0:00:02.460,0:00:06.870
Sometimes we given functions[br]that are actually products of

0:00:06.870,0:00:10.300
functions. That means two[br]functions multiplied together.

0:00:10.300,0:00:13.730
So an example would be Y equals

0:00:13.730,0:00:20.628
X squared. Times by the cosine[br]of 3 X so there we've got a

0:00:20.628,0:00:24.579
product X squared is one[br]function multiplied by the

0:00:24.579,0:00:27.213
cosine of 3 XA second function.

0:00:28.090,0:00:32.098
Usually we write this[br]as UV.

0:00:33.130,0:00:40.540
And there is a formula which we[br]can use to be able to

0:00:40.540,0:00:43.960
differentiate this DY by the X

0:00:43.960,0:00:46.610
is UDVIDX. Plus

0:00:47.270,0:00:51.245
V times by du by

0:00:51.245,0:00:55.850
DX. So every time we've[br]got a product, we can use

0:00:55.850,0:00:56.476
this formula.

0:00:57.570,0:00:59.685
We've identified you with X

0:00:59.685,0:01:04.570
squared. We've identified[br]V with calls 3X, So what

0:01:04.570,0:01:08.935
we need to do now is[br]write down those

0:01:08.935,0:01:12.330
derivatives. So do you[br]buy the X?

0:01:13.400,0:01:16.552
You, we've identified[br]as X squared. So do

0:01:16.552,0:01:19.310
you buy the X is 2 X.

0:01:21.380,0:01:28.968
DV by DX, we identified V as[br]costs 3X, so we need to write

0:01:28.968,0:01:36.556
down the derivative of DV by DX,[br]and we know that that will be

0:01:36.556,0:01:38.724
minus three sign 3X.

0:01:39.560,0:01:41.696
Now we can put these two.

0:01:42.450,0:01:47.866
Together with these two[br]into this formula. So

0:01:47.866,0:01:53.959
let's do that. Why by[br]DX is equal to.

0:01:55.260,0:01:56.010
You

0:01:57.520,0:01:58.660
X squared.

0:02:00.500,0:02:07.826
Times DV by the X so[br]it's times by minus three

0:02:07.826,0:02:09.158
sign 3X.

0:02:10.740,0:02:12.080
Plus the.

0:02:13.760,0:02:17.198
That's cause 3X.

0:02:17.850,0:02:21.144
Times by du

0:02:21.144,0:02:24.790
by DX2X. Now

0:02:24.790,0:02:29.972
this. Looks ugly and really we[br]need to tidy it up a little bit.

0:02:30.860,0:02:35.060
We want to write these terms in[br]a nice order, but at the same

0:02:35.060,0:02:38.660
time we want to try and identify[br]any common factors that there

0:02:38.660,0:02:44.058
are. The reason is that what you[br]might want to do is solve an

0:02:44.058,0:02:48.296
equation like this by putting it[br]equal to 0 to find solutions for

0:02:48.296,0:02:52.208
Maxima and minima. So it's[br]important to be able to spot the

0:02:52.208,0:02:56.120
common factors and take them[br]out. And if we look in this

0:02:56.120,0:03:00.032
term, we've gotten X squared,[br]and in this term we've gotten X,

0:03:00.032,0:03:04.270
so we actually got a common[br]factor of X, and we can take

0:03:04.270,0:03:09.486
that out and put it at the front[br]of the bracket so X and then a

0:03:09.486,0:03:15.030
bracket. Let's think what we've[br]got left here. We've got minus

0:03:15.030,0:03:20.622
three and an X, so we have minus[br]three X sign 3X.

0:03:21.420,0:03:27.440
Plus now we took out the X, so[br]we've got the two left to

0:03:27.440,0:03:33.890
multiply by the cost 3X, and so[br]we have two cars 3X and we just

0:03:33.890,0:03:35.610
close the bracket there.

0:03:38.470,0:03:42.097
We've got that finished and it's[br]in a nice factorize form so that

0:03:42.097,0:03:46.003
if we wanted to do something[br]more with it, as I said, find a

0:03:46.003,0:03:48.793
maximum or minimum we could go[br]on and do that.

0:03:49.730,0:03:52.397
Let's have a look[br]at another example.

0:03:53.530,0:03:56.154
This time, let's have a[br]look at one.

0:04:00.030,0:04:04.200
Where we've got all just[br]functions of X know trig

0:04:04.200,0:04:09.204
functions in just nice functions[br]of X except will make this one

0:04:09.204,0:04:14.625
have a fractional index will[br]make it be to the power 1/2. In

0:04:14.625,0:04:17.127
other words, it's a square root.

0:04:18.700,0:04:25.784
So why is a U times of[br]E? So let's try and get into

0:04:25.784,0:04:29.832
the habit of identifying these[br]functions very specifically.

0:04:32.420,0:04:36.947
So we've identified the[br]two bits that would be

0:04:36.947,0:04:40.468
multiplied together to[br]make the product U&V.

0:04:41.990,0:04:48.820
Now let's do that derivatives.[br]Do you buy the axis

0:04:48.820,0:04:50.869
three X squared?

0:04:51.660,0:04:55.896
Member multiplied by the index[br]and take one off it so that

0:04:55.896,0:04:58.014
gives us a three X squared.

0:04:58.880,0:05:00.488
TV by The X.

0:05:03.500,0:05:04.916
Now this one is going to be a

0:05:04.916,0:05:07.038
little bit different. We've got

0:05:07.038,0:05:10.936
a half there. We've gotta minus[br]X in there.

0:05:12.670,0:05:17.301
Making use of our idea of[br]function of a function, we

0:05:17.301,0:05:21.090
differentiate this inside, so[br]that's the derivative of minus.

0:05:21.090,0:05:22.774
X is minus one.

0:05:25.180,0:05:29.888
And then we must differentiate[br]this so we get a half.

0:05:30.500,0:05:35.901
Times 4 minus X and we[br]take one away from 1/2.

0:05:35.901,0:05:38.356
That gives us minus 1/2.

0:05:40.530,0:05:47.064
Let's quote our formula DY by[br]the X is equal to.

0:05:47.880,0:05:54.630
And we know that it's[br]UDV by the X plus

0:05:54.630,0:05:57.330
VU by The X.

0:06:00.400,0:06:03.689
And we can now plug in[br]the bits that we need,

0:06:03.689,0:06:05.483
so we you is X cubed.

0:06:08.100,0:06:15.660
Times by this minus 1/2[br]of 4 minus X to

0:06:15.660,0:06:17.928
the minus 1/2.

0:06:20.070,0:06:23.090
Plus VDU by DX.

0:06:23.650,0:06:28.557
That's 4 minus[br]X to the half.

0:06:29.870,0:06:34.500
Times by three X[br]squared.

0:06:36.130,0:06:39.382
Now again, this doesn't look a[br]nice lump of algebra really. We

0:06:39.382,0:06:43.176
might have to do something with[br]it later on, and if we did have

0:06:43.176,0:06:46.970
to do something with it later[br]on, we need it to look a lot

0:06:46.970,0:06:49.138
better. We need it to look a lot

0:06:49.138,0:06:53.090
nicer. So I'm going to go[br]into a new sheet and I'm

0:06:53.090,0:06:56.090
going to rewrite this at the[br]top of the page, and then

0:06:56.090,0:06:59.090
we're going to have a look at[br]how we might simplify it.

0:07:00.740,0:07:04.210
So the why by DX?

0:07:05.540,0:07:13.060
Equal 2X cubed times[br]minus half 4 minus

0:07:13.060,0:07:16.820
X to the minus

0:07:16.820,0:07:24.422
1/2. Plus now we have[br]V which was 4 minus X to the

0:07:24.422,0:07:31.086
half times EU by DX, which was[br]three X squared and we want to

0:07:31.086,0:07:32.990
put all this together.

0:07:35.980,0:07:39.796
It's this term which doesn't[br]look awfully good. Let's

0:07:39.796,0:07:47.088
remember that. To the power[br]minus 1/2 means that it's

0:07:47.088,0:07:54.098
in the denominator, so we[br]have minus X cubed over

0:07:54.098,0:08:00.407
2 * 4 minus X[br]to the half plus.

0:08:02.290,0:08:06.500
Haven't done anything with[br]this side yet, so we can

0:08:06.500,0:08:11.973
leave it as it is what we[br]want to do is put everything

0:08:11.973,0:08:17.446
over this denominator. 2 * 4[br]minus X to the half. We treat

0:08:17.446,0:08:22.077
this as though it's over a[br]denominator of one, so I'll

0:08:22.077,0:08:22.919
common denominator.

0:08:24.050,0:08:29.774
Will be 2, four minus X[br]to the half.

0:08:30.940,0:08:35.290
This stays just the same no[br]problem. We haven't done

0:08:35.290,0:08:40.075
anything with the denominator,[br]so we don't do anything with the

0:08:40.075,0:08:43.990
top. Plus this we have[br]multiplied a one here.

0:08:44.630,0:08:48.799
By this so we must multiply[br]everything here. So two times

0:08:48.799,0:08:54.105
the three is going to give us[br]six X squared, and then we have

0:08:54.105,0:08:59.411
4 minus X to the power half[br]multiplied by 4 minus X to the

0:08:59.411,0:09:04.717
power half. So that just gives[br]us 4 minus X, 'cause if we have

0:09:04.717,0:09:09.265
the half and half together,[br]that's one, and to the power one

0:09:09.265,0:09:12.676
is just traditionally written is[br]just 4 minus X.

0:09:14.510,0:09:20.426
Equals the denominator can stay[br]the same. We've now got it all

0:09:20.426,0:09:22.398
over this common denominator.

0:09:23.330,0:09:24.982
And we can look at what we've

0:09:24.982,0:09:29.440
got on the top. Again, we're[br]looking for the idea of a

0:09:29.440,0:09:33.400
common factor. Can we take out[br]a common factor? And yes, we

0:09:33.400,0:09:36.700
can. There's an X squared[br]there, and there's an X

0:09:36.700,0:09:40.990
squared inside that X cubed X[br]cubed is X squared times by X,

0:09:40.990,0:09:43.630
so we can pull out that X[br]squared.

0:09:46.100,0:09:51.590
Then what are we left with? If I[br]take out the X squared out of

0:09:51.590,0:09:57.080
this bracket? I've got 6 times[br]by 4 which is 24 and I've got 6

0:09:57.080,0:09:59.642
times by minus X which is minus

0:09:59.642,0:10:05.476
6. Thanks, but I mustn't forget[br]if I take an X squared out of

0:10:05.476,0:10:09.596
here. I've also got another[br]minus X left. So altogether

0:10:09.596,0:10:10.832
that's minus 7X.

0:10:11.900,0:10:16.100
And again, that's in a nice tidy[br]form, so that if we need too

0:10:16.100,0:10:19.700
next time, we can actually do[br]something else with it. We could

0:10:19.700,0:10:23.300
put this equal to 0 and take the[br]top and solve it.

0:10:26.490,0:10:28.878
We just look at one more

0:10:28.878,0:10:35.914
example. Let's take Y[br]equals 1 minus X cubed

0:10:35.914,0:10:40.940
all times by E to[br]the 2X.

0:10:42.550,0:10:49.298
First of all, let's identify our[br]U and our V, so will put that

0:10:49.298,0:10:54.600
you is going to be equal to 1[br]minus X cubed.

0:10:55.570,0:11:01.294
The V is going to be equal to[br]E to the 2X.

0:11:02.480,0:11:05.370
And now we differentiate you.

0:11:06.140,0:11:11.838
So we have DU by the X[br]is equal to the

0:11:11.838,0:11:17.018
derivative of one is 0[br]because one is a constant

0:11:17.018,0:11:21.680
and the derivative of[br]minus X cubed is minus

0:11:21.680,0:11:23.234
three X squared.

0:11:24.300,0:11:26.286
We want the derivative of V.

0:11:27.190,0:11:32.481
V is E to the 2X and remember[br]that in order to differentiate

0:11:32.481,0:11:35.330
the exponential function, we[br]differentiate the power.

0:11:36.010,0:11:42.029
And we multiply the derivative[br]of that power by E to the 2X

0:11:42.029,0:11:48.974
in this case, so that DV by the[br]X is equal to the derivative of

0:11:48.974,0:11:55.456
the power. The derivative of 2X,[br]which is just two times by E to

0:11:55.456,0:12:02.736
the 2X. And let's remember[br]our formula that if Y is

0:12:02.736,0:12:10.104
equal to U times by V[br]then D why by DX is

0:12:10.104,0:12:11.946
equal to you?

0:12:12.640,0:12:17.878
TV by the[br]X plus VD

0:12:17.878,0:12:20.497
you by X.

0:12:21.810,0:12:28.418
So we've got U. We've got DV[br]by DX. We've got V and we've

0:12:28.418,0:12:33.610
got du by DX here, so let's[br]make those substitutions DY

0:12:33.610,0:12:35.970
by ZX is equal to.

0:12:37.250,0:12:41.237
U to begin with, that's one[br]minus X cubed.

0:12:45.150,0:12:51.810
Times by and we want the[br]derivative of VDV by DX, so

0:12:51.810,0:12:55.695
that's here 2 times E to the

0:12:55.695,0:12:58.530
power 2X. Plus

0:12:59.720,0:13:02.672
and now we want V. That's E to

0:13:02.672,0:13:09.139
the 2X. And now we want the[br]ubaidi X, and that's minus three

0:13:09.139,0:13:14.407
X squared. That's times by minus[br]three X squared. And as we've

0:13:14.407,0:13:19.236
done before, let's look for any[br]common factors that there might

0:13:19.236,0:13:26.236
be. And here we've got a common[br]factor of E to the 2X in each

0:13:26.236,0:13:32.102
term, so let's take that out E[br]to the 2X, which will leave us

0:13:32.102,0:13:38.858
with. Well, here we've got 2[br]times by one minus X cubed, so

0:13:38.858,0:13:42.530
let's do that multiplication. So[br]we've got 2.

0:13:43.190,0:13:50.270
2 times by 1 -[br]2 times by X cubed.

0:13:51.350,0:13:56.782
And then from this one we've got[br]minus three X squared. So we put

0:13:56.782,0:14:01.438
that minus three X squared and[br]its usual. When you've got an

0:14:01.438,0:14:05.318
expression like this, a[br]polynomial in terms of X to

0:14:05.318,0:14:10.750
write it so that we've got the[br]powers of X in some kind of

0:14:10.750,0:14:15.018
order. And in this case I'll[br]write them in ascending order

0:14:15.018,0:14:20.450
from the smallest powers of X up[br]to the largest, so that would be

0:14:20.450,0:14:22.390
2 - 3 X squared.

0:14:22.510,0:14:26.850
Minus two X cubed and being[br]factorized. That derivative is

0:14:26.850,0:14:32.492
now in a position where we can[br]use it for other things, perhaps

0:14:32.492,0:14:35.964
to solve the why by DX equals 0.