WEBVTT
00:00:07.170 --> 00:00:08.906
PROFESSOR: Welcome
back to recitation.
00:00:08.906 --> 00:00:10.280
In this video
segment we're going
00:00:10.280 --> 00:00:12.020
to look at the chain rule.
00:00:12.020 --> 00:00:14.250
And specifically we're
going to answer a question
00:00:14.250 --> 00:00:16.420
that I've placed on the board.
00:00:16.420 --> 00:00:18.710
I want you to find
two values for theta
00:00:18.710 --> 00:00:21.620
so that d d theta of
the function cosine
00:00:21.620 --> 00:00:23.870
squared theta to
the fourth equals 0.
00:00:23.870 --> 00:00:26.430
At this point I should also
point out a few things.
00:00:26.430 --> 00:00:30.080
One, theta is a
variable in this case.
00:00:30.080 --> 00:00:31.730
And if you haven't
seen it before,
00:00:31.730 --> 00:00:33.960
frequently we use it
when we're dealing
00:00:33.960 --> 00:00:35.470
with trigonometric functions.
00:00:35.470 --> 00:00:39.490
Theta often represents the
variable that measures angle.
00:00:39.490 --> 00:00:41.430
So if you haven't
seen theta before,
00:00:41.430 --> 00:00:42.950
this is what it looks like.
00:00:42.950 --> 00:00:44.080
That's how you say it.
00:00:44.080 --> 00:00:46.910
And I'm actually doing the
same things we've done before.
00:00:46.910 --> 00:00:50.140
I'm taking the derivative with
respect to the variable theta,
00:00:50.140 --> 00:00:52.481
of a function of theta.
00:00:52.481 --> 00:00:53.980
The other thing I
want to point out,
00:00:53.980 --> 00:00:56.980
which I think we know
already, but just to be sure,
00:00:56.980 --> 00:00:59.360
is it that this squared
here, cosine squared theta
00:00:59.360 --> 00:01:03.150
to the fourth, means I'm taking
cosine of theta to the fourth
00:01:03.150 --> 00:01:04.820
and I'm actually squaring it.
00:01:04.820 --> 00:01:07.510
So with that
knowledge I would like
00:01:07.510 --> 00:01:10.520
us to use the chain rule to
find two values for theta
00:01:10.520 --> 00:01:13.340
where this derivative
is equal to 0.
00:01:13.340 --> 00:01:15.900
So I'll give you a
moment to take a stab
00:01:15.900 --> 00:01:18.690
at finding the derivative,
setting it equal to 0
00:01:18.690 --> 00:01:23.590
and finding some values for
theta that make this equation,
00:01:23.590 --> 00:01:25.190
the derivative equal 0.
00:01:25.190 --> 00:01:29.990
And we'll come back and then
I'll work it out as well.
00:01:29.990 --> 00:01:32.540
OK, so the first thing
we obviously need to do
00:01:32.540 --> 00:01:35.240
is be able to take the
derivative of this function
00:01:35.240 --> 00:01:36.360
on the left-hand side.
00:01:36.360 --> 00:01:39.480
And I should say it's a
little more complicated
00:01:39.480 --> 00:01:42.690
than an example you saw in the
lecture, because in the lecture
00:01:42.690 --> 00:01:44.950
you were given an example
with just two functions.
00:01:44.950 --> 00:01:47.400
So I want to write it a
little differently just
00:01:47.400 --> 00:01:49.780
to show very obviously what
the three functions are
00:01:49.780 --> 00:01:50.830
we're composing.
00:01:50.830 --> 00:01:53.400
So this function
I can rewrite it
00:01:53.400 --> 00:01:55.963
as cosine theta to the fourth
does that look like a 4,
00:01:55.963 --> 00:02:04.362
there we go and then I
square that whole thing.
00:02:04.362 --> 00:02:05.820
That's really what
the function is.
00:02:05.820 --> 00:02:07.040
OK?
00:02:07.040 --> 00:02:10.690
So we can see what is the
outermost function here?
00:02:10.690 --> 00:02:14.070
The outermost function
is actually the quantity
00:02:14.070 --> 00:02:15.150
of something squared.
00:02:15.150 --> 00:02:17.660
So the outermost
function is x squared.
00:02:17.660 --> 00:02:20.520
What's the next function
in, in this composition?
00:02:20.520 --> 00:02:23.450
The next function in
is the cosine function.
00:02:23.450 --> 00:02:27.120
And then the last function in is
the function taking something,
00:02:27.120 --> 00:02:28.537
raising it to the fourth.
00:02:28.537 --> 00:02:30.078
So it's very important
you understand
00:02:30.078 --> 00:02:32.040
sort of the
composition, which is
00:02:32.040 --> 00:02:35.430
the outermost function, which
is the innermost function?
00:02:35.430 --> 00:02:37.770
In order to do this chain rule.
00:02:37.770 --> 00:02:40.220
Now as you saw in
recitation, you
00:02:40.220 --> 00:02:44.850
were given the example dy-- you
had y as as a function of t--
00:02:44.850 --> 00:02:46.130
sorry, not in recitation.
00:02:46.130 --> 00:02:50.352
In the lecture, you were
given y as a function of t
00:02:50.352 --> 00:02:51.810
or a function of
x and then you had
00:02:51.810 --> 00:02:54.330
to put one other
variable in the middle.
00:02:54.330 --> 00:02:57.310
Here we're going to have a
composition of three functions.
00:02:57.310 --> 00:03:00.650
So we need to have two other
things sort of in the middle.
00:03:00.650 --> 00:03:02.340
So let's write this out.
00:03:02.340 --> 00:03:06.130
First, the outermost function,
we'll call the whole thing y.
00:03:06.130 --> 00:03:11.440
So y is equal to x squared
is the outermost function.
00:03:11.440 --> 00:03:14.690
So then this whole
thing is x now.
00:03:14.690 --> 00:03:19.305
So then we'll write x
is equal to cosine of w.
00:03:21.980 --> 00:03:27.602
And then w is equal to
theta to the fourth.
00:03:27.602 --> 00:03:28.600
OK.
00:03:28.600 --> 00:03:33.440
Again this is what you
saw before in the lecture.
00:03:33.440 --> 00:03:36.050
So you write the
outermost function.
00:03:36.050 --> 00:03:38.510
And then that
function is a function
00:03:38.510 --> 00:03:41.129
of cosine of another
one, cosine of w,
00:03:41.129 --> 00:03:42.920
and w is a function of
theta to the fourth.
00:03:42.920 --> 00:03:45.290
So if I put all
these back together,
00:03:45.290 --> 00:03:49.100
I have theta to the fourth in
here, and then I square that,
00:03:49.100 --> 00:03:51.420
and I come back to
the function I wanted.
00:03:51.420 --> 00:03:56.650
Now we know from the lecture
what we need to do to find,
00:03:56.650 --> 00:03:59.061
essentially, dy/d theta.
00:03:59.061 --> 00:04:00.310
That's what we're looking for.
00:04:03.750 --> 00:04:08.380
So dy/d theta, you'll see,
you remember from the lecture,
00:04:08.380 --> 00:04:18.149
should be dy/dx times
dx/dw times dw/d theta.
00:04:18.149 --> 00:04:19.524
So it's slightly
more complicated
00:04:19.524 --> 00:04:21.565
than we saw before because
there's one more term.
00:04:21.565 --> 00:04:22.970
OK.
00:04:22.970 --> 00:04:24.780
So now let's work
out what these are.
00:04:24.780 --> 00:04:27.820
Well dy/dx is fairly
straightforward.
00:04:27.820 --> 00:04:29.900
dy/dx is just 2x.
00:04:33.010 --> 00:04:35.330
And then dx/dw, well
what's the derivative
00:04:35.330 --> 00:04:36.300
of the cosine function?
00:04:36.300 --> 00:04:39.090
The derivative of
cosine is negative sine.
00:04:39.090 --> 00:04:45.310
So this is times negative
sine of w times--
00:04:45.310 --> 00:04:47.330
and what's dw/d theta?
00:04:47.330 --> 00:04:49.800
So w is the function
theta to the fourth.
00:04:49.800 --> 00:04:54.870
So dw/d theta is 4
theta to the third.
00:04:54.870 --> 00:04:58.670
Now when you look at
this you should remember,
00:04:58.670 --> 00:05:00.830
x we sort of inserted
into the problem
00:05:00.830 --> 00:05:03.527
to make the problem easier, and
w we inserted into the problem
00:05:03.527 --> 00:05:05.360
to make the problem
easier for us to follow.
00:05:05.360 --> 00:05:07.520
So we don't want all
these x's and w's.
00:05:07.520 --> 00:05:11.260
We want everything in terms
of, in terms of theta.
00:05:11.260 --> 00:05:12.390
But what is w?
00:05:12.390 --> 00:05:14.110
Well w is theta to the fourth.
00:05:14.110 --> 00:05:16.600
So I can replace that
by theta to the fourth.
00:05:16.600 --> 00:05:18.650
And what is x?
00:05:18.650 --> 00:05:20.590
x is cosine w.
00:05:20.590 --> 00:05:22.160
But w is theta to the fourth.
00:05:22.160 --> 00:05:24.355
So x is actually cosine
of theta to the fourth.
00:05:24.355 --> 00:05:26.000
OK?
00:05:26.000 --> 00:05:33.990
So I get 2 cosine of theta
to the fourth times negative
00:05:33.990 --> 00:05:38.060
sine-- w again is theta
to the fourth-- of theta
00:05:38.060 --> 00:05:43.710
to the fourth times
4 theta to the third.
00:05:43.710 --> 00:05:45.910
Let's make this nicer.
00:05:45.910 --> 00:05:47.590
I'll bring the
coefficient and the theta
00:05:47.590 --> 00:05:50.150
to the third in front and
this minus sign in front.
00:05:50.150 --> 00:05:55.700
I get negative 8 theta to
the third cosine of theta
00:05:55.700 --> 00:06:00.340
to fourth sine of
theta to the fourth.
00:06:00.340 --> 00:06:02.610
OK, and then the problem
asks to find where
00:06:02.610 --> 00:06:04.370
the derivative is equal to 0.
00:06:04.370 --> 00:06:05.502
Find two values.
00:06:05.502 --> 00:06:07.210
Now why did I ask you
to find two values?
00:06:07.210 --> 00:06:08.770
Because one is very easy.
00:06:08.770 --> 00:06:11.640
If this thing is set
equal to 0, one value
00:06:11.640 --> 00:06:13.190
should stand out
right away for theta
00:06:13.190 --> 00:06:15.750
that makes this product 0.
00:06:15.750 --> 00:06:17.840
And that is theta equals 0.
00:06:17.840 --> 00:06:20.320
So theta equals 0 is
our easiest answer.
00:06:26.690 --> 00:06:28.174
So if you didn't
do that, then you
00:06:28.174 --> 00:06:29.590
wanted the more
challenging stuff,
00:06:29.590 --> 00:06:32.310
you could have done
the other things also.
00:06:32.310 --> 00:06:34.570
So what about cosine
theta to the fourth?
00:06:34.570 --> 00:06:36.990
If we want a product of
three things to be 0,
00:06:36.990 --> 00:06:39.430
then at least one
of them has to be 0.
00:06:39.430 --> 00:06:42.410
So I could have set cosine
theta to the fourth equal to 0.
00:06:48.560 --> 00:06:51.870
And for what values
of-- an angle
00:06:51.870 --> 00:06:54.840
I should just say-- for what
values of theta to the fourth
00:06:54.840 --> 00:06:56.340
is this going to be 0?
00:06:56.340 --> 00:07:01.550
Well this means that
theta to the fourth
00:07:01.550 --> 00:07:06.682
is equal to either maybe pi over
2, or you could add another pi.
00:07:06.682 --> 00:07:07.182
OK?
00:07:07.182 --> 00:07:11.690
So we could say, well let's
just say that's one example.
00:07:11.690 --> 00:07:14.830
Theta to the fourth equals
pi over 2 would work right?
00:07:14.830 --> 00:07:18.230
Because cosine of pi
over 2 is equal to 0.
00:07:18.230 --> 00:07:20.200
OK?
00:07:20.200 --> 00:07:22.740
And so then we could
have said theta is
00:07:22.740 --> 00:07:29.390
equal to pi over 2 to the 1/4.
00:07:29.390 --> 00:07:31.470
That's another example there.
00:07:31.470 --> 00:07:34.797
We could have also done, we
could have added pi to this
00:07:34.797 --> 00:07:35.880
and gotten another answer.
00:07:35.880 --> 00:07:37.360
But I only asked for two.
00:07:37.360 --> 00:07:40.294
So I guess that I'm
allowed to stop there.