WEBVTT

00:00:06.860 --> 00:00:08.900
PROFESSOR: Welcome
back to recitation.

00:00:08.900 --> 00:00:11.090
In this video segment,
what I'd like us to do

00:00:11.090 --> 00:00:13.260
is work on this
following problem.

00:00:13.260 --> 00:00:17.941
Find d/dx of the integral from
0 to x squared, cosine t dt.

00:00:17.941 --> 00:00:19.940
I'm going to give you a
moment to think about it

00:00:19.940 --> 00:00:21.981
and then I'll come back
and show you how I do it.

00:00:30.690 --> 00:00:31.660
OK, welcome back.

00:00:31.660 --> 00:00:34.240
Hopefully you were able to
make some headway on this.

00:00:34.240 --> 00:00:37.740
Let's look at the problem and
see how we would break it down.

00:00:37.740 --> 00:00:39.990
Well we know from the
fundamental theorem of calculus

00:00:39.990 --> 00:00:42.540
that you saw in the
lecture, that if up here

00:00:42.540 --> 00:00:44.024
instead of x
squared we had an x,

00:00:44.024 --> 00:00:45.690
then the problem would
be easy to solve.

00:00:45.690 --> 00:00:48.470
We'd just use the fundamental
theorem of calculus, the answer

00:00:48.470 --> 00:00:50.547
would be cosine x.

00:00:50.547 --> 00:00:52.880
But of course, we don't have
an x, we have an x squared.

00:00:52.880 --> 00:00:54.490
That's why I gave
you this problem.

00:00:54.490 --> 00:00:57.270
And we need to figure out how
to solve this problem when

00:00:57.270 --> 00:01:00.090
there's a different function
up here besides just x.

00:01:00.090 --> 00:01:01.780
What we're going to
use is we're going

00:01:01.780 --> 00:01:03.960
to combine the fundamental
theorem of calculus

00:01:03.960 --> 00:01:05.470
and the chain rule.

00:01:05.470 --> 00:01:08.090
So, let's start off with
how we would do this

00:01:08.090 --> 00:01:11.790
if it were the integral
from 0 to x, as I mentioned.

00:01:11.790 --> 00:01:17.980
So we'll define
capital F of x to be

00:01:17.980 --> 00:01:21.300
equal to the integral
from 0 to x cosine t dt.

00:01:24.660 --> 00:01:31.901
And then we know that f prime
of x is equal to cosine x.

00:01:31.901 --> 00:01:34.400
Now the problem is, we don't
just have this, as I mentioned.

00:01:34.400 --> 00:01:38.870
What we actually have--
let me write this down--

00:01:38.870 --> 00:01:41.635
we have F of x squared.

00:01:41.635 --> 00:01:43.320
Right?

00:01:43.320 --> 00:01:46.350
That's what this--
sorry let me highlight

00:01:46.350 --> 00:01:52.680
what I mean-- this boxed
thing is F of x squared.

00:01:52.680 --> 00:01:56.100
So we took F of x and we
evaluated it at x squared.

00:01:56.100 --> 00:01:57.650
That's what we get in the box.

00:01:57.650 --> 00:02:01.050
And so if we want to find
d/dx of F of x squared,

00:02:01.050 --> 00:02:02.760
it really is just
the chain rule.

00:02:02.760 --> 00:02:05.670
We really want to think of this
as a composition of functions.

00:02:05.670 --> 00:02:08.550
The first, the outside
function is capital F

00:02:08.550 --> 00:02:11.140
and the inside
function is x squared.

00:02:11.140 --> 00:02:15.220
So just in general, how do we
think about the chain rule?

00:02:15.220 --> 00:02:17.520
Well remember what
we do-- let me

00:02:17.520 --> 00:02:19.420
come back here for a
second-- remember what

00:02:19.420 --> 00:02:22.200
we do is we take the derivative
of the outside function,

00:02:22.200 --> 00:02:24.491
we evaluate it at
the inside function

00:02:24.491 --> 00:02:26.740
and then we take the derivative
of the inside function

00:02:26.740 --> 00:02:29.060
and we multiply
those two together.

00:02:29.060 --> 00:02:31.440
So all I have to
do is figure out,

00:02:31.440 --> 00:02:33.160
what is the following thing?

00:02:33.160 --> 00:02:40.300
We know d/dx the
quantity F of x squared

00:02:40.300 --> 00:02:44.570
should be equal to
F prime evaluated

00:02:44.570 --> 00:02:47.544
at x squared times 2x.

00:02:47.544 --> 00:02:49.210
That's just what we
said earlier, right?

00:02:49.210 --> 00:02:52.050
It's the derivative
of F evaluated

00:02:52.050 --> 00:02:55.920
at x squared times the
derivative of x squared.

00:02:55.920 --> 00:02:58.180
So now I just have to
figure out what this is.

00:02:58.180 --> 00:03:00.030
Well let's go back
to the other side

00:03:00.030 --> 00:03:02.700
and see what we wrote
that F prime was.

00:03:02.700 --> 00:03:08.310
If we come over here, we see
F prime at x is just cosine x.

00:03:08.310 --> 00:03:13.665
So F prime at x squared is going
to be this function evaluated

00:03:13.665 --> 00:03:15.090
at x squared.

00:03:15.090 --> 00:03:17.730
That's just cosine of x squared.

00:03:17.730 --> 00:03:24.830
So we see over here, we just
get cosine x squared times 2x.

00:03:24.830 --> 00:03:27.070
And because I'm a
mathematician, I

00:03:27.070 --> 00:03:29.860
want to write the 2x in
front before I finish.

00:03:29.860 --> 00:03:31.670
Because otherwise
I get confused.

00:03:31.670 --> 00:03:35.830
So the answer here is just
2x times cosine x squared.

00:03:35.830 --> 00:03:37.830
Now I want to point out
really what we did here.

00:03:37.830 --> 00:03:39.980
This is the answer to
this particular problem,

00:03:39.980 --> 00:03:43.070
but we can now solve
problems in general,

00:03:43.070 --> 00:03:46.595
when I put any function up
here, any function of x up here.

00:03:46.595 --> 00:03:47.660
Right?

00:03:47.660 --> 00:03:50.130
Ultimately, all I did was I
used the fundamental theorem

00:03:50.130 --> 00:03:52.100
of calculus and the chain rule.

00:03:52.100 --> 00:03:54.100
So any function I
put up here, I can

00:03:54.100 --> 00:03:56.260
do exactly the same process.

00:03:56.260 --> 00:03:58.945
I would define F of x to
be this type of thing,

00:03:58.945 --> 00:04:01.195
the way we would define it
for the fundamental theorem

00:04:01.195 --> 00:04:02.440
of calculus.

00:04:02.440 --> 00:04:04.550
I would know what
F prime of x was.

00:04:04.550 --> 00:04:07.310
And then I would
have to evaluate F

00:04:07.310 --> 00:04:11.186
at a, at this function up
here, whatever I put up there.

00:04:11.186 --> 00:04:12.560
So in this case
it was x squared.

00:04:12.560 --> 00:04:14.200
I could have made
it natural log x.

00:04:14.200 --> 00:04:17.062
I could've made it some big
polynomial or something more

00:04:17.062 --> 00:04:17.562
complicated.

00:04:17.562 --> 00:04:19.580
Right?

00:04:19.580 --> 00:04:22.790
And once I do that, I just
follow this same process.

00:04:22.790 --> 00:04:24.505
Now, instead of
the x squared here

00:04:24.505 --> 00:04:25.880
I would have that
other function.

00:04:25.880 --> 00:04:27.920
So I'd evaluate capital
F at whatever function

00:04:27.920 --> 00:04:31.960
that is times the
derivative of that function.

00:04:31.960 --> 00:04:33.540
It's exactly the same process.

00:04:33.540 --> 00:04:40.860
So I want to point out that this
is a bigger situation than I

00:04:40.860 --> 00:04:43.070
had before, or a bigger
situation than just

00:04:43.070 --> 00:04:45.220
this little problem.

00:04:45.220 --> 00:04:46.762
So, just so you understand that.

00:04:46.762 --> 00:04:48.570
OK?

00:04:48.570 --> 00:04:50.920
So again, I just want
to say one more time.

00:04:50.920 --> 00:04:52.520
Now you know how
to solve problems

00:04:52.520 --> 00:04:54.540
where you have any other
function of x up here

00:04:54.540 --> 00:04:56.190
and you want to
take the derivative

00:04:56.190 --> 00:04:58.830
of this kind of
expression of an integral

00:04:58.830 --> 00:05:01.000
with another function
of x up there.

00:05:01.000 --> 00:05:03.203
All right, I think
I'll stop there.