the
bottom. u's have little
212
00:12:39,34 --> 00:12:41,16
round things on the bottom.
213
00:12:41,16 --> 00:12:44,98
What's the new value of u?
214
00:12:44,98 --> 00:12:50,99
The value of u at (x
215
00:12:50,99 --> 00:12:53,53
delta x) is (u
216
00:12:53,53 --> 00:12:55,55
delta u), right?
217
00:12:55,55 --> 00:12:56,4
That's what delta u is.
218
00:12:56,4 --> 00:13:01,07
It's the change in u when
x gets replaced by (x
219
00:13:01,07 --> 00:13:02,41
delta x).
220
00:13:02,41 --> 00:13:07,64
And the change in v, the
new value v, is (v
221
00:13:07,64 --> 00:13:09,7
delta v).
222
00:13:09,7 --> 00:13:13,13
So this is the new value of u
divided by the new value of v.
223
00:13:13,13 --> 00:13:16,13
That's the beginning.
224
00:13:16,13 --> 00:13:21,25
And then I subtract off the
old values, which are - u/v.
225
00:13:21,25 --> 00:13:26,1
This'll be easier to work out
when I write it out this way.
226
00:13:26,1 --> 00:13:27,75
So now, we'll cross
multiply, as I said.
227
00:13:27,75 --> 00:13:33,35
So I get (uv
228
00:13:33,35 --> 00:13:38,89
(delta u)v) minus, now I
cross multiply this way,
229
00:13:38,89 --> 00:13:46,33
you get (uv - u(delta v)).
230
00:13:46,33 --> 00:13:48,14
And I divide all this by (v
231
00:13:48,14 --> 00:13:49,98
delta v)u [Correction:___(
v___+___delta___v)v].
232
00:13:49,98 --> 00:13:52,84
233
00:13:52,84 --> 00:13:57,37
Okay, now the reason I like to
do it this way is that you see
234
00:13:57,37 --> 00:14:00,73
the cancellation happening
here. uv and uv occur twice
235
00:14:00,73 --> 00:14:02,19
and so I can cancel them.
236
00:14:02,19 --> 00:14:04,52
And I will, and I'll answer
these questions in a minute.
237
00:14:04,52 --> 00:14:06,26
Audience: [INAUDIBLE].
238
00:14:06,26 --> 00:14:11,34
Professor: Ooo, that's a v.
239
00:14:11,34 --> 00:14:14,03
All right.
240
00:14:14,03 --> 00:14:15,57
Good, anything else?
241
00:14:15,57 --> 00:14:16,75
That's what all hands were.
242
00:14:16,75 --> 00:14:17,88
Good.
243
00:14:17,88 --> 00:14:20,87
All right, so I cancel these
and what I'm left with then
244
00:14:20,87 --> 00:14:28,2
is (delta u)v - u(delta v)
and all this is over (v
245
00:14:28,2 --> 00:14:31,53
delta v)v.
246
00:14:31,53 --> 00:14:33,36
Ok, there's the difference.
247
00:14:33,36 --> 00:14:36,6
There's the change
in the quotient.
248
00:14:36,6 --> 00:14:39,58
The change in this function
is given by this formula.
249
00:14:39,58 --> 00:14:42,42
And now to compute the
derivative, I want to divide by
250
00:14:42,42 --> 00:14:45
delta x, and take the limit.
251
00:14:45 --> 00:14:51,65
So let's write that down,
delta(u/v)/delta x
252
00:14:51,65 --> 00:14:56,82
is this formula here
divided by delta x.
253
00:14:56,82 --> 00:15:00,42
And again, I'm going to put
the delta x under these
254
00:15:00,42 --> 00:15:02,33
delta u and delta v.
255
00:15:02,33 --> 00:15:02,78
Okay?
256
00:15:02,78 --> 00:15:06,08
I'm gonna put delta x in the
denominator, but I can think
257
00:15:06,08 --> 00:15:09,92
of that as dividing into
this factor and this factor.
258
00:15:09,92 --> 00:15:16,98
So this is ((delta u/ delta
x)v) - u(delta v/delta x)).
259
00:15:16,98 --> 00:15:21,16
260
00:15:21,16 --> 00:15:23,71
And all that is divided by
the same denominator, (v
261
00:15:23,71 --> 00:15:24,28
delta v)v.
262
00:15:24,28 --> 00:15:28,4
263
00:15:28,4 --> 00:15:28,97
Right?
264
00:15:28,97 --> 00:15:33,01
Put the delta x up in
the numerator there.
265
00:15:33,01 --> 00:15:37,83
Next up, take the limit
as delta x goes to 0.
266
00:15:37,83 --> 00:15:43,47
I get, by definition, the
derivative of (u/v).
267
00:15:43,47 --> 00:15:46,81
And on the right hand side,
well, this is the derivative
268
00:15:46,81 --> 00:15:51,3
u(du/ dx) right?
269
00:15:51,3 --> 00:15:51,49
Times v.
270
00:15:51,49 --> 00:16:00,42
See and then u times, and here
it's the derivative (dv/ dx).
271
00:16:00,42 --> 00:16:04,25
Now what about the denominator?
272
00:16:04,25 --> 00:16:09,56
So when delta x goes to
0, v stays the same,
273
00:16:09,56 --> 00:16:10,72
v stays the same.
274
00:16:10,72 --> 00:16:13,48
What happens to this delta v?
275
00:16:13,48 --> 00:16:17,97
It goes to 0, again,
because v is continuous.
276
00:16:17,97 --> 00:16:23,94
So again, delta v goes to 0
with delta x because they're
277
00:16:23,94 --> 00:16:25,36
continuous and you
just get (v*v).
278
00:16:25,36 --> 00:16:28,18
279
00:16:28,18 --> 00:16:31,02
I think that's the formula
I wrote down over there.
280
00:16:31,02 --> 00:16:31,7
(du/dx)v - u(dv/dx).
281
00:16:31,7 --> 00:16:35,51
282
00:16:35,51 --> 00:16:40,77
And all divided by the square
of the old denominator.
283
00:16:40,77 --> 00:16:42,16
Well, that's it.
284
00:16:42,16 --> 00:16:43,54
That's the quotient rule.
285
00:16:43,54 --> 00:16:44,52
Weird formula.
286
00:16:44,52 --> 00:16:46,16
Let's see an application.
287
00:16:46,16 --> 00:16:51,07
Let's see an example.
288
00:16:51,07 --> 00:16:54,68
So the example I'm going
to give is pretty simple.
289
00:16:54,68 --> 00:16:58,1
I'm going to take the
numerator to be just 1.
290
00:16:58,1 --> 00:17:02,79
So I'm gonna take u = 1.
291
00:17:02,79 --> 00:17:10,71
So now I'm differentiating 1
/ v, the reciprocal of a
292
00:17:10,71 --> 00:17:14,43
function; 1 over a function.
293
00:17:14,43 --> 00:17:16,88
Here's a copy of my rule.
294
00:17:16,88 --> 00:17:23,72
What's du/ dx in that case?
u is a constant, so that
295
00:17:23,72 --> 00:17:27,05
term is 0 in this rule.
296
00:17:27,05 --> 00:17:28,7
I don't have to
worry about this.
297
00:17:28,7 --> 00:17:31,65
I get a minus.
298
00:17:31,65 --> 00:17:36,8
And then u = 1, and dv/ dx.
299
00:17:36,8 --> 00:17:38,82
Well, v is whatever v is.
300
00:17:38,82 --> 00:17:40,79
I'll write dv/dx as v'.
301
00:17:40,79 --> 00:17:43,91
302
00:17:43,91 --> 00:17:45,52
And then I get a v^2
in the denominator.
303
00:17:45,52 --> 00:17:50,07
So that's the rule.
304
00:17:50,07 --> 00:17:51,38
I could write it as (v^-2)v'.
305
00:17:51,38 --> 00:17:56,84
306
00:17:56,84 --> 00:17:59,3
(-v'/v^2).
307
00:17:59,3 --> 00:18:03,73
That's the derivative of 1 / v.
308
00:18:03,73 --> 00:18:12,11
How about sub-example of that?
309
00:18:12,11 --> 00:18:15,84
I'm going to take the special
case where u = 1 again.
310
00:18:15,84 --> 00:18:16,77
And v = x^n.
311
00:18:16,77 --> 00:18:21
312
00:18:21 --> 00:18:25,63
And I'm gonna use the rule that
we developed earlier about
313
00:18:25,63 --> 00:18:29,08
the derivative of x^n.
314
00:18:29,08 --> 00:18:40,4
So what do I get here? d / dx
(1/x^n) is, I'm plugging into
315
00:18:40,4 --> 00:18:45,26
this formula here with v = x^n.
316
00:18:45,26 --> 00:18:51,58
So I get minus, uh, v^-2.
317
00:18:51,58 --> 00:18:57,25
If v = x^n, v^-2 is, by the
rule of exponents, x^-2n.
318
00:18:57,25 --> 00:19:01,43
319
00:19:01,43 --> 00:19:05,55
And then v' is the derivative
of x^n, which is (nx)^(n-1).
320
00:19:05,55 --> 00:19:10,15
321
00:19:10,15 --> 00:19:12,01
Ok, so let's put
these together.
322
00:19:12,01 --> 00:19:13,55
There's several
powers of x here.
323
00:19:13,55 --> 00:19:14,94
I can put them together.
324
00:19:14,94 --> 00:19:21,045
I get -nx ^ ((- 2n)
325
00:19:21,045 --> 00:19:22,33
(n - 1)).
326
00:19:22,33 --> 00:19:24,05
One of these n's cancels.
327
00:19:24,05 --> 00:19:29,26
And what I'm left
with is ((-n) - 1).
328
00:19:29,26 --> 00:19:32,55
So we've computed the
derivative of 1 / x^n,
329
00:19:32,55 --> 00:19:39,21
which I could also
write as x^-n, right?
330
00:19:39,21 --> 00:19:42,64
So I've computed the derivative
of negative powers of x.
331
00:19:42,64 --> 00:19:46,56
And this is the
formula that I get.
332
00:19:46,56 --> 00:19:52,405
If you think of this - n as a
unit, as a thing to itself, it
333
00:19:52,405 --> 00:19:54,31
occurs here in the exponent.
334
00:19:54,31 --> 00:19:59,89
It occurs here, and
it occurs here.
335
00:19:59,89 --> 00:20:01,82
So how does that compare
with the formula
336
00:20:01,82 --> 00:20:04,12
that we had up here?
337
00:20:04,12 --> 00:20:09,32
The derivative of a power of
x is that power times x to
338
00:20:09,32 --> 00:20:12,3
one less than that power.
339
00:20:12,3 --> 00:20:16,01
That's exactly the same as the
rule that I wrote down here.
340
00:20:16,01 --> 00:20:19,27
But the power here happens to
be a negative number, and the
341
00:20:19,27 --> 00:20:22,5
same negative number shows
up as a coefficient and
342
00:20:22,5 --> 00:20:23,96
there in the exponent.
343
00:20:23,96 --> 00:20:24,28
Yeah?
344
00:20:24,28 --> 00:20:30,44
Student: [INAUDIBLE].
345
00:20:30,44 --> 00:20:34,93
Professor: How did I do this?
346
00:20:34,93 --> 00:20:49,15
Student: [INAUDIBLE].
347
00:20:49,15 --> 00:20:55,99
Professor: Where did
that x^-2n come from?
348
00:20:55,99 --> 00:20:59,9
So I'm applying this rule.
349
00:20:59,9 --> 00:21:04,44
So the denominator in the
quotient rule is v^2.
350
00:21:04,44 --> 00:21:11,19
And v was x^n, so the
denominator is x^2n.
351
00:21:11,19 --> 00:21:12,65
And I decided to
write it as x^-2n.
352
00:21:12,65 --> 00:21:19,01
353
00:21:19,01 --> 00:21:22,08
So the green comments there...
354
00:21:22,08 --> 00:21:26,27
What they say is that I
can enlarge this rule.
355
00:21:26,27 --> 00:21:31,35
This exact same rule is true
for negative values of n, as
356
00:21:31,35 --> 00:21:36,31
well as positive values of n.
357
00:21:36,31 --> 00:21:42,43
So there's something new in
your list of rules that you
358
00:21:42,43 --> 00:21:46,67
can apply, of values
of the derivative.
359
00:21:46,67 --> 00:21:48,69
That standard rule is true
for negative as well
360
00:21:48,69 --> 00:21:51,12
as positive exponents.
361
00:21:51,12 --> 00:21:57,29
And that comes out
of a quotient rule.
362
00:21:57,29 --> 00:21:59,02
Okay, so we've done two rules.
363
00:21:59,02 --> 00:22:04,65
I've talked about the product
rule and the quotient rule.
364
00:22:04,65 --> 00:22:05,67
What's next?
365
00:22:05,67 --> 00:22:07,15
Let's see the chain rule.
366
00:22:07,15 --> 00:22:22,22
So this is a composition rule.
367
00:22:22,22 --> 00:22:25,02
So the kind of thing that I
have in mind, composition of
368
00:22:25,02 --> 00:22:28,21
functions is about
substitution.
369
00:22:28,21 --> 00:22:30,36
So the kind of function that
I have in mind is, for
370
00:22:30,36 --> 00:22:31,73
instance, y = (sin t)^10.
371
00:22:31,73 --> 00:22:39,7
372
00:22:39,7 --> 00:22:43,04
That's a new one.
373
00:22:43,04 --> 00:22:44,46
We haven't seen how to
differentiate that
374
00:22:44,46 --> 00:22:46,59
before, I think.
375
00:22:46,59 --> 00:22:50,6
This kind of power of a trig
function happens very often.
376
00:22:50,6 --> 00:22:53,54
You've seen them happen, as
well, I'm sure, already.
377
00:22:53,54 --> 00:22:58,02
And there's a little notational
switch that people use.
378
00:22:58,02 --> 00:22:59,32
They'll write sin^10(t).
379
00:22:59,32 --> 00:23:02,91
380
00:23:02,91 --> 00:23:05,63
But remember that when you
write sin^10(t), what you mean
381
00:23:05,63 --> 00:23:10,44
is take the sine of t, and then
take the tenth power of that.
382
00:23:10,44 --> 00:23:13,59
It's the meaning of sin^10(t).
383
00:23:13,59 --> 00:23:21,05
So the method of dealing with
this kind of composition of
384
00:23:21,05 --> 00:23:33,19
functions is to use
new variable names.
385
00:23:33,19 --> 00:23:36,83
What I mean is, I can
think of this (sin t)^10.
386
00:23:36,83 --> 00:23:39,71
387
00:23:39,71 --> 00:23:42,07
I can think of it it as
a two step process.
388
00:23:42,07 --> 00:23:44,16
First of all, I compute
the sine of t.
389
00:23:44,16 --> 00:23:47,45
And let's call the result x.
390
00:23:47,45 --> 00:23:50,15
There's the new variable name.
391
00:23:50,15 --> 00:23:53,34
And then, I express
y in terms of x.
392
00:23:53,34 --> 00:23:58,07
So y says take this and raise
it to the tenth power.
393
00:23:58,07 --> 00:23:59,36
In other words, y = x^10.
394
00:23:59,36 --> 00:24:03,4
395
00:24:03,4 --> 00:24:06,94
And then you plug x = sine of
t into that, and you get the
396
00:24:06,94 --> 00:24:10,59
formula for what y
is in terms of t.
397
00:24:10,59 --> 00:24:14,55
So it's good practice to
introduce new letters when
398
00:24:14,55 --> 00:24:17,06
they're convenient, and
this is one place where
399
00:24:17,06 --> 00:24:21,82
it's very convenient.
400
00:24:21,82 --> 00:24:25,07
So let's find a rule for
differentiating a composition,
401
00:24:25,07 --> 00:24:27,77
a function that can be
expressed by doing one function
402
00:24:27,77 --> 00:24:30,27
and then applying
another function.
403
00:24:30,27 --> 00:24:32,88
And here's the rule.
404
00:24:32,88 --> 00:24:35,05
Well, maybe I'll actually
derive this rule first, and
405
00:24:35,05 --> 00:24:37,42
then you'll see what it is.
406
00:24:37,42 --> 00:24:40,6
In fact, the rule is
very simple to derive.
407
00:24:40,6 --> 00:24:43,89
So this is a proof first, and
then we'll write down the rule.
408
00:24:43,89 --> 00:24:51,11
I'm interested in delta y /
delta t. y is a function of
409
00:24:51,11 --> 00:24:53,76
x. x is a function of t.
410
00:24:53,76 --> 00:24:56,95
And I'm interested in how
y changes with respect to
411
00:24:56,95 --> 00:25:00,85
t, with respect to the
original variable t.
412
00:25:00,85 --> 00:25:05,41
Well, because of that
intermediate variable, I can
413
00:25:05,41 --> 00:25:12,67
write this as ((delta y /
delta x) (delta x/ delta t)).
414
00:25:12,67 --> 00:25:15,33
It cancels, right?
415
00:25:15,33 --> 00:25:17,6
The delta x cancels.
416
00:25:17,6 --> 00:25:23,1
The change in that immediate
variable cancels out.
417
00:25:23,1 --> 00:25:26,12
This is just basic algebra.
418
00:25:26,12 --> 00:25:29,93
But what happens when I
let delta t get small?
419
00:25:29,93 --> 00:25:34,37
Well this give me dy /dt.
420
00:25:34,37 --> 00:25:42,22
On the right hand side,
I get (dy/dx) (dx/dt).
421
00:25:42,22 --> 00:25:44,43
So students will often
remember this rule.
422
00:25:44,43 --> 00:25:46,85
This is the rule, by saying
that you can cancel
423
00:25:46,85 --> 00:25:49,08
out for the dx's.
424
00:25:49,08 --> 00:25:51,86
And that's not so
far from the truth.
425
00:25:51,86 --> 00:25:55,16
That's a good way
to think of it.
426
00:25:55,16 --> 00:26:01,41
In other words, this is
the so-called chain rule.
427
00:26:01,41 --> 00:26:14,54
And it says that
differentiation of a
428
00:26:14,54 --> 00:26:26,69
composition is a product.
429
00:26:26,69 --> 00:26:34,91
It's just the product of
the two derivatives.
430
00:26:34,91 --> 00:26:39,57
So that's how you differentiate
a composite of two functions.
431
00:26:39,57 --> 00:26:42,07
And let's just do an example.
432
00:26:42,07 --> 00:26:44,69
Let's do this example.
433
00:26:44,69 --> 00:26:48,82
Let's see how that comes out.
434
00:26:48,82 --> 00:26:55,25
So let's differentiate,
what did I say?
435
00:26:55,25 --> 00:26:56,53
(sin t)^10.
436
00:26:56,53 --> 00:26:59,4
437
00:26:59,4 --> 00:27:01,856
Okay, there's an inside
function and an
438
00:27:01,856 --> 00:27:03,13
outside function.
439
00:27:03,13 --> 00:27:07,91
The inside function is
x as a function of t.
440
00:27:07,91 --> 00:27:12,08
This is the inside
function, and this is
441
00:27:12,08 --> 00:27:19,17
the outside function.
442
00:27:19,17 --> 00:27:22,59
So the rule says, first of
all let's differentiate
443
00:27:22,59 --> 00:27:23,55
the outside function.
444
00:27:23,55 --> 00:27:25,37
Take dy/ dx.
445
00:27:25,37 --> 00:27:29,2
Differentiate it with
respect to that variable x.
446
00:27:29,2 --> 00:27:31,02
The outside function
is the 10th power.
447
00:27:31,02 --> 00:27:34,64
What's it's derivative?
448
00:27:34,64 --> 00:27:42,44
So I get 10x^9.
449
00:27:42,44 --> 00:27:50,31
In this account, I'm using
this newly introduced
450
00:27:50,31 --> 00:27:53,99
variable named x.
451
00:27:53,99 --> 00:27:58,15
So the derivative of the
outside function is 10x^9.
452
00:27:58,15 --> 00:28:01,04
And then here's the inside
function, and the next thing I
453
00:28:01,04 --> 00:28:03,13
want to do is differentiate it.
454
00:28:03,13 --> 00:28:07,73
So what's dx /dt, d/dt (sine
t), the derivative of sine t?
455
00:28:07,73 --> 00:28:11,83
All right, that's cosine t.
456
00:28:11,83 --> 00:28:13,16
That's what the
chain gives you.
457
00:28:13,16 --> 00:28:18,155
This is correct, but since we
were the ones to introduce
458
00:28:18,155 --> 00:28:21,99
this notation x here, that
wasn't given to us in the
459
00:28:21,99 --> 00:28:24,56
original problem here.
460
00:28:24,56 --> 00:28:27,98
The last step in this process
should be to put back, to
461
00:28:27,98 --> 00:28:32,44
substitute back in what
x is in terms of t.
462
00:28:32,44 --> 00:28:35,32
So x = sin t.
463
00:28:35,32 --> 00:28:44,79
So that tells me that I
get 10(sin(t))^9, that's
464
00:28:44,79 --> 00:28:47,86
x^9, times the cos(t).
465
00:28:47,86 --> 00:28:50,86
Or the same thing
is sin^9(t)cos(t).
466
00:28:50,86 --> 00:28:56,04
467
00:28:56,04 --> 00:28:59,54
So there's an application
of the chain rule.
468
00:28:59,54 --> 00:29:02,29
You know, people often
wonder where the name
469
00:29:02,29 --> 00:29:03,17
chain rule comes from.
470
00:29:03,17 --> 00:29:06,34
I was just wondering
about that myself.
471
00:29:06,34 --> 00:29:15,23
So is it because it
chains you down?
472
00:29:15,23 --> 00:29:18,07
Is it like a chain fence?
473
00:29:18,07 --> 00:29:19,59
I decided what it is.
474
00:29:19,59 --> 00:29:24,22
It's because by using it, you
burst the chains of
475
00:29:24,22 --> 00:29:26,41
differentiation, and you can
differentiate many more
476
00:29:26,41 --> 00:29:28,04
functions using it.
477
00:29:28,04 --> 00:29:32,915
So when you want to think of
the chain rule, just think
478
00:29:32,915 --> 00:29:35,64
of that chain there.
479
00:29:35,64 --> 00:29:47,96
It lets you burst free.
480
00:29:47,96 --> 00:30:04,83
Let me give you another
application of the chain rule.
481
00:30:04,83 --> 00:30:16,27
Ready for this one?
482
00:30:16,27 --> 00:30:17,97
So I'd like to differentiate
the sin(10t).
483
00:30:17,97 --> 00:30:25,76
484
00:30:25,76 --> 00:30:27,44
Again, this is the composite
of two functions.
485
00:30:27,44 --> 00:30:30,22
What's the inside function?
486
00:30:30,22 --> 00:30:36,76
Okay, so I think I'll introduce
this new notation. x = 10t, and
487
00:30:36,76 --> 00:30:38,26
the outside function
is the sine.
488
00:30:38,26 --> 00:30:41,32
So y = sin x.
489
00:30:41,32 --> 00:30:46,66
So now the chain rule
says dy/ dt is...
490
00:30:46,66 --> 00:30:47,92
Okay, let's see.
491
00:30:47,92 --> 00:30:50,71
I take the derivative of
the outside function,
492
00:30:50,71 --> 00:30:54,24
and what's that?
493
00:30:54,24 --> 00:30:58,52
Sine' and we can substitute
because we know what sine' is.
494
00:30:58,52 --> 00:31:06,47
So I get cosine of whatever,
x, and then times what?
495
00:31:06,47 --> 00:31:11,4
Now I differentiate the inside
function, which is just 10.
496
00:31:11,4 --> 00:31:16,38
So I could write this
as 10cos of what?
497
00:31:16,38 --> 00:31:17,36
10t, x = 10t.
498
00:31:17,36 --> 00:31:20,26
499
00:31:20,26 --> 00:31:26,55
Now, once you get used to this,
this middle variable, you don't
500
00:31:26,55 --> 00:31:33,19
have to give a name for it.
501
00:31:33,19 --> 00:31:35,81
You can just to think about it
in your mind without actually
502
00:31:35,81 --> 00:31:44,89
writing it down,
d/dt (sin(10t)).
503
00:31:44,89 --> 00:31:47,98
504
00:31:47,98 --> 00:31:50,28
I'll just do it again without
introducing this middle
505
00:31:50,28 --> 00:31:52,24
variable explicitly.
506
00:31:52,24 --> 00:31:54,53
Think about it.
507
00:31:54,53 --> 00:31:58,1
I first of all differentiate
the outside function,
508
00:31:58,1 --> 00:31:59,74
and I get cosine.
509
00:31:59,74 --> 00:32:03,17
But I don't change the thing
that I'm plugging into it.
510
00:32:03,17 --> 00:32:08,56
It's still x that I'm
plugging into it. x is 10t.
511
00:32:08,56 --> 00:32:11,47
So let's just write 10t and
not worry about the name
512
00:32:11,47 --> 00:32:12,72
of that extra variable.
513
00:32:12,72 --> 00:32:15,51
If it confuses you,
introduce the new variable.
514
00:32:15,51 --> 00:32:18,18
And do it carefully
and slowly like this.
515
00:32:18,18 --> 00:32:20,95
But, quite quickly, I think
you'll get to be able to keep
516
00:32:20,95 --> 00:32:23,24
that step in your mind.
517
00:32:23,24 --> 00:32:24,16
I'm not quite done yet.
518
00:32:24,16 --> 00:32:27,08
I haven't differentiated
the inside function, the
519
00:32:27,08 --> 00:32:29,19
derivative of 10t = 10.
520
00:32:29,19 --> 00:32:33,25
So you get, again,
the same result.
521
00:32:33,25 --> 00:32:36,42
A little short cut that
you'll get used to.
522
00:32:36,42 --> 00:32:39,62
Really and truly, once you have
the chain rule, the world
523
00:32:39,62 --> 00:32:41,11
is yours to conquer.
524
00:32:41,11 --> 00:32:46,73
It puts you in a very,
very powerful position.
525
00:32:46,73 --> 00:32:50,21
Okay, well let's see.
526
00:32:50,21 --> 00:32:51,31
What have I covered today?
527
00:32:51,31 --> 00:32:57,37
I've talked about product rule,
quotient rule, composition.
528
00:32:57,37 --> 00:32:58,7
I should tell you
something about higher
529
00:32:58,7 --> 00:33:00,67
derivatives, as well.
530
00:33:00,67 --> 00:33:10,44
So let's do that.
531
00:33:10,44 --> 00:33:12,15
This is a simple story.
532
00:33:12,15 --> 00:33:14,95
Higher is kind of
a strange word.
533
00:33:14,95 --> 00:33:32,95
It just means differentiate
over and over again.
534
00:33:32,95 --> 00:33:34,6
All right, so let's see.
535
00:33:34,6 --> 00:33:39,55
If we have a function u or
u(x), please allow me to just
536
00:33:39,55 --> 00:33:45,01
write it as briefly as u.
537
00:33:45,01 --> 00:33:49,33
Well, this is a sort
of notational thing.
538
00:33:49,33 --> 00:33:51,78
I can differentiate
it and get u'.
539
00:33:51,78 --> 00:33:54,79
540
00:33:54,79 --> 00:33:55,9
That's a new function.
541
00:33:55,9 --> 00:33:57,68
Like if you started
with the sine, that's
542
00:33:57,68 --> 00:34:00,76
gonna be the cosine.
543
00:34:00,76 --> 00:34:03,57
A new function, so I can
differentiate it again.
544
00:34:03,57 --> 00:34:05,01
And the notation for
the differentiating
545
00:34:05,01 --> 00:34:07,47
of it again, is u''.
546
00:34:07,47 --> 00:34:12,93
So u'' is just u'
differentiated again.
547
00:34:12,93 --> 00:34:21,38
For example, if u = sin
x, so u' = cos(x).
548
00:34:21,38 --> 00:34:24,31
Has Professor Gerison
talked about what the
549
00:34:24,31 --> 00:34:26,58
derivative of cosine is?
550
00:34:26,58 --> 00:34:28,22
What is it?
551
00:34:28,22 --> 00:34:33,02
Ha, ok so u'' = - sin x.
552
00:34:33,02 --> 00:34:36,81
553
00:34:36,81 --> 00:34:38,93
Let me go on.
554
00:34:38,93 --> 00:34:42,97
What do you suppose u''' means?
555
00:34:42,97 --> 00:34:46,42
I guess it's the
derivative of u''.
556
00:34:46,42 --> 00:34:53,05
It's called the
third derivative.
557
00:34:53,05 --> 00:34:56,21
And u'' is called the
second derivative.
558
00:34:56,21 --> 00:34:59
And it's (u')'
differentiated again.
559
00:34:59 --> 00:35:03,68
So to compute u''' in this
example, what do I do?
560
00:35:03,68 --> 00:35:05,34
I differentiate that again.
561
00:35:05,34 --> 00:35:08,46
There's a constant term,
- 1, constant factor.
562
00:35:08,46 --> 00:35:09,95
That comes out.
563
00:35:09,95 --> 00:35:13,5
The derivative of sine is what?
564
00:35:13,5 --> 00:35:17,93
Okay, so u''' = - cos x.
565
00:35:17,93 --> 00:35:18,69
Let's do it again.
566
00:35:18,69 --> 00:35:21,89
Now after a while, you get
tired of writing these things.
567
00:35:21,89 --> 00:35:24,65
And so maybe I'll use
the notation u^(4).
568
00:35:24,65 --> 00:35:27,29
That's the fourth derivative.
569
00:35:27,29 --> 00:35:29,49
That's u''''.
570
00:35:29,49 --> 00:35:33,44
Or it's (u''')' the
fourth derivative.
571
00:35:33,44 --> 00:35:37,97
And what is that
in this example?
572
00:35:37,97 --> 00:35:42,01
Okay, the cosine has derivative
-sin , like you told me.
573
00:35:42,01 --> 00:35:45,883
And that -sin cancels with that
sine, and all together, I get
574
00:35:45,883 --> 00:35:47,64
sin x.
575
00:35:47,64 --> 00:35:48,94
That's pretty bizarre.
576
00:35:48,94 --> 00:35:51,32
When I differentiate the
function sine of x four
577
00:35:51,32 --> 00:35:56,92
times, I get back to
the sine of x again.
578
00:35:56,92 --> 00:36:00,29
That's the way it is.
579
00:36:00,29 --> 00:36:03,15
Now this notation, prime
prime prime prime,
580
00:36:03,15 --> 00:36:03,99
and things like that.
581
00:36:03,99 --> 00:36:13,65
There are different
variants of that notation.
582
00:36:13,65 --> 00:36:24,07
For example, that's
another notation.
583
00:36:24,07 --> 00:36:29,97
Well, you've used the notation
du/ dx before. u' could
584
00:36:29,97 --> 00:36:35,73
also be denoted du/ dx.
585
00:36:35,73 --> 00:36:40,485
I think we've already here,
today, used this way
586
00:36:40,485 --> 00:36:43,23
of rewriting du/ dx.
587
00:36:43,23 --> 00:36:47,66
I think when I was talking
about d/dt(uv) and so on, I
588
00:36:47,66 --> 00:36:52,36
pulled that d / dt outside
and put whatever function
589
00:36:52,36 --> 00:36:55,01
you're differentiating
over to the right.
590
00:36:55,01 --> 00:36:57,43
So that's just a
notational switch.
591
00:36:57,43 --> 00:36:58,11
It looks good.
592
00:36:58,11 --> 00:37:06,26
It looks like good
algebra doesn't it?
593
00:37:06,26 --> 00:37:09,86
But what it's doing is
regarding this notation
594
00:37:09,86 --> 00:37:12,41
as an operator.
595
00:37:12,41 --> 00:37:16,92
It's something you apply to a
function to get a new function.
596
00:37:16,92 --> 00:37:18,81
I apply it to the sine
function, and I get
597
00:37:18,81 --> 00:37:20,68
the cosine function.
598
00:37:20,68 --> 00:37:24,22
I apply it to x^2,
and I get 2x.
599
00:37:24,22 --> 00:37:31,32
This thing here, that symbol,
represents an operator, which
600
00:37:31,32 --> 00:37:40,34
you apply to a function.
601
00:37:40,34 --> 00:37:44,86
And the operator says, take the
function and differentiate it.
602
00:37:44,86 --> 00:37:47,91
So further notation that people
often use, is they give a
603
00:37:47,91 --> 00:37:49,46
different name to
that operator.
604
00:37:49,46 --> 00:37:52,27
And they'll write
capital D for it.
605
00:37:52,27 --> 00:38:02,98
So this is just using capital
D for the symbol d/dx.
606
00:38:02,98 --> 00:38:05,05
So in terms of that
notation, let's see.
607
00:38:05,05 --> 00:38:20,44
Let's write down what higher
derivatives look like.
608
00:38:20,44 --> 00:38:21,87
So let's see.
609
00:38:21,87 --> 00:38:23,09
That's what u' is.
610
00:38:23,09 --> 00:38:24,36
How about u''?
611
00:38:24,36 --> 00:38:28,89
Let's write that in terms
of the d/dx notation.
612
00:38:28,89 --> 00:38:31,71
Well I'm supposed to
differentiate u' right?
613
00:38:31,71 --> 00:38:40,92
So that's d/dx applied
to the function du/ dx.
614
00:38:40,92 --> 00:38:43,03
Differentiate the derivative.
615
00:38:43,03 --> 00:38:47,24
That's what I've done.
616
00:38:47,24 --> 00:38:54,35
Or I could write that as d/dx
applied to d/dx applied to u.
617
00:38:54,35 --> 00:38:57,85
Just pulling that u outside.
618
00:38:57,85 --> 00:38:59,57
So I'm doing d/dx twice.
619
00:38:59,57 --> 00:39:01,59
I'm doing that operator twice.
620
00:39:01,59 --> 00:39:08,03
I could write that as
(d/dx)^2 applied to u.
621
00:39:08,03 --> 00:39:15,17
Differentiate twice, and
do it to the function u.
622
00:39:15,17 --> 00:39:23,13
Or, I can write it as, now
this is a strange one.
623
00:39:23,13 --> 00:39:33,33
I could also write
as like that.
624
00:39:33,33 --> 00:39:36,63
It's getting stranger
and stranger, isn't it?
625
00:39:36,63 --> 00:39:40,77
This is definitely just a
kind of abuse of notation.
626
00:39:40,77 --> 00:39:46,03
But people will go even further
and write (d^2)u/dx^2.
627
00:39:46,03 --> 00:39:50,5
628
00:39:50,5 --> 00:39:52,19
So this is the strangest one.
629
00:39:52,19 --> 00:39:57,34
This identity quality is the
strangest one, because you
630
00:39:57,34 --> 00:40:01,33
may think that you're taking
d of the quantity x^2.
631
00:40:01,33 --> 00:40:03,93
But that's not what's intended.
632
00:40:03,93 --> 00:40:08,24
This is not d(x^2).
633
00:40:08,24 --> 00:40:12,75
What's intended is
the quantity dx^2.
634
00:40:12,75 --> 00:40:15,79
In this notation, which is very
common, what's intended by the
635
00:40:15,79 --> 00:40:18,25
denominator is the
quantity dx^2.
636
00:40:18,25 --> 00:40:23,63
It's part of this second
differentiation operator.
637
00:40:23,63 --> 00:40:26,9
So I've written a bunch of
equalities down here, and the
638
00:40:26,9 --> 00:40:29,28
only content to them is that
these are all different
639
00:40:29,28 --> 00:40:32,32
notations for the same thing.
640
00:40:32,32 --> 00:40:34,94
You'll see this notation
very commonly.
641
00:40:34,94 --> 00:40:37,95
So for instance the
third derivative is
642
00:40:37,95 --> 00:40:47,54
(d^3)u/dx^3, and so on.
643
00:40:47,54 --> 00:40:47,83
Sorry?
644
00:40:47,83 --> 00:40:59,49
Student: [INAUDIBLE].
645
00:40:59,49 --> 00:40:59,88
Professor: Yes, absolutely.
646
00:40:59,88 --> 00:41:05,5
Or an equally good notation is
to write the operator (D^3)u.
647
00:41:05,5 --> 00:41:09,4
648
00:41:09,4 --> 00:41:11,65
Absolutely.
649
00:41:11,65 --> 00:41:13,96
So I guess I should also write
over here when I was talking
650
00:41:13,96 --> 00:41:17,56
about d^2, the second
derivative, another notation
651
00:41:17,56 --> 00:41:20,82
is do the operator
capital D twice.
652
00:41:20,82 --> 00:41:22,82
Let's see an example of
how this can be applied.
653
00:41:22,82 --> 00:41:23,69
I'll answer this question.
654
00:41:23,69 --> 00:41:32,8
Student: [INAUDIBLE].
655
00:41:32,8 --> 00:41:34,86
Professor: Yeah, so the
question is whether the fourth
656
00:41:34,86 --> 00:41:37,54
derivative always gives you the
original function back,
657
00:41:37,54 --> 00:41:38,88
like what happened here.
658
00:41:38,88 --> 00:41:39,58
No.
659
00:41:39,58 --> 00:41:43,47
That's very, very special
to sines and cosines.
660
00:41:43,47 --> 00:41:45,2
All right?
661
00:41:45,2 --> 00:41:47,85
And, in fact, let's see
an example of that.
662
00:41:47,85 --> 00:41:50,92
I'll do a calculation.
663
00:41:50,92 --> 00:42:06,13
Let's calculate the nth
derivative of x^n.
664
00:42:06,13 --> 00:42:13,19
Okay, n is a number,
like 1, 2, 3, 4.
665
00:42:13,19 --> 00:42:13,72
Here we go.
666
00:42:13,72 --> 00:42:15,36
Let's do this.
667
00:42:15,36 --> 00:42:17,65
So, let's do this bit by bit.
668
00:42:17,65 --> 00:42:22,5
What's the first
derivative of x^n?
669
00:42:22,5 --> 00:42:24,09
So everybody knows this.
670
00:42:24,09 --> 00:42:27,83
I'm just using a new notation,
this capital D notation.
671
00:42:27,83 --> 00:42:30,52
So it's n x ^ (n -1).
672
00:42:30,52 --> 00:42:33,64
Now you know know, by the way,
n could be a negative number
673
00:42:33,64 --> 00:42:38,372
for that, but for now, for this
application, I wanna take n to
674
00:42:38,372 --> 00:42:43,07
be 1, 2, 3, and so on;
one of those numbers.
675
00:42:43,07 --> 00:42:44,55
Ok, we did one derivative.
676
00:42:44,55 --> 00:42:49,53
Let's compute the second
derivative of x ^ n.
677
00:42:49,53 --> 00:42:53,805
Well there's this n constant
that comes out, and then the
678
00:42:53,805 --> 00:42:59,98
exponent comes down, and
it gets reduced by 1.
679
00:42:59,98 --> 00:43:01,19
All right?
680
00:43:01,19 --> 00:43:03,78
Should I do one more?
681
00:43:03,78 --> 00:43:07,6
D^3 (x^n) = n(n-1).
682
00:43:07,6 --> 00:43:09,41
That's the constant from here.
683
00:43:09,41 --> 00:43:13,42
Times that exponent, (n -
2), times 1 less, (n -
684
00:43:13,42 --> 00:43:15,74
3) is the new exponent.
685
00:43:15,74 --> 00:43:26,43
Well, I keep on going until
I come to a new blackboard.
686
00:43:26,43 --> 00:43:29,49
Now, I think I'm going to stop
when I get to the n minus first
687
00:43:29,49 --> 00:43:35,37
derivative, so we can see
what's likely to happen.
688
00:43:35,37 --> 00:43:40,68
So when I took the third
derivative, I had the n
689
00:43:40,68 --> 00:43:43,31
minus third power of x.
690
00:43:43,31 --> 00:43:44,94
And when I took the second
derivative, I had the
691
00:43:44,94 --> 00:43:45,76
second power of x.
692
00:43:45,76 --> 00:43:49,6
So, I think what'll happen when
I have the n minus first
693
00:43:49,6 --> 00:43:53,51
derivative is I'll have the
first power of x left over.
694
00:43:53,51 --> 00:43:55,39
The powers of x
keep coming down.
695
00:43:55,39 --> 00:43:59,35
And what I've done it n - 1
times, I get the first power.
696
00:43:59,35 --> 00:44:04,23
And then I get a big constant
out in front here times more
697
00:44:04,23 --> 00:44:06,84
and more and more of these
smaller and smaller
698
00:44:06,84 --> 00:44:08,5
integers that come down.
699
00:44:08,5 --> 00:44:10,63
What's the last integer
that came down before
700
00:44:10,63 --> 00:44:17,46
I got x^1 here?
701
00:44:17,46 --> 00:44:19,39
Well, let's see.
702
00:44:19,39 --> 00:44:23,09
It's just 2, because this
x^1 occurred as the
703
00:44:23,09 --> 00:44:24,34
derivative of x^2.
704
00:44:24,34 --> 00:44:27,8
And the coefficient in
front of that is 2.
705
00:44:27,8 --> 00:44:29,73
So that's what you get.
706
00:44:29,73 --> 00:44:35,14
The numbers n( n - 1)...2)x^1.
707
00:44:35,14 --> 00:44:40,76
And now we can differentiate
one more time and calculate
708
00:44:40,76 --> 00:44:42,77
what (D^n)(x^n) is.
709
00:44:42,77 --> 00:44:46,49
So I get the same
number, n(n-1)...
710
00:44:46,49 --> 00:44:49,68
and so on and so on, times 2.
711
00:44:49,68 --> 00:44:52,5
And then I guess
I'll say times 1.
712
00:44:52,5 --> 00:44:55,79
Times, what's the
derivative of x ^ 1?
713
00:44:55,79 --> 00:44:58,64
1, so times 1.
714
00:44:58,64 --> 00:45:01,26
Time 1, times 1.
715
00:45:01,26 --> 00:45:10,49
Where this one means the
constant function 1.
716
00:45:10,49 --> 00:45:14,07
Does anyone know what
this number is called?
717
00:45:14,07 --> 00:45:15,11
That has a name.
718
00:45:15,11 --> 00:45:19,72
It's called n factorial.
719
00:45:19,72 --> 00:45:21,4
And it's written n!
720
00:45:21,4 --> 00:45:24,24
721
00:45:24,24 --> 00:45:28,83
And we just used an example
of mathematical induction.
722
00:45:28,83 --> 00:45:34,26
So the end result is
(D^n) (x^n) = n!
723
00:45:34,26 --> 00:45:37,75
constant.
724
00:45:37,75 --> 00:45:42,46
Okay that's a neat fact.
725
00:45:42,46 --> 00:45:45,55
Final question for the
lecture is what's D^n
726
00:45:45,55 --> 00:45:49,73
1 applied to x ^ n?
727
00:45:49,73 --> 00:45:50,85
Ha.
728
00:45:50,85 --> 00:45:54,34
Excellent.
729
00:45:54,34 --> 00:45:56,62
It's the derivative
of a constant.
730
00:45:56,62 --> 00:45:58,18
So it's 0.
731
00:45:58,18 --> 00:45:58,44
Okay.
732
00:45:58,44 --> 00:46:00,03
Thank you.
733
00:46:00,03 --> 00:46:02,245