WEBVTT

00:00:07.680 --> 00:00:09.860
DAVID JORDAN: Hello, and
welcome back to recitation.

00:00:09.860 --> 00:00:11.651
Today the problem I'd
like to work with you

00:00:11.651 --> 00:00:14.950
is about computing
partial derivatives

00:00:14.950 --> 00:00:16.930
and the total differential.

00:00:16.930 --> 00:00:20.760
So we have a function z which
is x squared plus y squared.

00:00:20.760 --> 00:00:23.100
So it depends on the
two variables x and y.

00:00:23.100 --> 00:00:25.890
Now the variables
x and y themselves

00:00:25.890 --> 00:00:29.210
depend on two auxiliary
variables, u and v.

00:00:29.210 --> 00:00:31.520
So that's the
setup that we have.

00:00:31.520 --> 00:00:35.500
So in part a, we just want to
compute the total differential

00:00:35.500 --> 00:00:38.420
dz in terms of dx and dy.

00:00:38.420 --> 00:00:41.220
So u and v aren't going
to enter into the picture.

00:00:41.220 --> 00:00:44.580
And then in part b,
we're going to compute

00:00:44.580 --> 00:00:47.470
the partial derivative
partial z partial u

00:00:47.470 --> 00:00:50.270
in two different ways.

00:00:50.270 --> 00:00:53.270
First, we're going to compute
it using the chain rule.

00:00:53.270 --> 00:00:57.940
And then we're going to compute
it using total differentials.

00:00:57.940 --> 00:01:00.380
And so we'll substitute
in some of the work

00:01:00.380 --> 00:01:02.610
that we had in a
to solve that part.

00:01:02.610 --> 00:01:06.690
So why don't you pause the video
now and work on the problem.

00:01:06.690 --> 00:01:08.440
We'll check back and
we'll do it together.

00:01:15.606 --> 00:01:16.480
Hi, and welcome back.

00:01:16.480 --> 00:01:17.230
Let's get started.

00:01:20.550 --> 00:01:23.130
So first, computing
a is not so bad.

00:01:25.614 --> 00:01:27.030
So we just need
to first remember,

00:01:27.030 --> 00:01:29.980
what does it mean to compute
the total differential?

00:01:29.980 --> 00:01:35.170
So the total
differential dz is just

00:01:35.170 --> 00:01:40.830
the partial derivative of z
in the x-direction dx plus

00:01:40.830 --> 00:01:43.670
z in the y-direction dy.

00:01:43.670 --> 00:01:44.430
OK?

00:01:44.430 --> 00:01:48.250
So now, looking at our
formula here for z,

00:01:48.250 --> 00:01:51.900
we have-- so the
partial derivative of z

00:01:51.900 --> 00:01:58.020
in the x-direction is
2x, so this is 2x dx.

00:01:58.020 --> 00:02:01.466
And the partial derivative
of z in the y is 2y,

00:02:01.466 --> 00:02:05.860
so we have 2y dy.

00:02:05.860 --> 00:02:09.310
OK, and that's all
we have to do for a.

00:02:09.310 --> 00:02:16.310
Now for b, we want to compute
this partial derivative

00:02:16.310 --> 00:02:17.510
in two different ways.

00:02:17.510 --> 00:02:18.780
First, using the chain rule.

00:02:18.780 --> 00:02:20.910
So let's remember what
the chain rule says.

00:02:20.910 --> 00:02:23.360
So whenever I think
about the chain rule,

00:02:23.360 --> 00:02:30.666
I like to draw this
dependency graph.

00:02:30.666 --> 00:02:31.165
OK?

00:02:31.165 --> 00:02:33.040
And this is just a
way for me to organize

00:02:33.040 --> 00:02:36.150
how the different variables
depend on one another.

00:02:36.150 --> 00:02:39.270
So at the top, we have z.

00:02:39.270 --> 00:02:47.620
And z is a function of
x and y, but x is itself

00:02:47.620 --> 00:02:53.000
a function of both u
and v, and y is also

00:02:53.000 --> 00:02:57.490
a function of u and v.
So z depends on x and y,

00:02:57.490 --> 00:03:00.380
and x and y each jointly
depend on u and v.

00:03:00.380 --> 00:03:01.770
So it's a little
bit complicated,

00:03:01.770 --> 00:03:03.540
the relationships here.

00:03:03.540 --> 00:03:06.620
So now, what the
chain rule says is

00:03:06.620 --> 00:03:11.620
that if we take a partial
derivative-- partial z

00:03:11.620 --> 00:03:16.250
partial u-- we have to go
through our dependency graph.

00:03:16.250 --> 00:03:19.020
Every way that we
can get from z to u,

00:03:19.020 --> 00:03:23.980
we get a term in our summation
for each one of those.

00:03:23.980 --> 00:03:27.240
So for instance, z
goes to x goes to u.

00:03:27.240 --> 00:03:32.420
So that means that we
have partial z partial x,

00:03:32.420 --> 00:03:36.130
partial x partial u.

00:03:36.130 --> 00:03:41.470
And then we can also go
z goes to y goes to u.

00:03:41.470 --> 00:03:51.000
And that will give us partial z
partial y, partial y partial u.

00:03:51.000 --> 00:03:54.380
And now these partials
are ones that we can just

00:03:54.380 --> 00:03:56.590
compute from our formulas.

00:03:56.590 --> 00:03:59.220
So for instance,
partial z partial x

00:03:59.220 --> 00:04:00.820
is 2x, which we computed.

00:04:03.990 --> 00:04:08.710
Now partial x partial u, we have
to remember that x is defined

00:04:08.710 --> 00:04:10.770
as u squared minus v squared.

00:04:10.770 --> 00:04:13.570
And so partial x
partial u, that's 2u.

00:04:18.480 --> 00:04:25.280
Partial z partial y, again,
is this 2y that we computed.

00:04:25.280 --> 00:04:33.360
And partial y partial u is v.
This v is just because u was

00:04:33.360 --> 00:04:35.980
u*v, and we take a partial
in the u-direction.

00:04:35.980 --> 00:04:36.480
OK.

00:04:39.590 --> 00:04:48.510
So altogether this
is 4u*x plus 2v*y,

00:04:48.510 --> 00:04:51.410
and that's our
partial derivative.

00:04:51.410 --> 00:04:55.500
So notice that, you know,
x is a function of u and v.

00:04:55.500 --> 00:04:58.780
So if I really wanted
to, I could substitute

00:04:58.780 --> 00:05:01.483
for x its formula for
u and v, but that's not

00:05:01.483 --> 00:05:02.191
really necessary.

00:05:02.191 --> 00:05:05.940
You know, what's interesting
about these problems is

00:05:05.940 --> 00:05:09.360
how the differentials
depend on one another,

00:05:09.360 --> 00:05:11.410
and I'm perfectly happy
with an answer that

00:05:11.410 --> 00:05:12.870
has mixed variables like this.

00:05:12.870 --> 00:05:14.070
That's fine.

00:05:14.070 --> 00:05:20.470
So now, let's go
over here and let's

00:05:20.470 --> 00:05:24.440
see if we can get the
same answer by using

00:05:24.440 --> 00:05:25.910
total differentials.

00:05:25.910 --> 00:05:28.210
Now, I have to say
that the chain rule

00:05:28.210 --> 00:05:34.450
that we used on the
previous problem,

00:05:34.450 --> 00:05:38.620
it's the quickest way to
do these sorts of things.

00:05:38.620 --> 00:05:42.250
I like to do total differentials
if I have some time

00:05:42.250 --> 00:05:44.750
to actually explore the problem
and get comfortable with it.

00:05:44.750 --> 00:05:47.007
I prefer to use total
differentials because I think

00:05:47.007 --> 00:05:48.090
it's a little bit clearer.

00:05:48.090 --> 00:05:51.520
Somehow, this chain
rule it's just, to me,

00:05:51.520 --> 00:05:55.090
it's just a prescription,
it's not an explanation.

00:05:55.090 --> 00:05:58.330
So why don't we compute
some total differentials.

00:05:58.330 --> 00:06:04.060
So we already saw-- let
me just repeat over here.

00:06:04.060 --> 00:06:09.630
We already saw that dz
is 2x dx plus 2y dy.

00:06:12.150 --> 00:06:14.600
OK.

00:06:14.600 --> 00:06:16.245
Now, we want to
use the fact that x

00:06:16.245 --> 00:06:18.780
is itself a function of
u and v. So that's what

00:06:18.780 --> 00:06:19.700
we need to do now.

00:06:19.700 --> 00:06:33.660
So that tells us that dx is 2u
du minus 2v dv in the same way.

00:06:33.660 --> 00:06:36.630
And dy.

00:06:36.630 --> 00:06:40.110
So remember, y was u*v.
So taking d of u*v,

00:06:40.110 --> 00:06:47.170
we get v du plus u dv.

00:06:47.170 --> 00:06:47.870
OK?

00:06:47.870 --> 00:06:49.540
So now, so what we've
done is we've just

00:06:49.540 --> 00:06:54.870
listed out all of the
total differentials.

00:06:54.870 --> 00:06:57.730
And the nice thing about
this is once you've

00:06:57.730 --> 00:07:01.860
done these computations,
now it's just substitution.

00:07:01.860 --> 00:07:05.950
So what we really want
to know is how does z

00:07:05.950 --> 00:07:09.180
depend on u and v. And
so all we need to do

00:07:09.180 --> 00:07:12.880
is substitute in our
formulas for dx here.

00:07:12.880 --> 00:07:17.240
So this tells us
that dz is-- OK,

00:07:17.240 --> 00:07:23.020
so we have 2x-- instead of
dx, we just plug in here--

00:07:23.020 --> 00:07:30.820
so we have 2u du minus 2v dv.

00:07:30.820 --> 00:07:32.830
So that was this term.

00:07:32.830 --> 00:07:40.990
And now we have
plus 2y-- and now

00:07:40.990 --> 00:07:47.791
we just plug in this--
so v du plus u dv.

00:07:47.791 --> 00:07:48.290
You see?

00:07:48.290 --> 00:07:50.779
It's just substitution.

00:07:50.779 --> 00:07:52.570
So then now, we just
expand everything out.

00:07:56.630 --> 00:07:58.570
And so we get-- OK,
so let's collect

00:07:58.570 --> 00:08:01.020
all the things involving du.

00:08:01.020 --> 00:08:05.850
So if we collect all the
things involving du, we have--

00:08:05.850 --> 00:08:15.890
2 times 2 times x times
u-- 4x*u plus 2y*v.

00:08:15.890 --> 00:08:17.240
This whole quantity times du.

00:08:19.900 --> 00:08:26.000
And then if we collect the
terms in dv, we have 2y*u.

00:08:26.000 --> 00:08:42.700
So that's coming from here, and
then we have a minus 4x*v. OK?

00:08:42.700 --> 00:08:47.550
And now what that tells
us is that-- so let's

00:08:47.550 --> 00:08:52.940
just remember that
one definition

00:08:52.940 --> 00:08:55.480
of the partial derivative
partial z partial u

00:08:55.480 --> 00:08:56.350
is this coefficient.

00:08:58.870 --> 00:09:02.740
So if I go over here, if we
write the total differential

00:09:02.740 --> 00:09:11.000
dz, we can write that as
partial z partial u du

00:09:11.000 --> 00:09:18.920
plus partial z partial v dv.

00:09:18.920 --> 00:09:20.280
Right?

00:09:20.280 --> 00:09:20.930
Well, look.

00:09:20.930 --> 00:09:23.830
What we have here
on these two sides

00:09:23.830 --> 00:09:25.500
is essentially the
same expression.

00:09:25.500 --> 00:09:27.090
So that means if
we want to compute

00:09:27.090 --> 00:09:32.590
partial z partial u, then that's
just equal to this coefficient

00:09:32.590 --> 00:09:34.300
here.

00:09:34.300 --> 00:09:41.150
So we get that partial z
partial u is 4x*u plus 2--

00:09:41.150 --> 00:09:44.810
that should be v.
One of those is an x.

00:09:44.810 --> 00:09:45.320
Let's see.

00:09:45.320 --> 00:09:47.650
So where did this come from.

00:09:47.650 --> 00:09:49.590
Yeah, one of those
is an x, sorry--

00:09:49.590 --> 00:09:50.570
SPEAKER 1: It's a y.

00:09:50.570 --> 00:09:53.110
DAVID JORDAN: --is a y.

00:09:53.110 --> 00:09:55.130
2v*y, OK.

00:09:55.130 --> 00:09:58.950
Now just as a sanity
check, why don't we

00:09:58.950 --> 00:10:00.620
go back to the
middle of the board,

00:10:00.620 --> 00:10:03.200
and we'll see that we
got the same thing.

00:10:03.200 --> 00:10:07.350
So 4x*u plus 2v*y, that's what
we concluded for partial z

00:10:07.350 --> 00:10:09.000
partial u.

00:10:09.000 --> 00:10:11.740
And then going back to the
middle of the board, that's we

00:10:11.740 --> 00:10:13.620
found again.

00:10:13.620 --> 00:10:17.690
So let's just go over
the two different methods

00:10:17.690 --> 00:10:18.510
and compare them.

00:10:18.510 --> 00:10:22.420
So if I'm in a rush to
do a computation-- maybe

00:10:22.420 --> 00:10:24.610
I'm taking an
exam-- I definitely

00:10:24.610 --> 00:10:28.160
think it's the quickest
to just compute,

00:10:28.160 --> 00:10:30.600
to figure out what the
dependency of the variable is,

00:10:30.600 --> 00:10:32.380
and I use this dependency graph.

00:10:32.380 --> 00:10:35.620
And then I just trace
all the paths from z

00:10:35.620 --> 00:10:40.260
to the independent variable
u that I'm interested in.

00:10:40.260 --> 00:10:44.070
And then I multiply all
the partial derivatives

00:10:44.070 --> 00:10:48.570
that correspond to each edge
and I get an expression.

00:10:48.570 --> 00:10:52.230
Now if I have more
time, then I really

00:10:52.230 --> 00:10:54.630
prefer to use the method
of total differentials

00:10:54.630 --> 00:10:57.290
that we did on the third board.

00:10:57.290 --> 00:10:59.981
I like it, because once you
do some simple calculus,

00:10:59.981 --> 00:11:02.765
and then after that it's
just, it's basic algebra.

00:11:07.900 --> 00:11:09.520
I find that I'm
less likely to make

00:11:09.520 --> 00:11:11.080
a mistake doing that method.

00:11:11.080 --> 00:11:12.724
But as you saw, it
involves computing

00:11:12.724 --> 00:11:14.640
a lot more derivatives
that we didn't actually

00:11:14.640 --> 00:11:15.800
use in the final answer.

00:11:15.800 --> 00:11:19.740
For instance, when we
computed total differentials,

00:11:19.740 --> 00:11:22.679
we got an expression
for partial z partial v

00:11:22.679 --> 00:11:24.220
at the end of the
day, even though we

00:11:24.220 --> 00:11:25.261
weren't asked to do that.

00:11:25.261 --> 00:11:29.140
So it's lengthier, but I
think more conceptually

00:11:29.140 --> 00:11:30.360
straightforward.

00:11:30.360 --> 00:11:32.860
So I think I'll
leave it at that.