0:00:01.440,0:00:06.156
In this video, we're going to be[br]looking at how we might

0:00:06.156,0:00:11.038
differentiate. Functions of Y[br]with respect to X.

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Let's begin by looking at an[br]equation like this.

0:00:18.060,0:00:23.702
X squared[br]Plus Y

0:00:23.702,0:00:27.005
squared. Minus

0:00:27.005,0:00:33.930
4X. +5 Y[br]minus 8 equals

0:00:33.930,0:00:40.109
0. Now here the X[br]is and wiser all tangled

0:00:40.109,0:00:43.175
together and the wise Y squared

0:00:43.175,0:00:48.506
and Y. It will be quite[br]difficult to rearrange this, so

0:00:48.506,0:00:51.194
it said Y equals a function of

0:00:51.194,0:00:58.594
X. We could perhaps given values[br]of X workout what values of why

0:00:58.594,0:01:01.204
were and thereby draw graph.

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But nevertheless, these things[br]are intimately connected, and

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differentiating something like[br]this is going to be much harder,

0:01:09.820,0:01:16.050
so it would seem than if it just[br]said why equals some function of

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X. But that's what we're going[br]to have a look at in this video.

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How can we differentiate a[br]function and equation that looks

0:01:24.045,0:01:27.180
like this where the X and wiser[br]all tangled up together?

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We're going to use something[br]known as the chain rule or

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function of a function. It's got[br]those two names. Chain rule an

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function of a function.

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I'm going to try and stick with[br]using the name chain rule, but

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you may know it as function of a

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function. There is a video which[br]covers this particular rule

0:01:48.938,0:01:51.800
explicitly. So I'm just going to

0:01:51.800,0:01:58.994
revise it. So let's take[br]an example such as Y

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equals. 5 + 2[br]X to the 10th power. And

0:02:06.316,0:02:11.932
let's say I want to[br]differentiate this. Then the

0:02:11.932,0:02:15.676
chain rule says that if I

0:02:15.676,0:02:21.493
port. You[br]equals 5

0:02:21.493,0:02:24.575
+ 2

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X. Then

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Why will be equal to you to[br]the power 10?

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And we can form[br]the why by DX

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by doing DY by[br]du times DU by

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DX and it's this.

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That is the chain rule.

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Or some of you may know it[br]function of a function.

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So this is what we've got to do.[br]We've got to work out the why by

0:03:04.290,0:03:08.630
DU and workout. Do you buy the[br]X? So I'm going to turn the

0:03:08.630,0:03:12.350
page. I'm going to virtually[br]write this out again in order to

0:03:12.350,0:03:16.070
make sure that I've got the[br]space in which to do it.

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So we begin with Y equals

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5. Plus 2X to the

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power 10. We put[br]you equals 5 +

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2 X and then[br]Y is equal to

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U to the power

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10. DY by the X[br]is given by the Y by

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EU times. Do you buy the[br]X and we can workout each

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of these two divided by DU&U[br]by DX. So let's have a

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look at that why by EU[br]is the derivative of U to

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the power 10 with respect to

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you. Which is 10 U to the power[br]9. Remember we multiply by the

0:04:19.478,0:04:25.442
index and take one of the index[br]to give us nine and so that's

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10. 5 + 2 X[br]to the Power 9 replacing EU

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by 5 + 2 X.

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And we can calculate you[br]by The X.

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That's the derivative of 5[br]+ 2 X.

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With respect to X, and that's[br]just two because the derivative

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of five 5 is a constant stats[br]zero, the derivative of two X is

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just two. So now we know what[br]divided by du is. It's this and

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we know what do you buy the[br]access. It's this so we can

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write those in.

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10 Times 5[br]+ 2 X to the

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9th. Times by two and[br]of course, the two times by 10

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gives us 20.

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So we've used our[br]chain rule in order

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to be able to[br]differentiate this function.

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So let me just[br]write our chain rule

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down here DY by[br]X is equal to

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Y by DU times[br]du by X.

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Now. Let's suppose that we

0:06:08.840,0:06:14.557
had zed. And said was a[br]function of Why?

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Then D zed by the[br]X would be equal to

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D zed. Bye.

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DY times DY

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by X. Using[br]our chain rule again.

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Let's take an example.[br]Let's say that zed is

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equal to Y squared.

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Then these Ed by the[br]X would be equal to.

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The derivative of Y squared[br]with respect to Y.

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Times DY by the X[br]and the derivative of Y

0:07:06.920,0:07:13.790
squared with respect to Y[br]is equal to two Y

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times DY by X.

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Notice what we seem to have done[br]is just differentiated. Why

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square root respect to Y and[br]multiplied by DY by the X? And

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I'll draw attention to that[br]again later on. For now, let's

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have a look at some examples.[br]Begin with this one Y squared.

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Plus X cubed.

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Minus Y cubed plus 6.

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Equals 3 Y. We want to[br]differentiate this with respect

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to X, so we're looking for the[br]derivative of Y squared with

0:08:01.920,0:08:03.450
respect to X.

0:08:03.980,0:08:11.068
Derivative of X cubed[br]with respect to X.

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Derivative of Y cubed with[br]respect to X.

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Derivative of the six with[br]respect to X and the derivative

0:08:22.558,0:08:29.682
of three Y with respect to X.[br]Not remember how we said we

0:08:29.682,0:08:36.806
would do this? We said we will[br]take the derivative of Y squared

0:08:36.806,0:08:38.998
with respect to Y.

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And multiply by DY by the X that[br]was our chain rule.

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Plus this is straightforward,[br]don't need to worry about this.

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This is the derivative of X[br]cubed with respect to X. We know

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that is 3 X squared multiplied[br]by the index and take one away

0:09:01.694,0:09:07.982
minus. And here again, we've got[br]to apply our chain rule. So

0:09:07.982,0:09:13.174
we've got the derivative of Y[br]cubed with respect to Y.

0:09:13.880,0:09:17.324
Times by DY. By

0:09:17.324,0:09:21.210
The X. Plus

0:09:22.020,0:09:25.972
Now the derivative of six well[br]six is just a constant, so we

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know its derivative is 0.

0:09:28.690,0:09:34.282
Equals and again we want the[br]derivative of three Y, so our

0:09:34.282,0:09:39.874
chain rule tells us this is the[br]derivative of three Y with

0:09:39.874,0:09:43.136
respect to Y times DY by DX.

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Now we can go back and work out[br]each of these derivatives with

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respect to Y, so that will give

0:09:51.190,0:09:57.140
us 2Y. DYIDX[br]plus three

0:09:57.140,0:10:04.052
X squared.[br]Minus the derivative of Y Cube

0:10:04.052,0:10:11.228
with respect to Y is 3 Y[br]squared divided by DX. We can

0:10:11.228,0:10:12.884
ignore the zero.

0:10:13.460,0:10:20.324
And the derivative of three Y[br]with respect to Y is 3 times

0:10:20.324,0:10:21.908
divided by DX.

0:10:22.930,0:10:29.235
Now. If we get together all the[br]terms that have a DY by the X in

0:10:29.235,0:10:35.660
them. Then, having done that, we[br]can sort out what divided by DX

0:10:35.660,0:10:40.940
actually is, so I'm going to[br]gather together all the terms in

0:10:40.940,0:10:48.420
do I buy DX over on this side of[br]the equation so I can keep .3 X

0:10:48.420,0:10:53.700
squared there on its own. So I[br]have three X squared equals.

0:10:54.600,0:11:01.739
Now here on this side I've[br]got 3D Y by X.

0:11:02.700,0:11:08.524
I'm going to take this away from[br]each side, so that's minus two

0:11:08.524,0:11:15.558
Y. DY by X and I'm going[br]to add this term to both sides

0:11:15.558,0:11:22.248
plus three Y squared DY by The[br]X. what I can see here is that

0:11:22.248,0:11:28.938
I've got a common factor of the[br]why by DX that I can take out,

0:11:28.938,0:11:33.844
so that's what we're going to do[br]next, so will have.

0:11:34.970,0:11:38.410
Three X squared equals.

0:11:39.920,0:11:45.951
Bracket. And taking out[br]that common factor of the why

0:11:45.951,0:11:51.117
by DX. So let's just go back[br]and have a lot. What do I

0:11:51.117,0:11:54.069
buy? DX was multiplying. It[br]was multiplying A3.

0:11:55.150,0:12:02.248
It was multiplying a minus two Y[br]and it was multiplying A plus

0:12:02.248,0:12:09.346
three Y squared, so it was[br]multiplying a 3 - 2 Y and

0:12:09.346,0:12:17.062
a plus. 3 Y squared. So now[br]we can get divided by DX on

0:12:17.062,0:12:23.494
its own if we divide throughout[br]by this expression sode, why by

0:12:23.494,0:12:29.390
DX is equal to three X squared[br]and dividing throughout dividing

0:12:29.390,0:12:35.286
both sides by 3 - 2 Y[br]plus three Y squared.

0:12:36.260,0:12:41.130
And then we've got our[br]expression for DY by DX.

0:12:41.820,0:12:44.816
That was a reasonably[br]straightforward example. The

0:12:44.816,0:12:49.096
work that many complications and[br]it followed very directly from

0:12:49.096,0:12:54.660
our first look at this. So now[br]let's look at a slightly more

0:12:54.660,0:12:58.512
complicated example, one where[br]in fact we've got other

0:12:58.512,0:13:04.574
functions. Of X&Y. So in this[br]case will start up with a sign

0:13:04.574,0:13:09.161
Y where we've got the Axis and[br]the wise actually combined

0:13:09.161,0:13:13.331
together, so we've got X[br]squared times by Y cubed.

0:13:15.040,0:13:22.540
Minus calls X and let's say[br]equals 2 Yi. Want to be

0:13:22.540,0:13:29.415
able to differentiate this with[br]respect to X, so that's the

0:13:29.415,0:13:36.290
derivative of sine Y with[br]respect to X plus the derivative

0:13:36.290,0:13:39.415
of X squared Y cubed.

0:13:40.010,0:13:47.990
With respect to X minus the[br]derivative of Cos X with respect

0:13:47.990,0:13:55.970
to X equals the derivative of[br]two Y with respect to X.

0:13:56.970,0:14:02.510
Now let's remember what our[br]function of a function rule

0:14:02.510,0:14:09.158
tells us that this is done[br]as the derivative of sine Y

0:14:09.158,0:14:14.144
with respect to Y times by[br]DY by DX.

0:14:16.340,0:14:21.870
This one. Bit of a problem[br]'cause This is X squared times

0:14:21.870,0:14:24.768
by Y cubed. So it's a product.

0:14:25.370,0:14:31.498
It's a U times by AV, so let me[br]just write a little U over the

0:14:31.498,0:14:33.796
top and a little V there.

0:14:34.670,0:14:41.276
You is X squared and[br]V is Y cubed.

0:14:41.810,0:14:43.518
So this is plus.

0:14:44.330,0:14:49.136
Let's remember how we[br]differentiate a product. We take

0:14:49.136,0:14:55.847
you. And we multiply it by[br]the derivative of V. That's the

0:14:55.847,0:14:59.903
derivative of Y cubed with[br]respect to X.

0:15:01.110,0:15:08.190
Plus the an we multiply that[br]by the derivative of U, which

0:15:08.190,0:15:13.500
in this case is the derivative[br]of X squared.

0:15:14.100,0:15:21.211
Minus now we can do this one.[br]The derivative of Cos X with

0:15:21.211,0:15:27.775
respect to X. The derivative of[br]causes minus sign, so a minus

0:15:27.775,0:15:34.339
and minus makes a plus sign. X[br]equals the derivative of and

0:15:34.339,0:15:40.903
again my chain rule tells me[br]that this is the derivative of

0:15:40.903,0:15:44.732
two Y with respect to Y times

0:15:44.732,0:15:48.508
by. Divide by The X.

0:15:50.150,0:15:54.605
So this has been much more[br]complicated, but notice how it

0:15:54.605,0:15:58.250
follows the standard rules that[br]we've already got for

0:15:58.250,0:16:01.895
differentiation. So now the[br]derivative of sine wired with

0:16:01.895,0:16:04.730
respect to Y is just cause why.

0:16:05.420,0:16:08.549
DY by X.

0:16:09.340,0:16:13.580
Plus X[br]squared

0:16:14.840,0:16:20.660
Times by now this will be the[br]derivative of Y cubed with

0:16:20.660,0:16:22.115
respect to Y.

0:16:22.660,0:16:26.698
Times by DY. By The[br]X.

0:16:28.100,0:16:35.250
Plus this one Y cubed times by[br]now the derivative of X squared

0:16:35.250,0:16:39.100
with respect to X is just 2X.

0:16:39.820,0:16:42.118
Plus sign X.

0:16:42.880,0:16:47.430
We've already done that one[br]equals and hear the derivative

0:16:47.430,0:16:53.800
of two Y with respect to Y.[br]These two times DY by The X.

0:16:55.080,0:17:02.140
Almost done now we still[br]got a little bit of

0:17:02.140,0:17:09.200
differentiation in here to do[br]so. Let's do that cause

0:17:09.200,0:17:16.260
YDY by DX plus I[br]think I can fairly safely

0:17:16.260,0:17:23.320
remove these brackets now. X[br]squared times 3 Y squared

0:17:23.320,0:17:28.968
DY by X +2 XY[br]cubed plus sign.

0:17:28.990,0:17:34.625
X equals 2 DY[br]by The X.

0:17:35.240,0:17:39.465
Where at the same stage as we[br]were last time we've got the

0:17:39.465,0:17:43.040
differentiation Dom and it's[br]this thing. The why by DX that

0:17:43.040,0:17:47.265
we want. So what we gotta do is[br]get all those terms that

0:17:47.265,0:17:51.490
involved why by DX on one side[br]of the equation and the other

0:17:51.490,0:17:55.390
terms on the other side. Now[br]these are the two terms that

0:17:55.390,0:17:59.940
don't have a divided by DX in[br]them, so I'm going to keep them

0:17:59.940,0:18:03.515
at this side. So it's two XY[br]cubed plus sign X.

0:18:05.780,0:18:11.128
Two, XY[br]cubed plus

0:18:11.128,0:18:13.802
sign X

0:18:13.802,0:18:20.312
equals. Let's just go back and[br]see what we've got. We've got a

0:18:20.312,0:18:22.242
2 divided by DX here.

0:18:22.280,0:18:28.370
2.[br]DY by The X.

0:18:29.090,0:18:34.576
And we're going to bring these[br]two terms over to this side by

0:18:34.576,0:18:39.218
taking them away from both[br]sides. So we're going to take

0:18:39.218,0:18:42.594
away 'cause why do I buy the X?

0:18:43.330,0:18:49.921
On both sides, minus cause YDY[br]by the X, and then we're going

0:18:49.921,0:18:56.512
to take away the other term.[br]This is the X squared times by

0:18:56.512,0:19:02.089
three Y squared divided by DX,[br]right that a little bit

0:19:02.089,0:19:08.680
differently. When I do it, so we[br]have minus three X squared Y

0:19:08.680,0:19:10.708
squared divided by X.

0:19:11.990,0:19:17.464
Again, we see we've got a set of[br]terms here, each with divided by

0:19:17.464,0:19:19.419
DX as a common factor.

0:19:19.920,0:19:26.526
Two, XY cubed plus sign[br]X is equal to.

0:19:27.500,0:19:33.704
Let's take out this common[br]factor of DY by The X.

0:19:34.300,0:19:39.085
Well, it's multiplying two, so[br]we've got a two there. It's

0:19:39.085,0:19:43.435
multiplying minus cause Y, so[br]we've gotta minus cause why

0:19:43.435,0:19:47.785
there? And it's multiplying[br]minus three X squared Y squared

0:19:47.785,0:19:53.005
minus three X squared Y squared.[br]And now finally we can get

0:19:53.005,0:19:57.790
divided by DX on its own,[br]because we can divide throughout

0:19:57.790,0:20:02.140
divide both sides of the[br]equation by what's in this

0:20:02.140,0:20:05.185
bracket. So we have two XY cubed

0:20:05.185,0:20:12.623
plus sign. X all over 2[br]minus cause Y minus three

0:20:12.623,0:20:15.531
X squared Y squared.

0:20:16.090,0:20:19.058
And there's our divide by[br]the eggs.

0:20:20.290,0:20:22.649
Now let's just look back at this

0:20:22.649,0:20:28.390
one. Look at all this[br]complicated differentiation that

0:20:28.390,0:20:30.490
we had here.

0:20:31.120,0:20:35.476
Now I went through it slowly and[br]carefully, but when we're doing

0:20:35.476,0:20:39.469
calculations on our own, we[br]might make slips. It would be

0:20:39.469,0:20:43.462
helpful if we could automate[br]some of this process so it

0:20:43.462,0:20:47.092
instead of writing down the[br]derivative of sine wave with

0:20:47.092,0:20:50.722
respect to X is and going[br]through the chain rule.

0:20:51.430,0:20:56.302
We went automatically to all we[br]need to do is differentiate sign

0:20:56.302,0:21:01.174
wired with respect to Y. That's[br]cause Y an multiplied by divided

0:21:01.174,0:21:06.452
by DX. So we automate we would[br]miss out at least these two

0:21:06.452,0:21:10.512
lines and go direct from there[br]to there. Similarly, the

0:21:10.512,0:21:15.790
derivative of two Y with respect[br]to X would be the derivative of

0:21:15.790,0:21:21.474
two Y with respect to Y two[br]times divided by DX. And so we

0:21:21.474,0:21:23.098
go direct from there.

0:21:23.120,0:21:28.688
To there, so let's have a look[br]at that in another example.

0:21:29.460,0:21:37.122
So we'll take[br]Y squared plus

0:21:37.122,0:21:44.784
X cubed minus[br]XY plus cause

0:21:44.784,0:21:50.780
Y. Equals note this time[br]we're going to go.

0:21:51.310,0:21:55.534
Direct to the differentiation,[br]we're not going to go by the

0:21:55.534,0:22:00.142
chain rule. We're going to use[br]it, of course, but we aren't

0:22:00.142,0:22:04.750
going to write it down, so we[br]want to differentiate this with

0:22:04.750,0:22:10.510
respect to X. So the first term[br]is Y squared, so we know that to

0:22:10.510,0:22:13.966
differentiate Y squared with[br]respect to X, we differentiate

0:22:13.966,0:22:16.270
this with respect to why that's

0:22:16.270,0:22:22.105
too why. And we multiply by[br]DY by DX.

0:22:22.690,0:22:28.722
Now we want the derivative of X[br]cubed with respect to X, so

0:22:28.722,0:22:31.140
that's 3X. Squared

0:22:31.920,0:22:33.950
Minus.

0:22:35.060,0:22:39.948
Now I am going to write this[br]down in full because it's the

0:22:39.948,0:22:44.836
derivative of XY with respect to[br]X. It's a product again, it's X

0:22:44.836,0:22:45.964
times by Y.

0:22:46.960,0:22:52.680
Plus, the derivative of Cos Y[br]with respect to X, which we do

0:22:52.680,0:22:58.400
as a derivative of cause Y with[br]respect to Y that's minus Sign

0:22:58.400,0:23:05.760
Y. Times DY by DX,[br]the derivative of 0 is

0:23:05.760,0:23:11.560
just zero. So let's get[br]these two terms together because

0:23:11.560,0:23:18.085
they both got the why by DX. So[br]I have two Y minus Sign Y.

0:23:18.180,0:23:24.668
Times Ty by DX[br]plus three X squared.

0:23:25.400,0:23:32.860
Minus. Now this is[br]a product. It is a U times

0:23:32.860,0:23:36.764
by AV. So we

0:23:36.764,0:23:43.880
want you. Times the derivative[br]of Y with respect to X, which

0:23:43.880,0:23:47.527
is just the wise by The X.

0:23:48.050,0:23:55.670
Plus V, which is Y Times[br]the derivative of X, which is

0:23:55.670,0:23:59.720
just one. Equals[br]0.

0:24:00.940,0:24:06.040
So we see here that we've got[br]another term now involving the

0:24:06.040,0:24:12.415
why by DX it's minus X, so we[br]can put that in the bracket so

0:24:12.415,0:24:15.815
we can have two Y minus sign Y

0:24:15.815,0:24:22.996
minus X. DY by the[br]X plus three X squared

0:24:22.996,0:24:30.216
and then minus this term[br]here minus Y equals 0.

0:24:31.690,0:24:37.670
If we take this over to the[br]other side, In other words, we

0:24:37.670,0:24:43.650
take this away from both sides[br]and add that to both sides. You

0:24:43.650,0:24:50.550
have two Y minus sign Y minus X[br]times DY by X is equal to.

0:24:50.550,0:24:56.070
Adding this one to both sides.[br]Why taking this one away from

0:24:56.070,0:24:58.370
both sides, we get that.

0:24:58.960,0:25:03.952
Now it's clear how we would[br]finish this off. We will take

0:25:03.952,0:25:07.696
this factor here and divide both[br]sides by it.

0:25:08.400,0:25:15.540
So we get DY by[br]the X was equal 2.

0:25:16.280,0:25:23.624
Just go back. We've got Y[br]minus three X squared to go

0:25:23.624,0:25:27.296
in the numerator on the top.

0:25:28.420,0:25:34.660
And this factor to go on the[br]bottom in the denominator two Y

0:25:34.660,0:25:36.100
minus Sign Y.

0:25:36.610,0:25:40.600
Minus.

0:25:40.600,0:25:44.500
X. And then

0:25:44.500,0:25:51.552
we have. Our derivative,[br]Let's just take one

0:25:51.552,0:25:55.995
more example.[br]Y

0:25:55.995,0:26:01.405
cubed[br]minus

0:26:01.405,0:26:04.860
X. Sign

0:26:04.860,0:26:08.786
Y. Plus Y squared

0:26:08.786,0:26:13.170
over X. Equals[br]8.

0:26:14.520,0:26:19.976
Again, all the axes and Wise[br]bundled up together if possible.

0:26:19.976,0:26:27.416
We want to be able to do this[br]directly. We want to be able to

0:26:27.416,0:26:31.880
differentiate it straight away[br]without going through the chain

0:26:31.880,0:26:38.328
rule. So the derivative of Y[br]cubed with respect to Y times by

0:26:38.328,0:26:44.776
DY by DX. So that's three Y[br]squared times DY by X minus.

0:26:44.840,0:26:51.171
We want the derivative of this.[br]This is a product so let's see

0:26:51.171,0:26:58.476
if we can do it all in one[br]go. Again you want X Times the

0:26:58.476,0:27:03.833
derivative of sine Y with[br]respect to X. That's X times

0:27:03.833,0:27:05.781
cause YDY by X.

0:27:06.310,0:27:12.790
And now we want sign Y times the[br]derivative of X so that sign Y

0:27:12.790,0:27:15.814
and the derivative of X is just

0:27:15.814,0:27:20.832
one. This one is a bit trickier.[br]This is a quotient Y squared

0:27:20.832,0:27:25.785
over X, so we want plus. Now[br]let's remember what we do with

0:27:25.785,0:27:32.920
the quotient. It's V which is on[br]the bottom that's X Times the

0:27:32.920,0:27:39.280
derivative of what's on the top.[br]The derivative of Y squared with

0:27:39.280,0:27:45.110
respect to X, which is the[br]derivative of Y squared with

0:27:45.110,0:27:52.000
respect to Y2Y times DY by the[br]X minus Y squared times the

0:27:52.000,0:27:57.300
derivative of V, which in this[br]case is just X.

0:27:57.300,0:28:03.470
All over V squared, which is[br]X squared equals 0.

0:28:04.430,0:28:06.418
Now this needs a little bit of

0:28:06.418,0:28:10.530
tidying up. We've got a[br]denominator here that we can

0:28:10.530,0:28:13.410
probably multiply out by, so[br]let's do that.

0:28:13.990,0:28:21.116
And do some tidying on the way,[br]so this will be three X squared

0:28:21.116,0:28:24.170
Y squared DY by The X.

0:28:25.070,0:28:32.966
Minus X cubed cause[br]YDY by the X

0:28:32.966,0:28:36.914
minus X squared Sign

0:28:36.914,0:28:44.182
Y. +2 XYDY by[br]X minus Y squared equals 0,

0:28:44.182,0:28:50.760
so I've multiplied throughout by[br]this X squared so the X

0:28:50.760,0:28:55.544
squared is appeared there,[br]multiplying that it's appeared

0:28:55.544,0:29:00.926
there inside that X cubed,[br]multiplying that it's appeared

0:29:00.926,0:29:07.504
there, multiplying the sign Y,[br]and it's gone. From here, 'cause

0:29:07.504,0:29:09.896
we've multiplied by so.

0:29:09.930,0:29:15.110
Multiplying and dividing by, in[br]effect, leaving this on changed.

0:29:15.110,0:29:21.844
Now let's get together all the[br]terms in DY by DX, so we

0:29:21.844,0:29:28.578
have the why by DX times this[br]term, three X squared Y squared.

0:29:29.100,0:29:33.000
This term minus X cubed

0:29:33.000,0:29:39.958
cause why? This[br]term +2 XY.

0:29:41.630,0:29:46.306
And here I've got minus X[br]squared sign Y or I think I want

0:29:46.306,0:29:50.982
to add that to the other side.[br]So that's plus X squared sign Y,

0:29:50.982,0:29:54.990
and here I've got minus Y[br]squared. Again, I think I want

0:29:54.990,0:29:59.666
to add that to both sides, so I[br]get plus Y squared over there.

0:30:00.340,0:30:08.060
Now why by X is[br]equal to X squared sign

0:30:08.060,0:30:15.780
Y plus Y squared in[br]the numerator and dividing by

0:30:15.780,0:30:22.728
this expression as the[br]denominator. Three X squared Y

0:30:22.728,0:30:28.904
squared minus X cubed cause[br]Y, +2 XY.

0:30:28.940,0:30:31.706
Notice how much shorter[br]automating that

0:30:31.706,0:30:35.394
Differentiation's made? What's[br]quite a complicated problem, and

0:30:35.394,0:30:40.465
that's something you want to[br]work at. Trying to automate your

0:30:40.465,0:30:45.536
differentiation so you don't[br]have to go through the rules and

0:30:45.536,0:30:47.841
write them down every time.