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Professor: I am Haynes
Miller, I am substituting
00:00:25.050 --> 00:00:26.510
for David Jerison today.
00:00:26.510 --> 00:00:41.880
So you have a substitute
teacher today.
00:00:41.880 --> 00:00:44.400
So I haven't been here
in this class with you
00:00:44.400 --> 00:00:47.580
so I'm not completely
sure where you are.
00:00:47.580 --> 00:00:52.170
I think you've just been
talking about differentiation
00:00:52.170 --> 00:00:56.840
and you've got some examples
of differentiation like these
00:00:56.840 --> 00:00:59.860
basic examples: the
derivative of x^n is nx^(n-1).
00:01:03.197 --> 00:01:05.280
But I think maybe you've
spent some time computing
00:01:05.280 --> 00:01:11.230
the derivative of the sine
function as well, recently.
00:01:11.230 --> 00:01:16.290
And I think you have
some rules for extending
00:01:16.290 --> 00:01:18.950
these calculations as well.
00:01:18.950 --> 00:01:24.080
For instance, I think you
know that if you differentiate
00:01:24.080 --> 00:01:27.770
a constant times a
function, what do you get?
00:01:27.770 --> 00:01:32.590
Student: [INAUDIBLE].
00:01:32.590 --> 00:01:36.670
Professor: The constant
comes outside like this.
00:01:36.670 --> 00:01:40.030
Or I could write (cu)' = cu'.
00:01:42.550 --> 00:01:45.350
That's this rule,
multiplying by a constant,
00:01:45.350 --> 00:01:58.870
and I think you also know
about differentiating a sum.
00:01:58.870 --> 00:02:03.650
Or I could write this
as (u + v)' = u' + v'.
00:02:06.870 --> 00:02:09.850
So I'm going to be using
those but today I'll
00:02:09.850 --> 00:02:12.290
talk about a collection
of other rules
00:02:12.290 --> 00:02:15.080
about how to deal with
a product of functions,
00:02:15.080 --> 00:02:18.180
a quotient of functions,
and, best of all,
00:02:18.180 --> 00:02:20.024
composition of functions.
00:02:20.024 --> 00:02:21.690
And then at the end,
I'll have something
00:02:21.690 --> 00:02:23.480
to say about higher derivatives.
00:02:23.480 --> 00:02:26.670
So that's the story for today.
00:02:26.670 --> 00:02:29.120
That's the program.
00:02:29.120 --> 00:02:43.360
So let's begin by talking
about the product rule.
00:02:43.360 --> 00:02:44.750
So the product
rule tells you how
00:02:44.750 --> 00:02:46.760
to differentiate a
product of functions,
00:02:46.760 --> 00:02:49.270
and I'll just give you
the rule, first of all.
00:02:49.270 --> 00:02:51.600
The rule is it's u'v + uv'.
00:02:57.280 --> 00:02:58.500
It's a little bit funny.
00:02:58.500 --> 00:03:02.320
Differentiating a
product gives you a sum.
00:03:02.320 --> 00:03:06.730
But let's see how that works
out in a particular example.
00:03:06.730 --> 00:03:08.270
For example, suppose
that I wanted
00:03:08.270 --> 00:03:11.280
to differentiate the product.
00:03:11.280 --> 00:03:13.890
Well, the product
of these two basic
00:03:13.890 --> 00:03:15.584
examples that we
just talked about.
00:03:15.584 --> 00:03:17.000
I'm going to use
the same variable
00:03:17.000 --> 00:03:20.790
in both cases instead of
different ones like I did here.
00:03:20.790 --> 00:03:23.230
So the derivative
of x^n times sin x.
00:03:28.430 --> 00:03:30.300
So this is a new thing.
00:03:30.300 --> 00:03:36.120
We couldn't do this without
using the product rule.
00:03:36.120 --> 00:03:39.670
So the first function is x^n
and the second one is sin x.
00:03:39.670 --> 00:03:41.740
And we're going to
apply this rule.
00:03:41.740 --> 00:03:49.450
So u is x^n. u' is, according
to the rule, nx^(n-1).
00:03:49.450 --> 00:03:56.029
And then I take v and write it
down the way it is, sine of x.
00:03:56.029 --> 00:03:57.320
And then I do it the other way.
00:03:57.320 --> 00:04:00.690
I take u the way
it is, that's x^n,
00:04:00.690 --> 00:04:05.260
and multiply it by the
derivative of v, v'.
00:04:05.260 --> 00:04:09.234
We just saw v' is cosine of x.
00:04:09.234 --> 00:04:11.520
So that's it.
00:04:11.520 --> 00:04:14.980
Obviously, you can
differentiate longer products,
00:04:14.980 --> 00:04:20.560
products of more things
by doing it one at a time.
00:04:20.560 --> 00:04:22.700
Let's see why this is true.
00:04:22.700 --> 00:04:25.730
I want to try to show you
why the product rule holds.
00:04:25.730 --> 00:04:31.490
So you have a standard way
of trying to understand this,
00:04:31.490 --> 00:04:34.590
and it involves looking at
the change in the function
00:04:34.590 --> 00:04:37.350
that you're interested
in differentiating.
00:04:37.350 --> 00:04:41.390
So I should look at how
much the product uv changes
00:04:41.390 --> 00:04:44.630
when x changes a little bit.
00:04:44.630 --> 00:04:47.120
Well, so how do
compute the change?
00:04:47.120 --> 00:04:49.330
Well, I write down the
value of the function
00:04:49.330 --> 00:04:56.160
at some new value
of x, x + delta x.
00:04:56.160 --> 00:04:58.410
Well, I better write
down the whole new value
00:04:58.410 --> 00:05:01.480
of the function, and
the function is uv.
00:05:01.480 --> 00:05:05.230
So the whole new
value looks like this.
00:05:05.230 --> 00:05:09.450
It's u(x + delta x)
times v(x + delta x).
00:05:09.450 --> 00:05:10.960
That's the new value.
00:05:10.960 --> 00:05:13.200
But what's the change
in the product?
00:05:13.200 --> 00:05:15.410
Well, I better subtract
off what the old value
00:05:15.410 --> 00:05:20.920
was, which is u(x) v(x).
00:05:20.920 --> 00:05:24.950
Okay, according to the
rule we're trying to prove,
00:05:24.950 --> 00:05:27.750
I have to get u' involved.
00:05:27.750 --> 00:05:31.420
So I want to involve the
change in u alone, by itself.
00:05:31.420 --> 00:05:32.990
Let's just try that.
00:05:32.990 --> 00:05:36.620
I see part of the formula for
the change in u right there.
00:05:36.620 --> 00:05:40.260
Let's see if we can get
the rest of it in place.
00:05:40.260 --> 00:05:46.330
So the change in x is
(u(x + delta x) - u(x).
00:05:46.330 --> 00:05:50.080
That's the change in x
[Correction: change in u].
00:05:50.080 --> 00:05:54.760
This part of it occurs up here,
multiplied by v(x + delta x),
00:05:54.760 --> 00:05:57.890
so let's put that in too.
00:05:57.890 --> 00:06:00.440
Now this equality sign
isn't very good right now.
00:06:00.440 --> 00:06:04.770
I've got this
product here so far,
00:06:04.770 --> 00:06:06.690
but I've introduced
something I don't like.
00:06:06.690 --> 00:06:09.800
I've introduced u times
v(x delta x), right?
00:06:09.800 --> 00:06:12.010
Minus that.
00:06:12.010 --> 00:06:17.120
So the next thing I'm gonna do
is correct that little defect
00:06:17.120 --> 00:06:24.620
by adding in u(x)
v(x + delta x).
00:06:24.620 --> 00:06:28.834
Okay, now I cancelled off
what was wrong with this line.
00:06:28.834 --> 00:06:30.500
But I'm still not
quite there, because I
00:06:30.500 --> 00:06:32.670
haven't put this in yet.
00:06:32.670 --> 00:06:38.420
So I better subtract off
uv, and then I'll be home.
00:06:38.420 --> 00:06:40.500
But I'm going to do
that in a clever way,
00:06:40.500 --> 00:06:43.900
because I noticed that
I already have a u here.
00:06:43.900 --> 00:06:46.620
So I'm gonna take
this factor of u
00:06:46.620 --> 00:06:48.560
and make it the
same as this factor.
00:06:48.560 --> 00:06:52.510
So I get u(x) times this,
minus u(x) times that.
00:06:52.510 --> 00:06:57.180
That's the same thing as
u times the difference.
00:06:57.180 --> 00:06:59.406
So that was a
little bit strange,
00:06:59.406 --> 00:07:01.030
but when you stand
back and look at it,
00:07:01.030 --> 00:07:04.280
you can see multiplied out,
the middle terms cancel.
00:07:04.280 --> 00:07:07.340
And you get the right answer.
00:07:07.340 --> 00:07:10.420
Well I like that because
it's involved the change in u
00:07:10.420 --> 00:07:16.810
and the change in v. So this
is equal to delta u times v(x +
00:07:16.810 --> 00:07:26.320
delta x) minus u(x) times
the change in v. Well,
00:07:26.320 --> 00:07:27.510
I'm almost there.
00:07:27.510 --> 00:07:30.050
The next step in computing
the derivative is
00:07:30.050 --> 00:07:42.910
take difference quotient,
divide this by delta x.
00:07:42.910 --> 00:07:50.690
So, (delta (uv)) /
(delta x) is well,
00:07:50.690 --> 00:08:03.150
I'll say (delta u / delta
x) times v(x + delta x).
00:08:03.150 --> 00:08:10.000
Have I made a mistake here?
00:08:10.000 --> 00:08:13.340
This plus magically became a
minus on the way down here,
00:08:13.340 --> 00:08:18.810
so I better fix that.
00:08:18.810 --> 00:08:23.260
Plus u times (delta
v) / (delta x).
00:08:23.260 --> 00:08:27.950
This u is this u over here.
00:08:27.950 --> 00:08:30.310
So I've just divided
this formula by delta x,
00:08:30.310 --> 00:08:34.520
and now I can take the limit
as x goes to 0, so this
00:08:34.520 --> 00:08:42.070
is as delta x goes to 0.
00:08:42.070 --> 00:08:46.480
This becomes the definition
of the derivative,
00:08:46.480 --> 00:08:51.790
and on this side, I
get du/dx times...
00:08:51.790 --> 00:08:57.400
now what happens to this
quantity when delta x goes
00:08:57.400 --> 00:09:02.750
to 0?
00:09:02.750 --> 00:09:05.710
So this quantity is getting
closer and closer to x.
00:09:05.710 --> 00:09:09.000
So what happens
to the value of v?
00:09:09.000 --> 00:09:14.030
It becomes equal to x of v.
That uses continuity of v. So,
00:09:14.030 --> 00:09:22.590
v(x + delta x) goes
to v(x) by continuity.
00:09:22.590 --> 00:09:26.586
So this gives me times v,
and then I have u times,
00:09:26.586 --> 00:09:30.680
and delta v / delta
x gives me dv/dx.
00:09:30.680 --> 00:09:31.819
And that's the formula.
00:09:31.819 --> 00:09:33.360
That's the formula
as I wrote it down
00:09:33.360 --> 00:09:35.660
at the beginning over here.
00:09:35.660 --> 00:09:39.170
The derivative of a product
is given by this sum.
00:09:39.170 --> 00:09:46.223
Yeah?
00:09:46.223 --> 00:09:48.056
Student: How did you
get from the first line
00:09:48.056 --> 00:09:49.514
to the second of
the long equation?
00:09:49.514 --> 00:09:51.390
Professor: From here to here?
00:09:51.390 --> 00:09:53.460
Student: Yes.
00:09:53.460 --> 00:09:56.380
Professor: So maybe it's easiest
to work backwards and verify
00:09:56.380 --> 00:09:59.800
that what I wrote
down is correct here.
00:09:59.800 --> 00:10:05.130
So, if you look there's a u
times v(x + delta x) there.
00:10:05.130 --> 00:10:07.840
And there's also one here.
00:10:07.840 --> 00:10:09.940
And they occur with
opposite signs.
00:10:09.940 --> 00:10:11.490
So they cancel.
00:10:11.490 --> 00:10:20.530
What's left is u(x + delta
x) v(x + delta x) - uv.
00:10:20.530 --> 00:10:29.120
And that's just
what I started with.
00:10:29.120 --> 00:10:33.920
Student: [INAUDIBLE]
They cancel right?
00:10:33.920 --> 00:10:37.430
Professor: I cancelled out
this term and this term,
00:10:37.430 --> 00:10:39.700
and what's left is the ends.
00:10:39.700 --> 00:10:41.490
Any other questions?
00:10:41.490 --> 00:10:49.660
Student: [INAUDIBLE].
00:10:49.660 --> 00:10:55.640
Professor: Well, I just
calculated what delta uv is,
00:10:55.640 --> 00:10:57.980
and now I'm gonna divide
that by delta x on my way
00:10:57.980 --> 00:11:00.250
to computing the derivative.
00:11:00.250 --> 00:11:07.760
And so I copied down the right
hand side and divided delta x.
00:11:07.760 --> 00:11:11.550
I just decided to divide the
delta u by delta x and delta v
00:11:11.550 --> 00:11:16.230
by delta x.
00:11:16.230 --> 00:11:16.990
Good.
00:11:16.990 --> 00:11:22.490
Anything else?
00:11:22.490 --> 00:11:24.260
So we have the
product rule here.
00:11:24.260 --> 00:11:26.980
The rule for differentiating
a product of two functions.
00:11:26.980 --> 00:11:28.325
This is making us stronger.
00:11:28.325 --> 00:11:29.700
There are many
more functions you
00:11:29.700 --> 00:11:31.420
can find derivatives of now.
00:11:31.420 --> 00:11:33.580
How about quotients?
00:11:33.580 --> 00:11:35.370
Let's find out how
to differentiate
00:11:35.370 --> 00:11:47.669
a quotient of two functions.
00:11:47.669 --> 00:11:50.210
Well again, I'll write down what
the answer is and then we'll
00:11:50.210 --> 00:11:52.370
try to verify it.
00:11:52.370 --> 00:11:55.192
So there's a quotient.
00:11:55.192 --> 00:11:56.150
Let me write this down.
00:11:56.150 --> 00:11:58.970
There's a quotient
of two functions.
00:11:58.970 --> 00:12:00.340
And here's the rule for it.
00:12:00.340 --> 00:12:02.910
I always have to think about
this and hope that I get it
00:12:02.910 --> 00:12:09.140
right. (u'v - uv') / v^2.
00:12:09.140 --> 00:12:11.900
This may be the craziest rule
you'll see in this course,
00:12:11.900 --> 00:12:14.330
but there it is.
00:12:14.330 --> 00:12:18.014
And I'll try to show you why
that's true and see an example.
00:12:18.014 --> 00:12:18.930
Yeah there was a hand?
00:12:18.930 --> 00:12:27.300
Student: [INAUDIBLE]
00:12:27.300 --> 00:12:33.040
Professor: What letters look
the same? u and v look the same?
00:12:33.040 --> 00:12:37.040
I'll try to make them
look more different.
00:12:37.040 --> 00:12:38.520
The v's have points
on the bottom.
00:12:38.520 --> 00:12:41.160
u's have little round
things on the bottom.
00:12:41.160 --> 00:12:44.980
What's the new value of u?
00:12:44.980 --> 00:12:55.442
The value of u at x + delta
x is u + delta u, right?
00:12:55.442 --> 00:12:56.400
That's what delta u is.
00:12:56.400 --> 00:13:01.327
It's the change in u when
x gets replaced by delta x
00:13:01.327 --> 00:13:02.410
[Correction: x + delta x].
00:13:02.410 --> 00:13:09.700
And the change in v, the
new value v, is v + delta v.
00:13:09.700 --> 00:13:13.130
So this is the new value of u
divided by the new value of v.
00:13:13.130 --> 00:13:16.130
That's the beginning.
00:13:16.130 --> 00:13:18.705
And then I subtract off
the old values, which
00:13:18.705 --> 00:13:22.910
are u minus v. This'll
be easier to work out
00:13:22.910 --> 00:13:26.084
when I write it out this way.
00:13:26.084 --> 00:13:27.750
So now, we'll cross
multiply, as I said.
00:13:27.750 --> 00:13:38.890
So I get uv + (delta u)v minus,
now I cross multiply this way,
00:13:38.890 --> 00:13:46.330
you get uv - u(delta v).
00:13:46.330 --> 00:13:49.980
And I divide all this
by (v + delta v)u.
00:13:52.840 --> 00:13:55.970
Okay, now the reason
I like to do it
00:13:55.970 --> 00:13:59.490
this way is that you see the
cancellation happening here. uv
00:13:59.490 --> 00:14:02.187
and uv occur twice and
so I can cancel them.
00:14:02.187 --> 00:14:04.520
And I will, and I'll answer
these questions in a minute.
00:14:04.520 --> 00:14:06.260
Audience: [INAUDIBLE].
00:14:06.260 --> 00:14:14.030
Professor: Ooh,
that's a v. All right.
00:14:14.030 --> 00:14:15.570
Good, anything else?
00:14:15.570 --> 00:14:16.750
That's what all hands were.
00:14:16.750 --> 00:14:17.880
Good.
00:14:17.880 --> 00:14:20.870
All right, so I cancel these
and what I'm left with then
00:14:20.870 --> 00:14:24.900
is delta u times v
minus u times delta v
00:14:24.900 --> 00:14:32.250
and all this is over v
+ delta v times v. Okay,
00:14:32.250 --> 00:14:33.360
there's the difference.
00:14:33.360 --> 00:14:36.600
There's the change
in the quotient.
00:14:36.600 --> 00:14:39.580
The change in this function
is given by this formula.
00:14:39.580 --> 00:14:41.570
And now to compute
the derivative,
00:14:41.570 --> 00:14:45.000
I want to divide by delta
x, and take the limit.
00:14:45.000 --> 00:14:53.560
So let's write that down,
delta(u/v)/delta x is this
00:14:53.560 --> 00:14:56.820
formula here divided by delta x.
00:14:56.820 --> 00:15:00.986
And again, I'm going to put
the delta x under these delta u
00:15:00.986 --> 00:15:02.780
and delta v. Okay?
00:15:02.780 --> 00:15:05.080
I'm gonna put delta
x in the denominator,
00:15:05.080 --> 00:15:07.150
but I can think of
that as dividing
00:15:07.150 --> 00:15:09.920
into this factor
and this factor.
00:15:09.920 --> 00:15:16.980
So this is (delta u/ delta
x)v - u(delta v/delta x).
00:15:21.130 --> 00:15:23.130
And all that is divided
by the same denominator,
00:15:23.130 --> 00:15:28.970
(v + delta v)v. Right?
00:15:28.970 --> 00:15:33.010
Put the delta x up in
the numerator there.
00:15:33.010 --> 00:15:37.830
Next up, take the limit
as delta x goes to 0.
00:15:37.830 --> 00:15:43.470
I get, by definition,
the derivative of (u/v).
00:15:43.470 --> 00:15:46.210
And on the right
hand side, well, this
00:15:46.210 --> 00:15:51.300
is the derivative du/dx right?
00:15:51.300 --> 00:15:55.570
Times v. See and then
u times, and here it's
00:15:55.570 --> 00:15:56.480
the derivative dv/dx.
00:16:00.420 --> 00:16:04.250
Now what about the denominator?
00:16:04.250 --> 00:16:10.220
So when delta x goes to 0,
v stays the same, v stays
00:16:10.220 --> 00:16:10.720
the same.
00:16:10.720 --> 00:16:13.480
What happens to this delta v?
00:16:13.480 --> 00:16:17.970
It goes to 0, again,
because v is continuous.
00:16:17.970 --> 00:16:23.330
So again, delta v
goes to 0 with delta x
00:16:23.330 --> 00:16:28.180
because they're continuous
and you just get v times v.
00:16:28.180 --> 00:16:30.867
I think that's the formula
I wrote down over there.
00:16:30.867 --> 00:16:31.700
(du/dx)v - u(dv/dx).
00:16:35.510 --> 00:16:40.770
And all divided by the square
of the old denominator.
00:16:40.770 --> 00:16:42.160
Well, that's it.
00:16:42.160 --> 00:16:43.540
That's the quotient rule.
00:16:43.540 --> 00:16:44.520
Weird formula.
00:16:44.520 --> 00:16:46.160
Let's see an application.
00:16:46.160 --> 00:16:51.070
Let's see an example.
00:16:51.070 --> 00:16:54.680
So the example I'm going
to give is pretty simple.
00:16:54.680 --> 00:16:58.100
I'm going to take the
numerator to be just 1.
00:16:58.100 --> 00:17:02.790
So I'm gonna take u = 1.
00:17:02.790 --> 00:17:07.580
So now I'm
differentiating 1 / v,
00:17:07.580 --> 00:17:14.430
the reciprocal of a
function; 1 over a function.
00:17:14.430 --> 00:17:16.880
Here's a copy of my rule.
00:17:16.880 --> 00:17:22.790
What's du/ dx in that
case? u is a constant,
00:17:22.790 --> 00:17:27.050
so that term is 0 in this rule.
00:17:27.050 --> 00:17:28.700
I don't have to
worry about this.
00:17:28.700 --> 00:17:31.650
I get a minus.
00:17:31.650 --> 00:17:36.800
And then u is 1, and dv/dx.
00:17:36.800 --> 00:17:38.820
Well, v is whatever v is.
00:17:38.820 --> 00:17:40.790
I'll write dv/dx as v'.
00:17:43.854 --> 00:17:45.520
And then I get a v^2
in the denominator.
00:17:45.520 --> 00:17:50.070
So that's the rule.
00:17:50.070 --> 00:17:51.380
I could write it as v^(-2) v'.
00:17:56.840 --> 00:17:59.300
Minus v' divided by v^2.
00:17:59.300 --> 00:18:03.730
That's the derivative of 1 / v.
00:18:03.730 --> 00:18:12.110
How about sub-example of that?
00:18:12.110 --> 00:18:15.840
I'm going to take the special
case where u = 1 again.
00:18:15.840 --> 00:18:16.770
And v is a power of x.
00:18:21.000 --> 00:18:25.630
And I'm gonna use the rule
that we developed earlier about
00:18:25.630 --> 00:18:29.080
the derivative of x^n.
00:18:29.080 --> 00:18:33.230
So what do I get here?
00:18:33.230 --> 00:18:42.645
d/dx (1/x^n) is, I'm plugging
into this formula here with v =
00:18:42.645 --> 00:18:45.260
x^n.
00:18:45.260 --> 00:18:51.580
So I get minus, uh, v^-2.
00:18:51.580 --> 00:18:57.250
If v = x^n, v^-2 is, by the
rule of exponents, x^(-2n).
00:19:01.430 --> 00:19:05.550
And then v' is the derivative
of x^n, which is nx^(n-1).
00:19:10.150 --> 00:19:12.010
Okay, so let's put
these together.
00:19:12.010 --> 00:19:13.550
There's several
powers of x here.
00:19:13.550 --> 00:19:14.940
I can put them together.
00:19:14.940 --> 00:19:22.330
I get -n x to the -2n + n - 1.
00:19:22.330 --> 00:19:23.936
One of these n's cancels.
00:19:23.936 --> 00:19:25.310
And what I'm left
with is -n - 1.
00:19:29.260 --> 00:19:32.550
So we've computed the
derivative of 1 / x^n,
00:19:32.550 --> 00:19:39.210
which I could also
write as x^-n, right?
00:19:39.210 --> 00:19:42.640
So I've computed the derivative
of negative powers of x.
00:19:42.640 --> 00:19:46.560
And this is the
formula that I get.
00:19:46.560 --> 00:19:51.990
If you think of this -n as a
unit, as a thing to itself,
00:19:51.990 --> 00:19:54.310
it occurs here in the exponent.
00:19:54.310 --> 00:19:59.890
It occurs here,
and it occurs here.
00:19:59.890 --> 00:20:01.820
So how does that
compare with the formula
00:20:01.820 --> 00:20:04.120
that we had up here?
00:20:04.120 --> 00:20:06.860
The derivative of
a power of x is
00:20:06.860 --> 00:20:12.300
that power times x to
one less than that power.
00:20:12.300 --> 00:20:16.010
That's exactly the same as the
rule that I wrote down here.
00:20:16.010 --> 00:20:19.050
But the power here happens
to be a negative number,
00:20:19.050 --> 00:20:22.360
and the same negative number
shows up as a coefficient
00:20:22.360 --> 00:20:23.780
and there in the exponent.
00:20:23.780 --> 00:20:24.280
Yeah?
00:20:24.280 --> 00:20:30.440
Student: [INAUDIBLE].
00:20:30.440 --> 00:20:34.930
Professor: How did I do this?
00:20:34.930 --> 00:20:49.150
Student: [INAUDIBLE].
00:20:49.150 --> 00:20:55.990
Professor: Where did
that x^(-2n) come from?
00:20:55.990 --> 00:20:59.900
So I'm applying this rule.
00:20:59.900 --> 00:21:04.440
So the denominator in
the quotient rule is v^2.
00:21:04.440 --> 00:21:11.109
And v was x^n, so the
denominator is x^(2n).
00:21:11.109 --> 00:21:12.650
And I decided to
write it as x^(-2n).
00:21:19.010 --> 00:21:22.080
So the green comments there...
00:21:22.080 --> 00:21:26.270
What they say is that I
can enlarge this rule.
00:21:26.270 --> 00:21:31.230
This exact same rule is true
for negative values of n,
00:21:31.230 --> 00:21:36.310
as well as positive values of n.
00:21:36.310 --> 00:21:40.300
So there's something
new in your list
00:21:40.300 --> 00:21:46.670
of rules that you can apply,
of values of the derivative.
00:21:46.670 --> 00:21:49.550
That standard rule is true for
negative as well as positive
00:21:49.550 --> 00:21:51.120
exponents.
00:21:51.120 --> 00:21:57.290
And that comes out
of a quotient rule.
00:21:57.290 --> 00:21:59.020
Okay, so we've done two rules.
00:21:59.020 --> 00:22:04.650
I've talked about the product
rule and the quotient rule.
00:22:04.650 --> 00:22:05.670
What's next?
00:22:05.670 --> 00:22:07.150
Let's see the chain rule.
00:22:07.150 --> 00:22:22.220
So this is a composition rule.
00:22:22.220 --> 00:22:24.890
So the kind of thing that
I have in mind, composition
00:22:24.890 --> 00:22:28.210
of functions is
about substitution.
00:22:28.210 --> 00:22:31.106
So the kind of function that I
have in mind is, for instance,
00:22:31.106 --> 00:22:31.730
y = (sin t)^10.
00:22:39.700 --> 00:22:42.695
That's a new one.
00:22:42.695 --> 00:22:44.820
We haven't seen how to
differentiate that before, I
00:22:44.820 --> 00:22:46.590
think.
00:22:46.590 --> 00:22:50.600
This kind of power of a trig
function happens very often.
00:22:50.600 --> 00:22:53.540
You've seen them happen,
as well, I'm sure, already.
00:22:53.540 --> 00:22:58.020
And there's a little notational
switch that people use.
00:22:58.020 --> 00:22:59.320
They'll write sin^10(t).
00:23:02.910 --> 00:23:05.100
But remember that when
you write sin^10(t),
00:23:05.100 --> 00:23:08.030
what you mean is
take the sine of t,
00:23:08.030 --> 00:23:10.440
and then take the
10th power of that.
00:23:10.440 --> 00:23:13.590
It's the meaning of sin^10(t).
00:23:13.590 --> 00:23:20.950
So the method of dealing
with this kind of composition
00:23:20.950 --> 00:23:33.190
of functions is to use
new variable names.
00:23:33.190 --> 00:23:36.830
What I mean is, I can
think of this (sin t)^10.
00:23:39.710 --> 00:23:42.070
I can think of it it
as a two step process.
00:23:42.070 --> 00:23:44.160
First of all, I
compute the sine of t.
00:23:44.160 --> 00:23:47.450
And let's call the result x.
00:23:47.450 --> 00:23:50.150
There's the new variable name.
00:23:50.150 --> 00:23:53.340
And then, I express
y in terms of x.
00:23:53.340 --> 00:23:58.070
So y says take this and
raise it to the tenth power.
00:23:58.070 --> 00:23:59.360
In other words, y = x^10.
00:24:03.400 --> 00:24:06.420
And then you plug x
= sin(t) into that,
00:24:06.420 --> 00:24:10.590
and you get the formula for
what y is in terms of t.
00:24:10.590 --> 00:24:14.550
So it's good practice to
introduce new letters when
00:24:14.550 --> 00:24:17.060
they're convenient, and
this is one place where
00:24:17.060 --> 00:24:21.820
it's very convenient.
00:24:21.820 --> 00:24:24.260
So let's find a rule
for differentiating
00:24:24.260 --> 00:24:25.860
a composition, a
function that can
00:24:25.860 --> 00:24:27.770
be expressed by
doing one function
00:24:27.770 --> 00:24:30.270
and then applying
another function.
00:24:30.270 --> 00:24:32.880
And here's the rule.
00:24:32.880 --> 00:24:34.930
Well, maybe I'll actually
derive this rule first,
00:24:34.930 --> 00:24:37.420
and then you'll see what it is.
00:24:37.420 --> 00:24:40.600
In fact, the rule is
very simple to derive.
00:24:40.600 --> 00:24:43.890
So this is a proof first, and
then we'll write down the rule.
00:24:43.890 --> 00:24:51.950
I'm interested in delta y /
delta t. y is a function of x.
00:24:51.950 --> 00:24:53.760
x is a function of t.
00:24:53.760 --> 00:24:56.850
And I'm interested in how
y changes with respect
00:24:56.850 --> 00:25:00.850
to t, with respect to
the original variable t.
00:25:00.850 --> 00:25:05.160
Well, because of that
intermediate variable,
00:25:05.160 --> 00:25:12.670
I can write this as (delta y /
delta x) (delta x / delta t).
00:25:12.670 --> 00:25:15.330
It cancels, right?
00:25:15.330 --> 00:25:17.600
The delta x cancels.
00:25:17.600 --> 00:25:23.100
The change in that immediate
variable cancels out.
00:25:23.100 --> 00:25:26.120
This is just basic algebra.
00:25:26.120 --> 00:25:29.930
But what happens when I
let delta t get small?
00:25:29.930 --> 00:25:31.410
Well this give me dy/dt.
00:25:34.370 --> 00:25:42.220
On the right-hand side,
I get (dy/dx) (dx/dt).
00:25:42.220 --> 00:25:44.430
So students will often
remember this rule.
00:25:44.430 --> 00:25:47.130
This is the rule, by saying
that you can cancel out
00:25:47.130 --> 00:25:49.080
for the dx's.
00:25:49.080 --> 00:25:51.860
And that's not so
far from the truth.
00:25:51.860 --> 00:25:55.160
That's a good way
to think of it.
00:25:55.160 --> 00:26:01.410
In other words, this is
the so-called chain rule.
00:26:01.410 --> 00:26:26.690
And it says that differentiation
of a composition is a product.
00:26:26.690 --> 00:26:34.910
It's just the product
of the two derivatives.
00:26:34.910 --> 00:26:39.570
So that's how you differentiate
a composite of two functions.
00:26:39.570 --> 00:26:42.070
And let's just do an example.
00:26:42.070 --> 00:26:44.690
Let's do this example.
00:26:44.690 --> 00:26:48.820
Let's see how that comes out.
00:26:48.820 --> 00:26:55.250
So let's differentiate,
what did I say?
00:26:55.250 --> 00:26:56.530
(sin t)^10.
00:26:59.400 --> 00:27:03.130
Okay, there's an inside function
and an outside function.
00:27:03.130 --> 00:27:07.910
The inside function is
x as a function of t.
00:27:07.910 --> 00:27:19.170
This is the inside function, and
this is the outside function.
00:27:19.170 --> 00:27:22.590
So the rule says, first
of all let's differentiate
00:27:22.590 --> 00:27:23.550
the outside function.
00:27:23.550 --> 00:27:25.370
Take dy/dx.
00:27:25.370 --> 00:27:29.200
Differentiate it with
respect to that variable x.
00:27:29.200 --> 00:27:31.020
The outside function
is the 10th power.
00:27:31.020 --> 00:27:34.640
What's its derivative?
00:27:34.640 --> 00:27:37.530
So I get 10x^9.
00:27:42.440 --> 00:27:51.090
In this account, I'm using
this newly introduced variable
00:27:51.090 --> 00:27:53.990
named x.
00:27:53.990 --> 00:27:58.150
So the derivative of the
outside function is 10x^9.
00:27:58.150 --> 00:28:00.360
And then here's the
inside function,
00:28:00.360 --> 00:28:03.130
and the next thing I want
to do is differentiate it.
00:28:03.130 --> 00:28:07.730
So what's dx/dt, d/dt (sin
t), the derivative of sine t?
00:28:07.730 --> 00:28:11.619
All right, that's cosine t.
00:28:11.619 --> 00:28:13.160
That's what the
chain rule gives you.
00:28:13.160 --> 00:28:17.490
This is correct, but
since we were the ones
00:28:17.490 --> 00:28:20.730
to introduce this
notation x here,
00:28:20.730 --> 00:28:24.560
that wasn't given to us in
the original problem here.
00:28:24.560 --> 00:28:26.400
The last step in
this process should
00:28:26.400 --> 00:28:28.980
be to put back,
to substitute back
00:28:28.980 --> 00:28:32.440
in what x is in terms of t.
00:28:32.440 --> 00:28:35.320
So x = sin t.
00:28:35.320 --> 00:28:45.980
So that tells me that I get
10(sin(t))^9, that's x^9,
00:28:45.980 --> 00:28:47.860
times the cos(t).
00:28:47.860 --> 00:28:50.860
Or the same thing
is sin^9(t)cos(t).
00:28:56.040 --> 00:28:59.540
So there's an application
of the chain rule.
00:28:59.540 --> 00:29:02.504
You know, people often
wonder where the name chain
00:29:02.504 --> 00:29:03.170
rule comes from.
00:29:03.170 --> 00:29:06.340
I was just wondering
about that myself.
00:29:06.340 --> 00:29:15.230
So is it because
it chains you down?
00:29:15.230 --> 00:29:18.070
Is it like a chain fence?
00:29:18.070 --> 00:29:19.590
I decided what it is.
00:29:19.590 --> 00:29:21.910
It's because by
using it, you burst
00:29:21.910 --> 00:29:25.880
the chains of differentiation,
and you can differentiate
00:29:25.880 --> 00:29:28.040
many more functions using it.
00:29:28.040 --> 00:29:31.553
So when you want to
think of the chain rule,
00:29:31.553 --> 00:29:35.640
just think of that chain there.
00:29:35.640 --> 00:29:47.960
It lets you burst free.
00:29:47.960 --> 00:30:04.830
Let me give you another
application of the chain rule.
00:30:04.830 --> 00:30:16.220
Ready for this one?
00:30:16.220 --> 00:30:17.970
So I'd like to
differentiate the sin(10t).
00:30:25.524 --> 00:30:27.440
Again, this is the
composite of two functions.
00:30:27.440 --> 00:30:30.220
What's the inside function?
00:30:30.220 --> 00:30:35.640
Okay, so I think I'll introduce
this new notation. x = 10t,
00:30:35.640 --> 00:30:38.260
and the outside
function is the sine.
00:30:38.260 --> 00:30:41.320
So y = sin x.
00:30:41.320 --> 00:30:46.660
So now the chain
rule says dy/dt is...
00:30:46.660 --> 00:30:47.920
Okay, let's see.
00:30:47.920 --> 00:30:50.710
I take the derivative
of the outside function,
00:30:50.710 --> 00:30:54.240
and what's that?
00:30:54.240 --> 00:30:56.470
Sine prime and we can
substitute because we
00:30:56.470 --> 00:30:58.520
know what sine prime is.
00:30:58.520 --> 00:31:06.470
So I get cosine of whatever,
x, and then times what?
00:31:06.470 --> 00:31:11.400
Now I differentiate the inside
function, which is just 10.
00:31:11.400 --> 00:31:16.380
So I could write this
as 10cos of what?
00:31:16.380 --> 00:31:17.360
10t, x is 10t.
00:31:20.260 --> 00:31:26.170
Now, once you get used to
this, this middle variable,
00:31:26.170 --> 00:31:33.190
you don't have to
give a name for it.
00:31:33.190 --> 00:31:35.150
You can just to think
about it in your mind
00:31:35.150 --> 00:31:44.890
without actually writing
it down, d/dt (sin(10t)).
00:31:47.980 --> 00:31:49.860
I'll just do it again
without introducing
00:31:49.860 --> 00:31:52.240
this middle variable explicitly.
00:31:52.240 --> 00:31:54.530
Think about it.
00:31:54.530 --> 00:31:58.100
I first of all differentiate
the outside function,
00:31:58.100 --> 00:31:59.740
and I get cosine.
00:31:59.740 --> 00:32:03.170
But I don't change the thing
that I'm plugging into it.
00:32:03.170 --> 00:32:08.560
It's still x that I'm
plugging into it. x is 10t.
00:32:08.560 --> 00:32:11.470
So let's just write 10t and
not worry about the name
00:32:11.470 --> 00:32:12.720
of that extra variable.
00:32:12.720 --> 00:32:15.510
If it confuses you,
introduce the new variable.
00:32:15.510 --> 00:32:18.180
And do it carefully
and slowly like this.
00:32:18.180 --> 00:32:19.970
But, quite quickly,
I think you'll
00:32:19.970 --> 00:32:23.202
get to be able to keep
that step in your mind.
00:32:23.202 --> 00:32:24.160
I'm not quite done yet.
00:32:24.160 --> 00:32:26.900
I haven't differentiated
the inside function,
00:32:26.900 --> 00:32:29.190
the derivative of 10t = 10.
00:32:29.190 --> 00:32:33.250
So you get, again,
the same result.
00:32:33.250 --> 00:32:36.420
A little shortcut that
you'll get used to.
00:32:36.420 --> 00:32:38.680
Really and truly, once
you have the chain rule,
00:32:38.680 --> 00:32:41.110
the world is yours to conquer.
00:32:41.110 --> 00:32:46.730
It puts you in a very,
very powerful position.
00:32:46.730 --> 00:32:50.210
Okay, well let's see.
00:32:50.210 --> 00:32:51.310
What have I covered today?
00:32:51.310 --> 00:32:57.370
I've talked about product rule,
quotient rule, composition.
00:32:57.370 --> 00:32:59.580
I should tell you something
about higher derivatives,
00:32:59.580 --> 00:33:00.670
as well.
00:33:00.670 --> 00:33:10.440
So let's do that.
00:33:10.440 --> 00:33:12.150
This is a simple story.
00:33:12.150 --> 00:33:14.950
Higher is kind of
a strange word.
00:33:14.950 --> 00:33:32.950
It just means differentiate
over and over again.
00:33:32.950 --> 00:33:34.600
All right, so let's see.
00:33:34.600 --> 00:33:38.510
If we have a function
u or u(x), please
00:33:38.510 --> 00:33:45.010
allow me to just write
it as briefly as u.
00:33:45.010 --> 00:33:49.330
Well, this is a sort
of notational thing.
00:33:49.330 --> 00:33:51.780
I can differentiate
it and get u'.
00:33:54.790 --> 00:33:55.900
That's a new function.
00:33:55.900 --> 00:33:57.680
Like if you started
with the sine, that's
00:33:57.680 --> 00:34:00.760
gonna be the cosine.
00:34:00.760 --> 00:34:03.570
A new function, so I can
differentiate it again.
00:34:03.570 --> 00:34:05.780
And the notation for the
differentiating of it again,
00:34:05.780 --> 00:34:07.470
is u prime prime.
00:34:07.470 --> 00:34:12.930
So u'' is just u'
differentiated again.
00:34:12.930 --> 00:34:21.380
For example, if u is the sine
of x, so u' is the cosine of x.
00:34:21.380 --> 00:34:24.150
Has Professor Jerison
talked about what
00:34:24.150 --> 00:34:26.580
the derivative of cosine is?
00:34:26.580 --> 00:34:28.220
What is it?
00:34:28.220 --> 00:34:33.020
Ha, okay so u'' is -sin x.
00:34:36.810 --> 00:34:38.930
Let me go on.
00:34:38.930 --> 00:34:42.970
What do you suppose u''' means?
00:34:42.970 --> 00:34:46.420
I guess it's the
derivative of u''.
00:34:46.420 --> 00:34:53.050
It's called the
third derivative.
00:34:53.050 --> 00:34:56.210
And u'' is called the
second derivative.
00:34:56.210 --> 00:34:59.000
And it's u''
differentiated again.
00:34:59.000 --> 00:35:03.680
So to compute u''' in this
example, what do I do?
00:35:03.680 --> 00:35:05.340
I differentiate that again.
00:35:05.340 --> 00:35:08.460
There's a constant term,
-1, constant factor.
00:35:08.460 --> 00:35:09.950
That comes out.
00:35:09.950 --> 00:35:13.500
The derivative of sine is what?
00:35:13.500 --> 00:35:17.930
Okay, so u''' = -cos x.
00:35:17.930 --> 00:35:18.690
Let's do it again.
00:35:18.690 --> 00:35:21.890
Now after a while, you get
tired of writing these things.
00:35:21.890 --> 00:35:24.650
And so maybe I'll use
the notation u^(4).
00:35:24.650 --> 00:35:27.290
That's the fourth derivative.
00:35:27.290 --> 00:35:29.490
That's u''''.
00:35:29.490 --> 00:35:33.440
Or it's (u''') differentiated
again, the fourth derivative.
00:35:33.440 --> 00:35:37.970
And what is that
in this example?
00:35:37.970 --> 00:35:41.290
Okay, the cosine has
derivative minus the sine,
00:35:41.290 --> 00:35:42.010
like you told me.
00:35:42.010 --> 00:35:44.430
And that minus sign
cancels with that sign,
00:35:44.430 --> 00:35:47.640
and all together, I get sin x.
00:35:47.640 --> 00:35:48.940
That's pretty bizarre.
00:35:48.940 --> 00:35:51.720
When I differentiate the
function sine of x four times,
00:35:51.720 --> 00:35:56.920
I get back to the
sine of x again.
00:35:56.920 --> 00:36:00.290
That's the way it is.
00:36:00.290 --> 00:36:03.491
Now this notation, prime
prime prime prime, and things
00:36:03.491 --> 00:36:03.990
like that.
00:36:03.990 --> 00:36:13.650
There are different
variants of that notation.
00:36:13.650 --> 00:36:24.070
For example, that's
another notation.
00:36:24.070 --> 00:36:29.320
Well, you've used the
notation du/dx before. u'
00:36:29.320 --> 00:36:30.630
could also be denoted du/dx.
00:36:35.730 --> 00:36:38.460
I think we've
already here, today,
00:36:38.460 --> 00:36:43.230
used this way of
rewriting du/dx.
00:36:43.230 --> 00:36:48.150
I think when I was talking about
d/dt(uv) and so on, I pulled
00:36:48.150 --> 00:36:52.360
that d/dt outside and
put whatever function
00:36:52.360 --> 00:36:55.010
you're differentiating
over to the right.
00:36:55.010 --> 00:36:57.430
So that's just a
notational switch.
00:36:57.430 --> 00:36:58.110
It looks good.
00:36:58.110 --> 00:37:06.260
It looks like good
algebra doesn't it?
00:37:06.260 --> 00:37:12.410
But what it's doing is regarding
this notation as an operator.
00:37:12.410 --> 00:37:16.920
It's something you apply to a
function to get a new function.
00:37:16.920 --> 00:37:20.680
I apply it to the sine function,
and I get the cosine function.
00:37:20.680 --> 00:37:24.220
I apply it to x^2, and I get 2x.
00:37:24.220 --> 00:37:31.140
This thing here, that symbol,
represents an operator,
00:37:31.140 --> 00:37:40.340
which you apply to a function.
00:37:40.340 --> 00:37:44.860
And the operator says, take the
function and differentiate it.
00:37:44.860 --> 00:37:47.330
So further notation
that people often use,
00:37:47.330 --> 00:37:49.460
is they give a different
name to that operator.
00:37:49.460 --> 00:37:52.270
And they'll write
capital D for it.
00:37:52.270 --> 00:38:02.980
So this is just using capital
D for the symbol d/dx.
00:38:02.980 --> 00:38:05.050
So in terms of that
notation, let's see.
00:38:05.050 --> 00:38:20.440
Let's write down what higher
derivatives look like.
00:38:20.440 --> 00:38:21.870
So let's see.
00:38:21.870 --> 00:38:23.090
That's what u' is.
00:38:23.090 --> 00:38:24.360
How about u''?
00:38:24.360 --> 00:38:28.890
Let's write that in terms
of the d/dx notation.
00:38:28.890 --> 00:38:31.710
Well I'm supposed to
differentiate u' right?
00:38:31.710 --> 00:38:35.590
So that's d/dx applied
to the function du/dx.
00:38:40.920 --> 00:38:43.030
Differentiate the derivative.
00:38:43.030 --> 00:38:47.240
That's what I've done.
00:38:47.240 --> 00:38:54.350
Or I could write that as d/dx
applied to d/dx applied to u.
00:38:54.350 --> 00:38:57.850
Just pulling that u outside.
00:38:57.850 --> 00:38:59.570
So I'm doing d/dx twice.
00:38:59.570 --> 00:39:01.590
I'm doing that operator twice.
00:39:01.590 --> 00:39:08.030
I could write that as
(d/dx)^2 applied to u.
00:39:08.030 --> 00:39:15.170
Differentiate twice, and
do it to the function u.
00:39:15.170 --> 00:39:23.130
Or, I can write it as,
now this is a strange one.
00:39:23.130 --> 00:39:33.330
I could also write
as-- like that.
00:39:33.330 --> 00:39:36.630
It's getting stranger
and stranger, isn't it?
00:39:36.630 --> 00:39:40.770
This is definitely just a
kind of abuse of notation.
00:39:40.770 --> 00:39:45.030
But people will go even
further and write d squared
00:39:45.030 --> 00:39:46.030
u divided by dx squared.
00:39:50.500 --> 00:39:52.190
So this is the strangest one.
00:39:52.190 --> 00:39:56.190
This identity quality
is the strangest one,
00:39:56.190 --> 00:40:00.130
because you may think that
you're taking d of the quantity
00:40:00.130 --> 00:40:01.330
x^2.
00:40:01.330 --> 00:40:03.930
But that's not what's intended.
00:40:03.930 --> 00:40:08.240
This is not d(x^2).
00:40:08.240 --> 00:40:12.750
What's intended is the
quantity dx squared.
00:40:12.750 --> 00:40:14.630
In this notation,
which is very common,
00:40:14.630 --> 00:40:16.410
what's intended
by the denominator
00:40:16.410 --> 00:40:18.250
is the quantity dx squared.
00:40:18.250 --> 00:40:23.630
It's part of this second
differentiation operator.
00:40:23.630 --> 00:40:26.240
So I've written a bunch
of equalities down here,
00:40:26.240 --> 00:40:28.570
and the only content
to them is that these
00:40:28.570 --> 00:40:32.320
are all different notations
for the same thing.
00:40:32.320 --> 00:40:34.940
You'll see this
notation very commonly.
00:40:34.940 --> 00:40:37.050
So for instance the
third derivative
00:40:37.050 --> 00:40:47.330
is d cubed u divided
by dx cubed, and so on.
00:40:47.330 --> 00:40:47.830
Sorry?
00:40:47.830 --> 00:40:58.755
Student: [INAUDIBLE].
00:40:58.755 --> 00:40:59.880
Professor: Yes, absolutely.
00:40:59.880 --> 00:41:04.417
Or an equally good notation is
to write the operator capital
00:41:04.417 --> 00:41:05.500
D, done three times, to u.
00:41:09.400 --> 00:41:11.502
Absolutely.
00:41:11.502 --> 00:41:13.960
So I guess I should also write
over here when I was talking
00:41:13.960 --> 00:41:16.180
about d^2, the
second derivative,
00:41:16.180 --> 00:41:20.820
another notation is do the
operator capital D twice.
00:41:20.820 --> 00:41:22.820
Let's see an example of
how this can be applied.
00:41:22.820 --> 00:41:23.903
I'll answer this question.
00:41:23.903 --> 00:41:32.582
Student: [INAUDIBLE].
00:41:32.582 --> 00:41:34.040
Professor: Yeah,
so the question is
00:41:34.040 --> 00:41:36.230
whether the fourth
derivative always gives you
00:41:36.230 --> 00:41:38.880
the original function back,
like what happened here.
00:41:38.880 --> 00:41:39.580
No.
00:41:39.580 --> 00:41:43.470
That's very, very special
to sines and cosines.
00:41:43.470 --> 00:41:45.200
All right?
00:41:45.200 --> 00:41:47.850
And, in fact, let's
see an example of that.
00:41:47.850 --> 00:41:50.920
I'll do a calculation.
00:41:50.920 --> 00:42:06.130
Let's calculate the
nth derivative of x^n.
00:42:06.130 --> 00:42:13.190
Okay, n is a number,
like 1, 2, 3, 4.
00:42:13.190 --> 00:42:13.720
Here we go.
00:42:13.720 --> 00:42:15.360
Let's do this.
00:42:15.360 --> 00:42:17.650
So, let's do this bit by bit.
00:42:17.650 --> 00:42:22.500
What's the first
derivative of x^n?
00:42:22.500 --> 00:42:24.090
So everybody knows this.
00:42:24.090 --> 00:42:27.830
I'm just using a new notation,
this capital D notation.
00:42:27.830 --> 00:42:30.520
So it's nx^(n-1).
00:42:30.520 --> 00:42:34.380
Now know, by the way, n could
be a negative number for that,
00:42:34.380 --> 00:42:37.250
but for now, for
this application,
00:42:37.250 --> 00:42:41.280
I wanna take n to be
1, 2, 3, and so on;
00:42:41.280 --> 00:42:43.070
one of those numbers.
00:42:43.070 --> 00:42:44.550
Okay, we did one derivative.
00:42:44.550 --> 00:42:49.530
Let's compute the second
derivative of x^n.
00:42:49.530 --> 00:42:52.070
Well there's this n
constant that comes out,
00:42:52.070 --> 00:42:59.980
and then the exponent comes
down, and it gets reduced by 1.
00:42:59.980 --> 00:43:01.190
All right?
00:43:01.190 --> 00:43:03.780
Should I do one more?
00:43:03.780 --> 00:43:07.600
D^3 (x^n) is n(n-1).
00:43:07.600 --> 00:43:09.410
That's the constant from here.
00:43:09.410 --> 00:43:13.220
Times that exponent,
n - 2, times 1 less, n
00:43:13.220 --> 00:43:15.740
- 3 is the new exponent.
00:43:15.740 --> 00:43:26.430
Well, I keep on going until
I come to a new blackboard.
00:43:26.430 --> 00:43:28.100
Now, I think I'm
going to stop when
00:43:28.100 --> 00:43:29.980
I get to the n minus
first derivative,
00:43:29.980 --> 00:43:35.370
so we can see what's
likely to happen.
00:43:35.370 --> 00:43:38.970
So when I took the
third derivative,
00:43:38.970 --> 00:43:42.957
I had the n minus
third power of x.
00:43:42.957 --> 00:43:44.540
And when I took the
second derivative,
00:43:44.540 --> 00:43:45.760
I had the second power of x.
00:43:45.760 --> 00:43:48.310
So, I think what'll
happen when I
00:43:48.310 --> 00:43:49.730
have the n minus
first derivative
00:43:49.730 --> 00:43:53.510
is I'll have the first
power of x left over.
00:43:53.510 --> 00:43:55.390
The powers of x
keep coming down.
00:43:55.390 --> 00:43:59.350
And what I've done it n - 1
times, I get the first power.
00:43:59.350 --> 00:44:04.230
And then I get a big constant
out in front here times more
00:44:04.230 --> 00:44:07.450
and more and more of these
smaller and smaller integers
00:44:07.450 --> 00:44:08.500
that come down.
00:44:08.500 --> 00:44:12.310
What's the last integer that
came down before I got x^1
00:44:12.310 --> 00:44:17.460
here?
00:44:17.460 --> 00:44:19.390
Well, let's see.
00:44:19.390 --> 00:44:23.320
It's just 2, because this x^1
occurred as the derivative
00:44:23.320 --> 00:44:24.340
of x^2.
00:44:24.340 --> 00:44:27.800
And the coefficient
in front of that is 2.
00:44:27.800 --> 00:44:29.730
So that's what you get.
00:44:29.730 --> 00:44:35.140
The numbers n, n-1, and so
on down to 2, times x^1.
00:44:35.140 --> 00:44:41.560
And now we can differentiate
one more time and calculate what
00:44:41.560 --> 00:44:42.770
D^n x^n is.
00:44:42.770 --> 00:44:48.070
So I get the same number, n
times n-1 and so on and so on,
00:44:48.070 --> 00:44:49.680
times 2.
00:44:49.680 --> 00:44:52.500
And then I guess
I'll say times 1.
00:44:52.500 --> 00:44:58.640
Times, what's the derivative
of x^1= 1, so times 1.
00:44:58.640 --> 00:45:01.260
Time 1, times 1.
00:45:01.260 --> 00:45:10.490
Where this one means
the constant function 1.
00:45:10.490 --> 00:45:14.070
Does anyone know what
this number is called?
00:45:14.070 --> 00:45:15.110
That has a name.
00:45:15.110 --> 00:45:19.720
It's called n factorial.
00:45:19.720 --> 00:45:21.400
And it's written n
exclamation point.
00:45:24.240 --> 00:45:28.830
And we just used an example
of mathematical induction.
00:45:28.830 --> 00:45:37.750
So the end result is
D^n x^n is n!, constant.
00:45:37.750 --> 00:45:42.460
Okay that's a neat fact.
00:45:42.460 --> 00:45:47.570
Final question for the lecture
is what's D^(n + 1) applied
00:45:47.570 --> 00:45:49.730
to x^n?
00:45:49.730 --> 00:45:50.850
Ha.
00:45:50.850 --> 00:45:54.340
Excellent.
00:45:54.340 --> 00:45:56.620
It's the derivative
of a constant.
00:45:56.620 --> 00:45:57.940
So it's 0.
00:45:57.940 --> 00:45:58.440
Okay.
00:45:58.440 --> 00:45:59.980
Thank you.