[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.46,0:00:06.87,Default,,0000,0000,0000,,Sometimes we given functions\Nthat are actually products of
Dialogue: 0,0:00:06.87,0:00:10.30,Default,,0000,0000,0000,,functions. That means two\Nfunctions multiplied together.
Dialogue: 0,0:00:10.30,0:00:13.73,Default,,0000,0000,0000,,So an example would be Y equals
Dialogue: 0,0:00:13.73,0:00:20.63,Default,,0000,0000,0000,,X squared. Times by the cosine\Nof 3 X so there we've got a
Dialogue: 0,0:00:20.63,0:00:24.58,Default,,0000,0000,0000,,product X squared is one\Nfunction multiplied by the
Dialogue: 0,0:00:24.58,0:00:27.21,Default,,0000,0000,0000,,cosine of 3 XA second function.
Dialogue: 0,0:00:28.09,0:00:32.10,Default,,0000,0000,0000,,Usually we write this\Nas UV.
Dialogue: 0,0:00:33.13,0:00:40.54,Default,,0000,0000,0000,,And there is a formula which we\Ncan use to be able to
Dialogue: 0,0:00:40.54,0:00:43.96,Default,,0000,0000,0000,,differentiate this DY by the X
Dialogue: 0,0:00:43.96,0:00:46.61,Default,,0000,0000,0000,,is UDVIDX. Plus
Dialogue: 0,0:00:47.27,0:00:51.24,Default,,0000,0000,0000,,V times by du by
Dialogue: 0,0:00:51.24,0:00:55.85,Default,,0000,0000,0000,,DX. So every time we've\Ngot a product, we can use
Dialogue: 0,0:00:55.85,0:00:56.48,Default,,0000,0000,0000,,this formula.
Dialogue: 0,0:00:57.57,0:00:59.68,Default,,0000,0000,0000,,We've identified you with X
Dialogue: 0,0:00:59.68,0:01:04.57,Default,,0000,0000,0000,,squared. We've identified\NV with calls 3X, So what
Dialogue: 0,0:01:04.57,0:01:08.94,Default,,0000,0000,0000,,we need to do now is\Nwrite down those
Dialogue: 0,0:01:08.94,0:01:12.33,Default,,0000,0000,0000,,derivatives. So do you\Nbuy the X?
Dialogue: 0,0:01:13.40,0:01:16.55,Default,,0000,0000,0000,,You, we've identified\Nas X squared. So do
Dialogue: 0,0:01:16.55,0:01:19.31,Default,,0000,0000,0000,,you buy the X is 2 X.
Dialogue: 0,0:01:21.38,0:01:28.97,Default,,0000,0000,0000,,DV by DX, we identified V as\Ncosts 3X, so we need to write
Dialogue: 0,0:01:28.97,0:01:36.56,Default,,0000,0000,0000,,down the derivative of DV by DX,\Nand we know that that will be
Dialogue: 0,0:01:36.56,0:01:38.72,Default,,0000,0000,0000,,minus three sign 3X.
Dialogue: 0,0:01:39.56,0:01:41.70,Default,,0000,0000,0000,,Now we can put these two.
Dialogue: 0,0:01:42.45,0:01:47.87,Default,,0000,0000,0000,,Together with these two\Ninto this formula. So
Dialogue: 0,0:01:47.87,0:01:53.96,Default,,0000,0000,0000,,let's do that. Why by\NDX is equal to.
Dialogue: 0,0:01:55.26,0:01:56.01,Default,,0000,0000,0000,,You
Dialogue: 0,0:01:57.52,0:01:58.66,Default,,0000,0000,0000,,X squared.
Dialogue: 0,0:02:00.50,0:02:07.83,Default,,0000,0000,0000,,Times DV by the X so\Nit's times by minus three
Dialogue: 0,0:02:07.83,0:02:09.16,Default,,0000,0000,0000,,sign 3X.
Dialogue: 0,0:02:10.74,0:02:12.08,Default,,0000,0000,0000,,Plus the.
Dialogue: 0,0:02:13.76,0:02:17.20,Default,,0000,0000,0000,,That's cause 3X.
Dialogue: 0,0:02:17.85,0:02:21.14,Default,,0000,0000,0000,,Times by du
Dialogue: 0,0:02:21.14,0:02:24.79,Default,,0000,0000,0000,,by DX2X. Now
Dialogue: 0,0:02:24.79,0:02:29.97,Default,,0000,0000,0000,,this. Looks ugly and really we\Nneed to tidy it up a little bit.
Dialogue: 0,0:02:30.86,0:02:35.06,Default,,0000,0000,0000,,We want to write these terms in\Na nice order, but at the same
Dialogue: 0,0:02:35.06,0:02:38.66,Default,,0000,0000,0000,,time we want to try and identify\Nany common factors that there
Dialogue: 0,0:02:38.66,0:02:44.06,Default,,0000,0000,0000,,are. The reason is that what you\Nmight want to do is solve an
Dialogue: 0,0:02:44.06,0:02:48.30,Default,,0000,0000,0000,,equation like this by putting it\Nequal to 0 to find solutions for
Dialogue: 0,0:02:48.30,0:02:52.21,Default,,0000,0000,0000,,Maxima and minima. So it's\Nimportant to be able to spot the
Dialogue: 0,0:02:52.21,0:02:56.12,Default,,0000,0000,0000,,common factors and take them\Nout. And if we look in this
Dialogue: 0,0:02:56.12,0:03:00.03,Default,,0000,0000,0000,,term, we've gotten X squared,\Nand in this term we've gotten X,
Dialogue: 0,0:03:00.03,0:03:04.27,Default,,0000,0000,0000,,so we actually got a common\Nfactor of X, and we can take
Dialogue: 0,0:03:04.27,0:03:09.49,Default,,0000,0000,0000,,that out and put it at the front\Nof the bracket so X and then a
Dialogue: 0,0:03:09.49,0:03:15.03,Default,,0000,0000,0000,,bracket. Let's think what we've\Ngot left here. We've got minus
Dialogue: 0,0:03:15.03,0:03:20.62,Default,,0000,0000,0000,,three and an X, so we have minus\Nthree X sign 3X.
Dialogue: 0,0:03:21.42,0:03:27.44,Default,,0000,0000,0000,,Plus now we took out the X, so\Nwe've got the two left to
Dialogue: 0,0:03:27.44,0:03:33.89,Default,,0000,0000,0000,,multiply by the cost 3X, and so\Nwe have two cars 3X and we just
Dialogue: 0,0:03:33.89,0:03:35.61,Default,,0000,0000,0000,,close the bracket there.
Dialogue: 0,0:03:38.47,0:03:42.10,Default,,0000,0000,0000,,We've got that finished and it's\Nin a nice factorize form so that
Dialogue: 0,0:03:42.10,0:03:46.00,Default,,0000,0000,0000,,if we wanted to do something\Nmore with it, as I said, find a
Dialogue: 0,0:03:46.00,0:03:48.79,Default,,0000,0000,0000,,maximum or minimum we could go\Non and do that.
Dialogue: 0,0:03:49.73,0:03:52.40,Default,,0000,0000,0000,,Let's have a look\Nat another example.
Dialogue: 0,0:03:53.53,0:03:56.15,Default,,0000,0000,0000,,This time, let's have a\Nlook at one.
Dialogue: 0,0:04:00.03,0:04:04.20,Default,,0000,0000,0000,,Where we've got all just\Nfunctions of X know trig
Dialogue: 0,0:04:04.20,0:04:09.20,Default,,0000,0000,0000,,functions in just nice functions\Nof X except will make this one
Dialogue: 0,0:04:09.20,0:04:14.62,Default,,0000,0000,0000,,have a fractional index will\Nmake it be to the power 1/2. In
Dialogue: 0,0:04:14.62,0:04:17.13,Default,,0000,0000,0000,,other words, it's a square root.
Dialogue: 0,0:04:18.70,0:04:25.78,Default,,0000,0000,0000,,So why is a U times of\NE? So let's try and get into
Dialogue: 0,0:04:25.78,0:04:29.83,Default,,0000,0000,0000,,the habit of identifying these\Nfunctions very specifically.
Dialogue: 0,0:04:32.42,0:04:36.95,Default,,0000,0000,0000,,So we've identified the\Ntwo bits that would be
Dialogue: 0,0:04:36.95,0:04:40.47,Default,,0000,0000,0000,,multiplied together to\Nmake the product U&V.
Dialogue: 0,0:04:41.99,0:04:48.82,Default,,0000,0000,0000,,Now let's do that derivatives.\NDo you buy the axis
Dialogue: 0,0:04:48.82,0:04:50.87,Default,,0000,0000,0000,,three X squared?
Dialogue: 0,0:04:51.66,0:04:55.90,Default,,0000,0000,0000,,Member multiplied by the index\Nand take one off it so that
Dialogue: 0,0:04:55.90,0:04:58.01,Default,,0000,0000,0000,,gives us a three X squared.
Dialogue: 0,0:04:58.88,0:05:00.49,Default,,0000,0000,0000,,TV by The X.
Dialogue: 0,0:05:03.50,0:05:04.92,Default,,0000,0000,0000,,Now this one is going to be a
Dialogue: 0,0:05:04.92,0:05:07.04,Default,,0000,0000,0000,,little bit different. We've got
Dialogue: 0,0:05:07.04,0:05:10.94,Default,,0000,0000,0000,,a half there. We've gotta minus\NX in there.
Dialogue: 0,0:05:12.67,0:05:17.30,Default,,0000,0000,0000,,Making use of our idea of\Nfunction of a function, we
Dialogue: 0,0:05:17.30,0:05:21.09,Default,,0000,0000,0000,,differentiate this inside, so\Nthat's the derivative of minus.
Dialogue: 0,0:05:21.09,0:05:22.77,Default,,0000,0000,0000,,X is minus one.
Dialogue: 0,0:05:25.18,0:05:29.89,Default,,0000,0000,0000,,And then we must differentiate\Nthis so we get a half.
Dialogue: 0,0:05:30.50,0:05:35.90,Default,,0000,0000,0000,,Times 4 minus X and we\Ntake one away from 1/2.
Dialogue: 0,0:05:35.90,0:05:38.36,Default,,0000,0000,0000,,That gives us minus 1/2.
Dialogue: 0,0:05:40.53,0:05:47.06,Default,,0000,0000,0000,,Let's quote our formula DY by\Nthe X is equal to.
Dialogue: 0,0:05:47.88,0:05:54.63,Default,,0000,0000,0000,,And we know that it's\NUDV by the X plus
Dialogue: 0,0:05:54.63,0:05:57.33,Default,,0000,0000,0000,,VU by The X.
Dialogue: 0,0:06:00.40,0:06:03.69,Default,,0000,0000,0000,,And we can now plug in\Nthe bits that we need,
Dialogue: 0,0:06:03.69,0:06:05.48,Default,,0000,0000,0000,,so we you is X cubed.
Dialogue: 0,0:06:08.10,0:06:15.66,Default,,0000,0000,0000,,Times by this minus 1/2\Nof 4 minus X to
Dialogue: 0,0:06:15.66,0:06:17.93,Default,,0000,0000,0000,,the minus 1/2.
Dialogue: 0,0:06:20.07,0:06:23.09,Default,,0000,0000,0000,,Plus VDU by DX.
Dialogue: 0,0:06:23.65,0:06:28.56,Default,,0000,0000,0000,,That's 4 minus\NX to the half.
Dialogue: 0,0:06:29.87,0:06:34.50,Default,,0000,0000,0000,,Times by three X\Nsquared.
Dialogue: 0,0:06:36.13,0:06:39.38,Default,,0000,0000,0000,,Now again, this doesn't look a\Nnice lump of algebra really. We
Dialogue: 0,0:06:39.38,0:06:43.18,Default,,0000,0000,0000,,might have to do something with\Nit later on, and if we did have
Dialogue: 0,0:06:43.18,0:06:46.97,Default,,0000,0000,0000,,to do something with it later\Non, we need it to look a lot
Dialogue: 0,0:06:46.97,0:06:49.14,Default,,0000,0000,0000,,better. We need it to look a lot
Dialogue: 0,0:06:49.14,0:06:53.09,Default,,0000,0000,0000,,nicer. So I'm going to go\Ninto a new sheet and I'm
Dialogue: 0,0:06:53.09,0:06:56.09,Default,,0000,0000,0000,,going to rewrite this at the\Ntop of the page, and then
Dialogue: 0,0:06:56.09,0:06:59.09,Default,,0000,0000,0000,,we're going to have a look at\Nhow we might simplify it.
Dialogue: 0,0:07:00.74,0:07:04.21,Default,,0000,0000,0000,,So the why by DX?
Dialogue: 0,0:07:05.54,0:07:13.06,Default,,0000,0000,0000,,Equal 2X cubed times\Nminus half 4 minus
Dialogue: 0,0:07:13.06,0:07:16.82,Default,,0000,0000,0000,,X to the minus
Dialogue: 0,0:07:16.82,0:07:24.42,Default,,0000,0000,0000,,1/2. Plus now we have\NV which was 4 minus X to the
Dialogue: 0,0:07:24.42,0:07:31.09,Default,,0000,0000,0000,,half times EU by DX, which was\Nthree X squared and we want to
Dialogue: 0,0:07:31.09,0:07:32.99,Default,,0000,0000,0000,,put all this together.
Dialogue: 0,0:07:35.98,0:07:39.80,Default,,0000,0000,0000,,It's this term which doesn't\Nlook awfully good. Let's
Dialogue: 0,0:07:39.80,0:07:47.09,Default,,0000,0000,0000,,remember that. To the power\Nminus 1/2 means that it's
Dialogue: 0,0:07:47.09,0:07:54.10,Default,,0000,0000,0000,,in the denominator, so we\Nhave minus X cubed over
Dialogue: 0,0:07:54.10,0:08:00.41,Default,,0000,0000,0000,,2 * 4 minus X\Nto the half plus.
Dialogue: 0,0:08:02.29,0:08:06.50,Default,,0000,0000,0000,,Haven't done anything with\Nthis side yet, so we can
Dialogue: 0,0:08:06.50,0:08:11.97,Default,,0000,0000,0000,,leave it as it is what we\Nwant to do is put everything
Dialogue: 0,0:08:11.97,0:08:17.45,Default,,0000,0000,0000,,over this denominator. 2 * 4\Nminus X to the half. We treat
Dialogue: 0,0:08:17.45,0:08:22.08,Default,,0000,0000,0000,,this as though it's over a\Ndenominator of one, so I'll
Dialogue: 0,0:08:22.08,0:08:22.92,Default,,0000,0000,0000,,common denominator.
Dialogue: 0,0:08:24.05,0:08:29.77,Default,,0000,0000,0000,,Will be 2, four minus X\Nto the half.
Dialogue: 0,0:08:30.94,0:08:35.29,Default,,0000,0000,0000,,This stays just the same no\Nproblem. We haven't done
Dialogue: 0,0:08:35.29,0:08:40.08,Default,,0000,0000,0000,,anything with the denominator,\Nso we don't do anything with the
Dialogue: 0,0:08:40.08,0:08:43.99,Default,,0000,0000,0000,,top. Plus this we have\Nmultiplied a one here.
Dialogue: 0,0:08:44.63,0:08:48.80,Default,,0000,0000,0000,,By this so we must multiply\Neverything here. So two times
Dialogue: 0,0:08:48.80,0:08:54.10,Default,,0000,0000,0000,,the three is going to give us\Nsix X squared, and then we have
Dialogue: 0,0:08:54.10,0:08:59.41,Default,,0000,0000,0000,,4 minus X to the power half\Nmultiplied by 4 minus X to the
Dialogue: 0,0:08:59.41,0:09:04.72,Default,,0000,0000,0000,,power half. So that just gives\Nus 4 minus X, 'cause if we have
Dialogue: 0,0:09:04.72,0:09:09.26,Default,,0000,0000,0000,,the half and half together,\Nthat's one, and to the power one
Dialogue: 0,0:09:09.26,0:09:12.68,Default,,0000,0000,0000,,is just traditionally written is\Njust 4 minus X.
Dialogue: 0,0:09:14.51,0:09:20.43,Default,,0000,0000,0000,,Equals the denominator can stay\Nthe same. We've now got it all
Dialogue: 0,0:09:20.43,0:09:22.40,Default,,0000,0000,0000,,over this common denominator.
Dialogue: 0,0:09:23.33,0:09:24.98,Default,,0000,0000,0000,,And we can look at what we've
Dialogue: 0,0:09:24.98,0:09:29.44,Default,,0000,0000,0000,,got on the top. Again, we're\Nlooking for the idea of a
Dialogue: 0,0:09:29.44,0:09:33.40,Default,,0000,0000,0000,,common factor. Can we take out\Na common factor? And yes, we
Dialogue: 0,0:09:33.40,0:09:36.70,Default,,0000,0000,0000,,can. There's an X squared\Nthere, and there's an X
Dialogue: 0,0:09:36.70,0:09:40.99,Default,,0000,0000,0000,,squared inside that X cubed X\Ncubed is X squared times by X,
Dialogue: 0,0:09:40.99,0:09:43.63,Default,,0000,0000,0000,,so we can pull out that X\Nsquared.
Dialogue: 0,0:09:46.10,0:09:51.59,Default,,0000,0000,0000,,Then what are we left with? If I\Ntake out the X squared out of
Dialogue: 0,0:09:51.59,0:09:57.08,Default,,0000,0000,0000,,this bracket? I've got 6 times\Nby 4 which is 24 and I've got 6
Dialogue: 0,0:09:57.08,0:09:59.64,Default,,0000,0000,0000,,times by minus X which is minus
Dialogue: 0,0:09:59.64,0:10:05.48,Default,,0000,0000,0000,,6. Thanks, but I mustn't forget\Nif I take an X squared out of
Dialogue: 0,0:10:05.48,0:10:09.60,Default,,0000,0000,0000,,here. I've also got another\Nminus X left. So altogether
Dialogue: 0,0:10:09.60,0:10:10.83,Default,,0000,0000,0000,,that's minus 7X.
Dialogue: 0,0:10:11.90,0:10:16.10,Default,,0000,0000,0000,,And again, that's in a nice tidy\Nform, so that if we need too
Dialogue: 0,0:10:16.10,0:10:19.70,Default,,0000,0000,0000,,next time, we can actually do\Nsomething else with it. We could
Dialogue: 0,0:10:19.70,0:10:23.30,Default,,0000,0000,0000,,put this equal to 0 and take the\Ntop and solve it.
Dialogue: 0,0:10:26.49,0:10:28.88,Default,,0000,0000,0000,,We just look at one more
Dialogue: 0,0:10:28.88,0:10:35.91,Default,,0000,0000,0000,,example. Let's take Y\Nequals 1 minus X cubed
Dialogue: 0,0:10:35.91,0:10:40.94,Default,,0000,0000,0000,,all times by E to\Nthe 2X.
Dialogue: 0,0:10:42.55,0:10:49.30,Default,,0000,0000,0000,,First of all, let's identify our\NU and our V, so will put that
Dialogue: 0,0:10:49.30,0:10:54.60,Default,,0000,0000,0000,,you is going to be equal to 1\Nminus X cubed.
Dialogue: 0,0:10:55.57,0:11:01.29,Default,,0000,0000,0000,,The V is going to be equal to\NE to the 2X.
Dialogue: 0,0:11:02.48,0:11:05.37,Default,,0000,0000,0000,,And now we differentiate you.
Dialogue: 0,0:11:06.14,0:11:11.84,Default,,0000,0000,0000,,So we have DU by the X\Nis equal to the
Dialogue: 0,0:11:11.84,0:11:17.02,Default,,0000,0000,0000,,derivative of one is 0\Nbecause one is a constant
Dialogue: 0,0:11:17.02,0:11:21.68,Default,,0000,0000,0000,,and the derivative of\Nminus X cubed is minus
Dialogue: 0,0:11:21.68,0:11:23.23,Default,,0000,0000,0000,,three X squared.
Dialogue: 0,0:11:24.30,0:11:26.29,Default,,0000,0000,0000,,We want the derivative of V.
Dialogue: 0,0:11:27.19,0:11:32.48,Default,,0000,0000,0000,,V is E to the 2X and remember\Nthat in order to differentiate
Dialogue: 0,0:11:32.48,0:11:35.33,Default,,0000,0000,0000,,the exponential function, we\Ndifferentiate the power.
Dialogue: 0,0:11:36.01,0:11:42.03,Default,,0000,0000,0000,,And we multiply the derivative\Nof that power by E to the 2X
Dialogue: 0,0:11:42.03,0:11:48.97,Default,,0000,0000,0000,,in this case, so that DV by the\NX is equal to the derivative of
Dialogue: 0,0:11:48.97,0:11:55.46,Default,,0000,0000,0000,,the power. The derivative of 2X,\Nwhich is just two times by E to
Dialogue: 0,0:11:55.46,0:12:02.74,Default,,0000,0000,0000,,the 2X. And let's remember\Nour formula that if Y is
Dialogue: 0,0:12:02.74,0:12:10.10,Default,,0000,0000,0000,,equal to U times by V\Nthen D why by DX is
Dialogue: 0,0:12:10.10,0:12:11.95,Default,,0000,0000,0000,,equal to you?
Dialogue: 0,0:12:12.64,0:12:17.88,Default,,0000,0000,0000,,TV by the\NX plus VD
Dialogue: 0,0:12:17.88,0:12:20.50,Default,,0000,0000,0000,,you by X.
Dialogue: 0,0:12:21.81,0:12:28.42,Default,,0000,0000,0000,,So we've got U. We've got DV\Nby DX. We've got V and we've
Dialogue: 0,0:12:28.42,0:12:33.61,Default,,0000,0000,0000,,got du by DX here, so let's\Nmake those substitutions DY
Dialogue: 0,0:12:33.61,0:12:35.97,Default,,0000,0000,0000,,by ZX is equal to.
Dialogue: 0,0:12:37.25,0:12:41.24,Default,,0000,0000,0000,,U to begin with, that's one\Nminus X cubed.
Dialogue: 0,0:12:45.15,0:12:51.81,Default,,0000,0000,0000,,Times by and we want the\Nderivative of VDV by DX, so
Dialogue: 0,0:12:51.81,0:12:55.70,Default,,0000,0000,0000,,that's here 2 times E to the
Dialogue: 0,0:12:55.70,0:12:58.53,Default,,0000,0000,0000,,power 2X. Plus
Dialogue: 0,0:12:59.72,0:13:02.67,Default,,0000,0000,0000,,and now we want V. That's E to
Dialogue: 0,0:13:02.67,0:13:09.14,Default,,0000,0000,0000,,the 2X. And now we want the\Nubaidi X, and that's minus three
Dialogue: 0,0:13:09.14,0:13:14.41,Default,,0000,0000,0000,,X squared. That's times by minus\Nthree X squared. And as we've
Dialogue: 0,0:13:14.41,0:13:19.24,Default,,0000,0000,0000,,done before, let's look for any\Ncommon factors that there might
Dialogue: 0,0:13:19.24,0:13:26.24,Default,,0000,0000,0000,,be. And here we've got a common\Nfactor of E to the 2X in each
Dialogue: 0,0:13:26.24,0:13:32.10,Default,,0000,0000,0000,,term, so let's take that out E\Nto the 2X, which will leave us
Dialogue: 0,0:13:32.10,0:13:38.86,Default,,0000,0000,0000,,with. Well, here we've got 2\Ntimes by one minus X cubed, so
Dialogue: 0,0:13:38.86,0:13:42.53,Default,,0000,0000,0000,,let's do that multiplication. So\Nwe've got 2.
Dialogue: 0,0:13:43.19,0:13:50.27,Default,,0000,0000,0000,,2 times by 1 -\N2 times by X cubed.
Dialogue: 0,0:13:51.35,0:13:56.78,Default,,0000,0000,0000,,And then from this one we've got\Nminus three X squared. So we put
Dialogue: 0,0:13:56.78,0:14:01.44,Default,,0000,0000,0000,,that minus three X squared and\Nits usual. When you've got an
Dialogue: 0,0:14:01.44,0:14:05.32,Default,,0000,0000,0000,,expression like this, a\Npolynomial in terms of X to
Dialogue: 0,0:14:05.32,0:14:10.75,Default,,0000,0000,0000,,write it so that we've got the\Npowers of X in some kind of
Dialogue: 0,0:14:10.75,0:14:15.02,Default,,0000,0000,0000,,order. And in this case I'll\Nwrite them in ascending order
Dialogue: 0,0:14:15.02,0:14:20.45,Default,,0000,0000,0000,,from the smallest powers of X up\Nto the largest, so that would be
Dialogue: 0,0:14:20.45,0:14:22.39,Default,,0000,0000,0000,,2 - 3 X squared.
Dialogue: 0,0:14:22.51,0:14:26.85,Default,,0000,0000,0000,,Minus two X cubed and being\Nfactorized. That derivative is
Dialogue: 0,0:14:26.85,0:14:32.49,Default,,0000,0000,0000,,now in a position where we can\Nuse it for other things, perhaps
Dialogue: 0,0:14:32.49,0:14:35.96,Default,,0000,0000,0000,,to solve the why by DX equals 0.