[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.44,0:00:06.16,Default,,0000,0000,0000,,In this video, we're going to be\Nlooking at how we might
Dialogue: 0,0:00:06.16,0:00:11.04,Default,,0000,0000,0000,,differentiate. Functions of Y\Nwith respect to X.
Dialogue: 0,0:00:11.80,0:00:17.39,Default,,0000,0000,0000,,Let's begin by looking at an\Nequation like this.
Dialogue: 0,0:00:18.06,0:00:23.70,Default,,0000,0000,0000,,X squared\NPlus Y
Dialogue: 0,0:00:23.70,0:00:27.00,Default,,0000,0000,0000,,squared. Minus
Dialogue: 0,0:00:27.00,0:00:33.93,Default,,0000,0000,0000,,4X. +5 Y\Nminus 8 equals
Dialogue: 0,0:00:33.93,0:00:40.11,Default,,0000,0000,0000,,0. Now here the X\Nis and wiser all tangled
Dialogue: 0,0:00:40.11,0:00:43.18,Default,,0000,0000,0000,,together and the wise Y squared
Dialogue: 0,0:00:43.18,0:00:48.51,Default,,0000,0000,0000,,and Y. It will be quite\Ndifficult to rearrange this, so
Dialogue: 0,0:00:48.51,0:00:51.19,Default,,0000,0000,0000,,it said Y equals a function of
Dialogue: 0,0:00:51.19,0:00:58.59,Default,,0000,0000,0000,,X. We could perhaps given values\Nof X workout what values of why
Dialogue: 0,0:00:58.59,0:01:01.20,Default,,0000,0000,0000,,were and thereby draw graph.
Dialogue: 0,0:01:01.81,0:01:05.37,Default,,0000,0000,0000,,But nevertheless, these things\Nare intimately connected, and
Dialogue: 0,0:01:05.37,0:01:09.82,Default,,0000,0000,0000,,differentiating something like\Nthis is going to be much harder,
Dialogue: 0,0:01:09.82,0:01:16.05,Default,,0000,0000,0000,,so it would seem than if it just\Nsaid why equals some function of
Dialogue: 0,0:01:16.05,0:01:21.20,Default,,0000,0000,0000,,X. But that's what we're going\Nto have a look at in this video.
Dialogue: 0,0:01:21.20,0:01:24.04,Default,,0000,0000,0000,,How can we differentiate a\Nfunction and equation that looks
Dialogue: 0,0:01:24.04,0:01:27.18,Default,,0000,0000,0000,,like this where the X and wiser\Nall tangled up together?
Dialogue: 0,0:01:27.77,0:01:31.55,Default,,0000,0000,0000,,We're going to use something\Nknown as the chain rule or
Dialogue: 0,0:01:31.55,0:01:35.68,Default,,0000,0000,0000,,function of a function. It's got\Nthose two names. Chain rule an
Dialogue: 0,0:01:35.68,0:01:37.06,Default,,0000,0000,0000,,function of a function.
Dialogue: 0,0:01:37.76,0:01:41.22,Default,,0000,0000,0000,,I'm going to try and stick with\Nusing the name chain rule, but
Dialogue: 0,0:01:41.22,0:01:43.35,Default,,0000,0000,0000,,you may know it as function of a
Dialogue: 0,0:01:43.35,0:01:48.94,Default,,0000,0000,0000,,function. There is a video which\Ncovers this particular rule
Dialogue: 0,0:01:48.94,0:01:51.80,Default,,0000,0000,0000,,explicitly. So I'm just going to
Dialogue: 0,0:01:51.80,0:01:58.99,Default,,0000,0000,0000,,revise it. So let's take\Nan example such as Y
Dialogue: 0,0:01:58.99,0:02:06.32,Default,,0000,0000,0000,,equals. 5 + 2\NX to the 10th power. And
Dialogue: 0,0:02:06.32,0:02:11.93,Default,,0000,0000,0000,,let's say I want to\Ndifferentiate this. Then the
Dialogue: 0,0:02:11.93,0:02:15.68,Default,,0000,0000,0000,,chain rule says that if I
Dialogue: 0,0:02:15.68,0:02:21.49,Default,,0000,0000,0000,,port. You\Nequals 5
Dialogue: 0,0:02:21.49,0:02:24.58,Default,,0000,0000,0000,,+ 2
Dialogue: 0,0:02:24.58,0:02:27.86,Default,,0000,0000,0000,,X. Then
Dialogue: 0,0:02:29.15,0:02:34.33,Default,,0000,0000,0000,,Why will be equal to you to\Nthe power 10?
Dialogue: 0,0:02:34.83,0:02:41.25,Default,,0000,0000,0000,,And we can form\Nthe why by DX
Dialogue: 0,0:02:41.25,0:02:47.68,Default,,0000,0000,0000,,by doing DY by\Ndu times DU by
Dialogue: 0,0:02:47.68,0:02:50.89,Default,,0000,0000,0000,,DX and it's this.
Dialogue: 0,0:02:51.56,0:02:54.17,Default,,0000,0000,0000,,That is the chain rule.
Dialogue: 0,0:02:54.70,0:02:58.43,Default,,0000,0000,0000,,Or some of you may know it\Nfunction of a function.
Dialogue: 0,0:02:59.33,0:03:04.29,Default,,0000,0000,0000,,So this is what we've got to do.\NWe've got to work out the why by
Dialogue: 0,0:03:04.29,0:03:08.63,Default,,0000,0000,0000,,DU and workout. Do you buy the\NX? So I'm going to turn the
Dialogue: 0,0:03:08.63,0:03:12.35,Default,,0000,0000,0000,,page. I'm going to virtually\Nwrite this out again in order to
Dialogue: 0,0:03:12.35,0:03:16.07,Default,,0000,0000,0000,,make sure that I've got the\Nspace in which to do it.
Dialogue: 0,0:03:17.06,0:03:20.80,Default,,0000,0000,0000,,So we begin with Y equals
Dialogue: 0,0:03:20.80,0:03:24.42,Default,,0000,0000,0000,,5. Plus 2X to the
Dialogue: 0,0:03:24.42,0:03:31.26,Default,,0000,0000,0000,,power 10. We put\Nyou equals 5 +
Dialogue: 0,0:03:31.26,0:03:38.06,Default,,0000,0000,0000,,2 X and then\NY is equal to
Dialogue: 0,0:03:38.06,0:03:41.46,Default,,0000,0000,0000,,U to the power
Dialogue: 0,0:03:41.46,0:03:48.86,Default,,0000,0000,0000,,10. DY by the X\Nis given by the Y by
Dialogue: 0,0:03:48.86,0:03:55.86,Default,,0000,0000,0000,,EU times. Do you buy the\NX and we can workout each
Dialogue: 0,0:03:55.86,0:04:02.85,Default,,0000,0000,0000,,of these two divided by DU&U\Nby DX. So let's have a
Dialogue: 0,0:04:02.85,0:04:09.85,Default,,0000,0000,0000,,look at that why by EU\Nis the derivative of U to
Dialogue: 0,0:04:09.85,0:04:13.35,Default,,0000,0000,0000,,the power 10 with respect to
Dialogue: 0,0:04:13.35,0:04:19.48,Default,,0000,0000,0000,,you. Which is 10 U to the power\N9. Remember we multiply by the
Dialogue: 0,0:04:19.48,0:04:25.44,Default,,0000,0000,0000,,index and take one of the index\Nto give us nine and so that's
Dialogue: 0,0:04:25.44,0:04:32.89,Default,,0000,0000,0000,,10. 5 + 2 X\Nto the Power 9 replacing EU
Dialogue: 0,0:04:32.89,0:04:36.07,Default,,0000,0000,0000,,by 5 + 2 X.
Dialogue: 0,0:04:36.97,0:04:41.69,Default,,0000,0000,0000,,And we can calculate you\Nby The X.
Dialogue: 0,0:04:42.80,0:04:48.96,Default,,0000,0000,0000,,That's the derivative of 5\N+ 2 X.
Dialogue: 0,0:04:49.30,0:04:54.22,Default,,0000,0000,0000,,With respect to X, and that's\Njust two because the derivative
Dialogue: 0,0:04:54.22,0:05:00.48,Default,,0000,0000,0000,,of five 5 is a constant stats\Nzero, the derivative of two X is
Dialogue: 0,0:05:00.48,0:05:07.08,Default,,0000,0000,0000,,just two. So now we know what\Ndivided by du is. It's this and
Dialogue: 0,0:05:07.08,0:05:12.66,Default,,0000,0000,0000,,we know what do you buy the\Naccess. It's this so we can
Dialogue: 0,0:05:12.66,0:05:13.94,Default,,0000,0000,0000,,write those in.
Dialogue: 0,0:05:14.85,0:05:22.13,Default,,0000,0000,0000,,10 Times 5\N+ 2 X to the
Dialogue: 0,0:05:22.13,0:05:29.41,Default,,0000,0000,0000,,9th. Times by two and\Nof course, the two times by 10
Dialogue: 0,0:05:29.41,0:05:30.99,Default,,0000,0000,0000,,gives us 20.
Dialogue: 0,0:05:31.57,0:05:38.16,Default,,0000,0000,0000,,So we've used our\Nchain rule in order
Dialogue: 0,0:05:38.16,0:05:43.93,Default,,0000,0000,0000,,to be able to\Ndifferentiate this function.
Dialogue: 0,0:05:44.73,0:05:51.15,Default,,0000,0000,0000,,So let me just\Nwrite our chain rule
Dialogue: 0,0:05:51.15,0:05:57.58,Default,,0000,0000,0000,,down here DY by\NX is equal to
Dialogue: 0,0:05:57.58,0:06:03.20,Default,,0000,0000,0000,,Y by DU times\Ndu by X.
Dialogue: 0,0:06:05.08,0:06:08.84,Default,,0000,0000,0000,,Now. Let's suppose that we
Dialogue: 0,0:06:08.84,0:06:14.56,Default,,0000,0000,0000,,had zed. And said was a\Nfunction of Why?
Dialogue: 0,0:06:15.36,0:06:22.12,Default,,0000,0000,0000,,Then D zed by the\NX would be equal to
Dialogue: 0,0:06:22.12,0:06:24.98,Default,,0000,0000,0000,,D zed. Bye.
Dialogue: 0,0:06:25.50,0:06:29.42,Default,,0000,0000,0000,,DY times DY
Dialogue: 0,0:06:29.42,0:06:35.58,Default,,0000,0000,0000,,by X. Using\Nour chain rule again.
Dialogue: 0,0:06:36.41,0:06:42.05,Default,,0000,0000,0000,,Let's take an example.\NLet's say that zed is
Dialogue: 0,0:06:42.05,0:06:44.56,Default,,0000,0000,0000,,equal to Y squared.
Dialogue: 0,0:06:45.65,0:06:52.42,Default,,0000,0000,0000,,Then these Ed by the\NX would be equal to.
Dialogue: 0,0:06:53.24,0:06:59.25,Default,,0000,0000,0000,,The derivative of Y squared\Nwith respect to Y.
Dialogue: 0,0:07:00.05,0:07:06.92,Default,,0000,0000,0000,,Times DY by the X\Nand the derivative of Y
Dialogue: 0,0:07:06.92,0:07:13.79,Default,,0000,0000,0000,,squared with respect to Y\Nis equal to two Y
Dialogue: 0,0:07:13.79,0:07:16.54,Default,,0000,0000,0000,,times DY by X.
Dialogue: 0,0:07:17.54,0:07:23.26,Default,,0000,0000,0000,,Notice what we seem to have done\Nis just differentiated. Why
Dialogue: 0,0:07:23.26,0:07:30.02,Default,,0000,0000,0000,,square root respect to Y and\Nmultiplied by DY by the X? And
Dialogue: 0,0:07:30.02,0:07:35.74,Default,,0000,0000,0000,,I'll draw attention to that\Nagain later on. For now, let's
Dialogue: 0,0:07:35.74,0:07:41.98,Default,,0000,0000,0000,,have a look at some examples.\NBegin with this one Y squared.
Dialogue: 0,0:07:42.57,0:07:45.93,Default,,0000,0000,0000,,Plus X cubed.
Dialogue: 0,0:07:46.78,0:07:50.15,Default,,0000,0000,0000,,Minus Y cubed plus 6.
Dialogue: 0,0:07:50.70,0:07:55.80,Default,,0000,0000,0000,,Equals 3 Y. We want to\Ndifferentiate this with respect
Dialogue: 0,0:07:55.80,0:08:01.92,Default,,0000,0000,0000,,to X, so we're looking for the\Nderivative of Y squared with
Dialogue: 0,0:08:01.92,0:08:03.45,Default,,0000,0000,0000,,respect to X.
Dialogue: 0,0:08:03.98,0:08:11.07,Default,,0000,0000,0000,,Derivative of X cubed\Nwith respect to X.
Dialogue: 0,0:08:11.07,0:08:15.37,Default,,0000,0000,0000,,Derivative of Y cubed with\Nrespect to X.
Dialogue: 0,0:08:16.53,0:08:22.56,Default,,0000,0000,0000,,Derivative of the six with\Nrespect to X and the derivative
Dialogue: 0,0:08:22.56,0:08:29.68,Default,,0000,0000,0000,,of three Y with respect to X.\NNot remember how we said we
Dialogue: 0,0:08:29.68,0:08:36.81,Default,,0000,0000,0000,,would do this? We said we will\Ntake the derivative of Y squared
Dialogue: 0,0:08:36.81,0:08:38.100,Default,,0000,0000,0000,,with respect to Y.
Dialogue: 0,0:08:39.75,0:08:45.43,Default,,0000,0000,0000,,And multiply by DY by the X that\Nwas our chain rule.
Dialogue: 0,0:08:46.07,0:08:50.41,Default,,0000,0000,0000,,Plus this is straightforward,\Ndon't need to worry about this.
Dialogue: 0,0:08:50.41,0:08:56.05,Default,,0000,0000,0000,,This is the derivative of X\Ncubed with respect to X. We know
Dialogue: 0,0:08:56.05,0:09:01.69,Default,,0000,0000,0000,,that is 3 X squared multiplied\Nby the index and take one away
Dialogue: 0,0:09:01.69,0:09:07.98,Default,,0000,0000,0000,,minus. And here again, we've got\Nto apply our chain rule. So
Dialogue: 0,0:09:07.98,0:09:13.17,Default,,0000,0000,0000,,we've got the derivative of Y\Ncubed with respect to Y.
Dialogue: 0,0:09:13.88,0:09:17.32,Default,,0000,0000,0000,,Times by DY. By
Dialogue: 0,0:09:17.32,0:09:21.21,Default,,0000,0000,0000,,The X. Plus
Dialogue: 0,0:09:22.02,0:09:25.97,Default,,0000,0000,0000,,Now the derivative of six well\Nsix is just a constant, so we
Dialogue: 0,0:09:25.97,0:09:27.49,Default,,0000,0000,0000,,know its derivative is 0.
Dialogue: 0,0:09:28.69,0:09:34.28,Default,,0000,0000,0000,,Equals and again we want the\Nderivative of three Y, so our
Dialogue: 0,0:09:34.28,0:09:39.87,Default,,0000,0000,0000,,chain rule tells us this is the\Nderivative of three Y with
Dialogue: 0,0:09:39.87,0:09:43.14,Default,,0000,0000,0000,,respect to Y times DY by DX.
Dialogue: 0,0:09:43.77,0:09:48.59,Default,,0000,0000,0000,,Now we can go back and work out\Neach of these derivatives with
Dialogue: 0,0:09:48.59,0:09:51.19,Default,,0000,0000,0000,,respect to Y, so that will give
Dialogue: 0,0:09:51.19,0:09:57.14,Default,,0000,0000,0000,,us 2Y. DYIDX\Nplus three
Dialogue: 0,0:09:57.14,0:10:04.05,Default,,0000,0000,0000,,X squared.\NMinus the derivative of Y Cube
Dialogue: 0,0:10:04.05,0:10:11.23,Default,,0000,0000,0000,,with respect to Y is 3 Y\Nsquared divided by DX. We can
Dialogue: 0,0:10:11.23,0:10:12.88,Default,,0000,0000,0000,,ignore the zero.
Dialogue: 0,0:10:13.46,0:10:20.32,Default,,0000,0000,0000,,And the derivative of three Y\Nwith respect to Y is 3 times
Dialogue: 0,0:10:20.32,0:10:21.91,Default,,0000,0000,0000,,divided by DX.
Dialogue: 0,0:10:22.93,0:10:29.24,Default,,0000,0000,0000,,Now. If we get together all the\Nterms that have a DY by the X in
Dialogue: 0,0:10:29.24,0:10:35.66,Default,,0000,0000,0000,,them. Then, having done that, we\Ncan sort out what divided by DX
Dialogue: 0,0:10:35.66,0:10:40.94,Default,,0000,0000,0000,,actually is, so I'm going to\Ngather together all the terms in
Dialogue: 0,0:10:40.94,0:10:48.42,Default,,0000,0000,0000,,do I buy DX over on this side of\Nthe equation so I can keep .3 X
Dialogue: 0,0:10:48.42,0:10:53.70,Default,,0000,0000,0000,,squared there on its own. So I\Nhave three X squared equals.
Dialogue: 0,0:10:54.60,0:11:01.74,Default,,0000,0000,0000,,Now here on this side I've\Ngot 3D Y by X.
Dialogue: 0,0:11:02.70,0:11:08.52,Default,,0000,0000,0000,,I'm going to take this away from\Neach side, so that's minus two
Dialogue: 0,0:11:08.52,0:11:15.56,Default,,0000,0000,0000,,Y. DY by X and I'm going\Nto add this term to both sides
Dialogue: 0,0:11:15.56,0:11:22.25,Default,,0000,0000,0000,,plus three Y squared DY by The\NX. what I can see here is that
Dialogue: 0,0:11:22.25,0:11:28.94,Default,,0000,0000,0000,,I've got a common factor of the\Nwhy by DX that I can take out,
Dialogue: 0,0:11:28.94,0:11:33.84,Default,,0000,0000,0000,,so that's what we're going to do\Nnext, so will have.
Dialogue: 0,0:11:34.97,0:11:38.41,Default,,0000,0000,0000,,Three X squared equals.
Dialogue: 0,0:11:39.92,0:11:45.95,Default,,0000,0000,0000,,Bracket. And taking out\Nthat common factor of the why
Dialogue: 0,0:11:45.95,0:11:51.12,Default,,0000,0000,0000,,by DX. So let's just go back\Nand have a lot. What do I
Dialogue: 0,0:11:51.12,0:11:54.07,Default,,0000,0000,0000,,buy? DX was multiplying. It\Nwas multiplying A3.
Dialogue: 0,0:11:55.15,0:12:02.25,Default,,0000,0000,0000,,It was multiplying a minus two Y\Nand it was multiplying A plus
Dialogue: 0,0:12:02.25,0:12:09.35,Default,,0000,0000,0000,,three Y squared, so it was\Nmultiplying a 3 - 2 Y and
Dialogue: 0,0:12:09.35,0:12:17.06,Default,,0000,0000,0000,,a plus. 3 Y squared. So now\Nwe can get divided by DX on
Dialogue: 0,0:12:17.06,0:12:23.49,Default,,0000,0000,0000,,its own if we divide throughout\Nby this expression sode, why by
Dialogue: 0,0:12:23.49,0:12:29.39,Default,,0000,0000,0000,,DX is equal to three X squared\Nand dividing throughout dividing
Dialogue: 0,0:12:29.39,0:12:35.29,Default,,0000,0000,0000,,both sides by 3 - 2 Y\Nplus three Y squared.
Dialogue: 0,0:12:36.26,0:12:41.13,Default,,0000,0000,0000,,And then we've got our\Nexpression for DY by DX.
Dialogue: 0,0:12:41.82,0:12:44.82,Default,,0000,0000,0000,,That was a reasonably\Nstraightforward example. The
Dialogue: 0,0:12:44.82,0:12:49.10,Default,,0000,0000,0000,,work that many complications and\Nit followed very directly from
Dialogue: 0,0:12:49.10,0:12:54.66,Default,,0000,0000,0000,,our first look at this. So now\Nlet's look at a slightly more
Dialogue: 0,0:12:54.66,0:12:58.51,Default,,0000,0000,0000,,complicated example, one where\Nin fact we've got other
Dialogue: 0,0:12:58.51,0:13:04.57,Default,,0000,0000,0000,,functions. Of X&Y. So in this\Ncase will start up with a sign
Dialogue: 0,0:13:04.57,0:13:09.16,Default,,0000,0000,0000,,Y where we've got the Axis and\Nthe wise actually combined
Dialogue: 0,0:13:09.16,0:13:13.33,Default,,0000,0000,0000,,together, so we've got X\Nsquared times by Y cubed.
Dialogue: 0,0:13:15.04,0:13:22.54,Default,,0000,0000,0000,,Minus calls X and let's say\Nequals 2 Yi. Want to be
Dialogue: 0,0:13:22.54,0:13:29.42,Default,,0000,0000,0000,,able to differentiate this with\Nrespect to X, so that's the
Dialogue: 0,0:13:29.42,0:13:36.29,Default,,0000,0000,0000,,derivative of sine Y with\Nrespect to X plus the derivative
Dialogue: 0,0:13:36.29,0:13:39.42,Default,,0000,0000,0000,,of X squared Y cubed.
Dialogue: 0,0:13:40.01,0:13:47.99,Default,,0000,0000,0000,,With respect to X minus the\Nderivative of Cos X with respect
Dialogue: 0,0:13:47.99,0:13:55.97,Default,,0000,0000,0000,,to X equals the derivative of\Ntwo Y with respect to X.
Dialogue: 0,0:13:56.97,0:14:02.51,Default,,0000,0000,0000,,Now let's remember what our\Nfunction of a function rule
Dialogue: 0,0:14:02.51,0:14:09.16,Default,,0000,0000,0000,,tells us that this is done\Nas the derivative of sine Y
Dialogue: 0,0:14:09.16,0:14:14.14,Default,,0000,0000,0000,,with respect to Y times by\NDY by DX.
Dialogue: 0,0:14:16.34,0:14:21.87,Default,,0000,0000,0000,,This one. Bit of a problem\N'cause This is X squared times
Dialogue: 0,0:14:21.87,0:14:24.77,Default,,0000,0000,0000,,by Y cubed. So it's a product.
Dialogue: 0,0:14:25.37,0:14:31.50,Default,,0000,0000,0000,,It's a U times by AV, so let me\Njust write a little U over the
Dialogue: 0,0:14:31.50,0:14:33.80,Default,,0000,0000,0000,,top and a little V there.
Dialogue: 0,0:14:34.67,0:14:41.28,Default,,0000,0000,0000,,You is X squared and\NV is Y cubed.
Dialogue: 0,0:14:41.81,0:14:43.52,Default,,0000,0000,0000,,So this is plus.
Dialogue: 0,0:14:44.33,0:14:49.14,Default,,0000,0000,0000,,Let's remember how we\Ndifferentiate a product. We take
Dialogue: 0,0:14:49.14,0:14:55.85,Default,,0000,0000,0000,,you. And we multiply it by\Nthe derivative of V. That's the
Dialogue: 0,0:14:55.85,0:14:59.90,Default,,0000,0000,0000,,derivative of Y cubed with\Nrespect to X.
Dialogue: 0,0:15:01.11,0:15:08.19,Default,,0000,0000,0000,,Plus the an we multiply that\Nby the derivative of U, which
Dialogue: 0,0:15:08.19,0:15:13.50,Default,,0000,0000,0000,,in this case is the derivative\Nof X squared.
Dialogue: 0,0:15:14.10,0:15:21.21,Default,,0000,0000,0000,,Minus now we can do this one.\NThe derivative of Cos X with
Dialogue: 0,0:15:21.21,0:15:27.78,Default,,0000,0000,0000,,respect to X. The derivative of\Ncauses minus sign, so a minus
Dialogue: 0,0:15:27.78,0:15:34.34,Default,,0000,0000,0000,,and minus makes a plus sign. X\Nequals the derivative of and
Dialogue: 0,0:15:34.34,0:15:40.90,Default,,0000,0000,0000,,again my chain rule tells me\Nthat this is the derivative of
Dialogue: 0,0:15:40.90,0:15:44.73,Default,,0000,0000,0000,,two Y with respect to Y times
Dialogue: 0,0:15:44.73,0:15:48.51,Default,,0000,0000,0000,,by. Divide by The X.
Dialogue: 0,0:15:50.15,0:15:54.60,Default,,0000,0000,0000,,So this has been much more\Ncomplicated, but notice how it
Dialogue: 0,0:15:54.60,0:15:58.25,Default,,0000,0000,0000,,follows the standard rules that\Nwe've already got for
Dialogue: 0,0:15:58.25,0:16:01.90,Default,,0000,0000,0000,,differentiation. So now the\Nderivative of sine wired with
Dialogue: 0,0:16:01.90,0:16:04.73,Default,,0000,0000,0000,,respect to Y is just cause why.
Dialogue: 0,0:16:05.42,0:16:08.55,Default,,0000,0000,0000,,DY by X.
Dialogue: 0,0:16:09.34,0:16:13.58,Default,,0000,0000,0000,,Plus X\Nsquared
Dialogue: 0,0:16:14.84,0:16:20.66,Default,,0000,0000,0000,,Times by now this will be the\Nderivative of Y cubed with
Dialogue: 0,0:16:20.66,0:16:22.12,Default,,0000,0000,0000,,respect to Y.
Dialogue: 0,0:16:22.66,0:16:26.70,Default,,0000,0000,0000,,Times by DY. By The\NX.
Dialogue: 0,0:16:28.10,0:16:35.25,Default,,0000,0000,0000,,Plus this one Y cubed times by\Nnow the derivative of X squared
Dialogue: 0,0:16:35.25,0:16:39.10,Default,,0000,0000,0000,,with respect to X is just 2X.
Dialogue: 0,0:16:39.82,0:16:42.12,Default,,0000,0000,0000,,Plus sign X.
Dialogue: 0,0:16:42.88,0:16:47.43,Default,,0000,0000,0000,,We've already done that one\Nequals and hear the derivative
Dialogue: 0,0:16:47.43,0:16:53.80,Default,,0000,0000,0000,,of two Y with respect to Y.\NThese two times DY by The X.
Dialogue: 0,0:16:55.08,0:17:02.14,Default,,0000,0000,0000,,Almost done now we still\Ngot a little bit of
Dialogue: 0,0:17:02.14,0:17:09.20,Default,,0000,0000,0000,,differentiation in here to do\Nso. Let's do that cause
Dialogue: 0,0:17:09.20,0:17:16.26,Default,,0000,0000,0000,,YDY by DX plus I\Nthink I can fairly safely
Dialogue: 0,0:17:16.26,0:17:23.32,Default,,0000,0000,0000,,remove these brackets now. X\Nsquared times 3 Y squared
Dialogue: 0,0:17:23.32,0:17:28.97,Default,,0000,0000,0000,,DY by X +2 XY\Ncubed plus sign.
Dialogue: 0,0:17:28.99,0:17:34.62,Default,,0000,0000,0000,,X equals 2 DY\Nby The X.
Dialogue: 0,0:17:35.24,0:17:39.46,Default,,0000,0000,0000,,Where at the same stage as we\Nwere last time we've got the
Dialogue: 0,0:17:39.46,0:17:43.04,Default,,0000,0000,0000,,differentiation Dom and it's\Nthis thing. The why by DX that
Dialogue: 0,0:17:43.04,0:17:47.26,Default,,0000,0000,0000,,we want. So what we gotta do is\Nget all those terms that
Dialogue: 0,0:17:47.26,0:17:51.49,Default,,0000,0000,0000,,involved why by DX on one side\Nof the equation and the other
Dialogue: 0,0:17:51.49,0:17:55.39,Default,,0000,0000,0000,,terms on the other side. Now\Nthese are the two terms that
Dialogue: 0,0:17:55.39,0:17:59.94,Default,,0000,0000,0000,,don't have a divided by DX in\Nthem, so I'm going to keep them
Dialogue: 0,0:17:59.94,0:18:03.52,Default,,0000,0000,0000,,at this side. So it's two XY\Ncubed plus sign X.
Dialogue: 0,0:18:05.78,0:18:11.13,Default,,0000,0000,0000,,Two, XY\Ncubed plus
Dialogue: 0,0:18:11.13,0:18:13.80,Default,,0000,0000,0000,,sign X
Dialogue: 0,0:18:13.80,0:18:20.31,Default,,0000,0000,0000,,equals. Let's just go back and\Nsee what we've got. We've got a
Dialogue: 0,0:18:20.31,0:18:22.24,Default,,0000,0000,0000,,2 divided by DX here.
Dialogue: 0,0:18:22.28,0:18:28.37,Default,,0000,0000,0000,,2.\NDY by The X.
Dialogue: 0,0:18:29.09,0:18:34.58,Default,,0000,0000,0000,,And we're going to bring these\Ntwo terms over to this side by
Dialogue: 0,0:18:34.58,0:18:39.22,Default,,0000,0000,0000,,taking them away from both\Nsides. So we're going to take
Dialogue: 0,0:18:39.22,0:18:42.59,Default,,0000,0000,0000,,away 'cause why do I buy the X?
Dialogue: 0,0:18:43.33,0:18:49.92,Default,,0000,0000,0000,,On both sides, minus cause YDY\Nby the X, and then we're going
Dialogue: 0,0:18:49.92,0:18:56.51,Default,,0000,0000,0000,,to take away the other term.\NThis is the X squared times by
Dialogue: 0,0:18:56.51,0:19:02.09,Default,,0000,0000,0000,,three Y squared divided by DX,\Nright that a little bit
Dialogue: 0,0:19:02.09,0:19:08.68,Default,,0000,0000,0000,,differently. When I do it, so we\Nhave minus three X squared Y
Dialogue: 0,0:19:08.68,0:19:10.71,Default,,0000,0000,0000,,squared divided by X.
Dialogue: 0,0:19:11.99,0:19:17.46,Default,,0000,0000,0000,,Again, we see we've got a set of\Nterms here, each with divided by
Dialogue: 0,0:19:17.46,0:19:19.42,Default,,0000,0000,0000,,DX as a common factor.
Dialogue: 0,0:19:19.92,0:19:26.53,Default,,0000,0000,0000,,Two, XY cubed plus sign\NX is equal to.
Dialogue: 0,0:19:27.50,0:19:33.70,Default,,0000,0000,0000,,Let's take out this common\Nfactor of DY by The X.
Dialogue: 0,0:19:34.30,0:19:39.08,Default,,0000,0000,0000,,Well, it's multiplying two, so\Nwe've got a two there. It's
Dialogue: 0,0:19:39.08,0:19:43.44,Default,,0000,0000,0000,,multiplying minus cause Y, so\Nwe've gotta minus cause why
Dialogue: 0,0:19:43.44,0:19:47.78,Default,,0000,0000,0000,,there? And it's multiplying\Nminus three X squared Y squared
Dialogue: 0,0:19:47.78,0:19:53.00,Default,,0000,0000,0000,,minus three X squared Y squared.\NAnd now finally we can get
Dialogue: 0,0:19:53.00,0:19:57.79,Default,,0000,0000,0000,,divided by DX on its own,\Nbecause we can divide throughout
Dialogue: 0,0:19:57.79,0:20:02.14,Default,,0000,0000,0000,,divide both sides of the\Nequation by what's in this
Dialogue: 0,0:20:02.14,0:20:05.18,Default,,0000,0000,0000,,bracket. So we have two XY cubed
Dialogue: 0,0:20:05.18,0:20:12.62,Default,,0000,0000,0000,,plus sign. X all over 2\Nminus cause Y minus three
Dialogue: 0,0:20:12.62,0:20:15.53,Default,,0000,0000,0000,,X squared Y squared.
Dialogue: 0,0:20:16.09,0:20:19.06,Default,,0000,0000,0000,,And there's our divide by\Nthe eggs.
Dialogue: 0,0:20:20.29,0:20:22.65,Default,,0000,0000,0000,,Now let's just look back at this
Dialogue: 0,0:20:22.65,0:20:28.39,Default,,0000,0000,0000,,one. Look at all this\Ncomplicated differentiation that
Dialogue: 0,0:20:28.39,0:20:30.49,Default,,0000,0000,0000,,we had here.
Dialogue: 0,0:20:31.12,0:20:35.48,Default,,0000,0000,0000,,Now I went through it slowly and\Ncarefully, but when we're doing
Dialogue: 0,0:20:35.48,0:20:39.47,Default,,0000,0000,0000,,calculations on our own, we\Nmight make slips. It would be
Dialogue: 0,0:20:39.47,0:20:43.46,Default,,0000,0000,0000,,helpful if we could automate\Nsome of this process so it
Dialogue: 0,0:20:43.46,0:20:47.09,Default,,0000,0000,0000,,instead of writing down the\Nderivative of sine wave with
Dialogue: 0,0:20:47.09,0:20:50.72,Default,,0000,0000,0000,,respect to X is and going\Nthrough the chain rule.
Dialogue: 0,0:20:51.43,0:20:56.30,Default,,0000,0000,0000,,We went automatically to all we\Nneed to do is differentiate sign
Dialogue: 0,0:20:56.30,0:21:01.17,Default,,0000,0000,0000,,wired with respect to Y. That's\Ncause Y an multiplied by divided
Dialogue: 0,0:21:01.17,0:21:06.45,Default,,0000,0000,0000,,by DX. So we automate we would\Nmiss out at least these two
Dialogue: 0,0:21:06.45,0:21:10.51,Default,,0000,0000,0000,,lines and go direct from there\Nto there. Similarly, the
Dialogue: 0,0:21:10.51,0:21:15.79,Default,,0000,0000,0000,,derivative of two Y with respect\Nto X would be the derivative of
Dialogue: 0,0:21:15.79,0:21:21.47,Default,,0000,0000,0000,,two Y with respect to Y two\Ntimes divided by DX. And so we
Dialogue: 0,0:21:21.47,0:21:23.10,Default,,0000,0000,0000,,go direct from there.
Dialogue: 0,0:21:23.12,0:21:28.69,Default,,0000,0000,0000,,To there, so let's have a look\Nat that in another example.
Dialogue: 0,0:21:29.46,0:21:37.12,Default,,0000,0000,0000,,So we'll take\NY squared plus
Dialogue: 0,0:21:37.12,0:21:44.78,Default,,0000,0000,0000,,X cubed minus\NXY plus cause
Dialogue: 0,0:21:44.78,0:21:50.78,Default,,0000,0000,0000,,Y. Equals note this time\Nwe're going to go.
Dialogue: 0,0:21:51.31,0:21:55.53,Default,,0000,0000,0000,,Direct to the differentiation,\Nwe're not going to go by the
Dialogue: 0,0:21:55.53,0:22:00.14,Default,,0000,0000,0000,,chain rule. We're going to use\Nit, of course, but we aren't
Dialogue: 0,0:22:00.14,0:22:04.75,Default,,0000,0000,0000,,going to write it down, so we\Nwant to differentiate this with
Dialogue: 0,0:22:04.75,0:22:10.51,Default,,0000,0000,0000,,respect to X. So the first term\Nis Y squared, so we know that to
Dialogue: 0,0:22:10.51,0:22:13.97,Default,,0000,0000,0000,,differentiate Y squared with\Nrespect to X, we differentiate
Dialogue: 0,0:22:13.97,0:22:16.27,Default,,0000,0000,0000,,this with respect to why that's
Dialogue: 0,0:22:16.27,0:22:22.10,Default,,0000,0000,0000,,too why. And we multiply by\NDY by DX.
Dialogue: 0,0:22:22.69,0:22:28.72,Default,,0000,0000,0000,,Now we want the derivative of X\Ncubed with respect to X, so
Dialogue: 0,0:22:28.72,0:22:31.14,Default,,0000,0000,0000,,that's 3X. Squared
Dialogue: 0,0:22:31.92,0:22:33.95,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:22:35.06,0:22:39.95,Default,,0000,0000,0000,,Now I am going to write this\Ndown in full because it's the
Dialogue: 0,0:22:39.95,0:22:44.84,Default,,0000,0000,0000,,derivative of XY with respect to\NX. It's a product again, it's X
Dialogue: 0,0:22:44.84,0:22:45.96,Default,,0000,0000,0000,,times by Y.
Dialogue: 0,0:22:46.96,0:22:52.68,Default,,0000,0000,0000,,Plus, the derivative of Cos Y\Nwith respect to X, which we do
Dialogue: 0,0:22:52.68,0:22:58.40,Default,,0000,0000,0000,,as a derivative of cause Y with\Nrespect to Y that's minus Sign
Dialogue: 0,0:22:58.40,0:23:05.76,Default,,0000,0000,0000,,Y. Times DY by DX,\Nthe derivative of 0 is
Dialogue: 0,0:23:05.76,0:23:11.56,Default,,0000,0000,0000,,just zero. So let's get\Nthese two terms together because
Dialogue: 0,0:23:11.56,0:23:18.08,Default,,0000,0000,0000,,they both got the why by DX. So\NI have two Y minus Sign Y.
Dialogue: 0,0:23:18.18,0:23:24.67,Default,,0000,0000,0000,,Times Ty by DX\Nplus three X squared.
Dialogue: 0,0:23:25.40,0:23:32.86,Default,,0000,0000,0000,,Minus. Now this is\Na product. It is a U times
Dialogue: 0,0:23:32.86,0:23:36.76,Default,,0000,0000,0000,,by AV. So we
Dialogue: 0,0:23:36.76,0:23:43.88,Default,,0000,0000,0000,,want you. Times the derivative\Nof Y with respect to X, which
Dialogue: 0,0:23:43.88,0:23:47.53,Default,,0000,0000,0000,,is just the wise by The X.
Dialogue: 0,0:23:48.05,0:23:55.67,Default,,0000,0000,0000,,Plus V, which is Y Times\Nthe derivative of X, which is
Dialogue: 0,0:23:55.67,0:23:59.72,Default,,0000,0000,0000,,just one. Equals\N0.
Dialogue: 0,0:24:00.94,0:24:06.04,Default,,0000,0000,0000,,So we see here that we've got\Nanother term now involving the
Dialogue: 0,0:24:06.04,0:24:12.42,Default,,0000,0000,0000,,why by DX it's minus X, so we\Ncan put that in the bracket so
Dialogue: 0,0:24:12.42,0:24:15.82,Default,,0000,0000,0000,,we can have two Y minus sign Y
Dialogue: 0,0:24:15.82,0:24:22.100,Default,,0000,0000,0000,,minus X. DY by the\NX plus three X squared
Dialogue: 0,0:24:22.100,0:24:30.22,Default,,0000,0000,0000,,and then minus this term\Nhere minus Y equals 0.
Dialogue: 0,0:24:31.69,0:24:37.67,Default,,0000,0000,0000,,If we take this over to the\Nother side, In other words, we
Dialogue: 0,0:24:37.67,0:24:43.65,Default,,0000,0000,0000,,take this away from both sides\Nand add that to both sides. You
Dialogue: 0,0:24:43.65,0:24:50.55,Default,,0000,0000,0000,,have two Y minus sign Y minus X\Ntimes DY by X is equal to.
Dialogue: 0,0:24:50.55,0:24:56.07,Default,,0000,0000,0000,,Adding this one to both sides.\NWhy taking this one away from
Dialogue: 0,0:24:56.07,0:24:58.37,Default,,0000,0000,0000,,both sides, we get that.
Dialogue: 0,0:24:58.96,0:25:03.95,Default,,0000,0000,0000,,Now it's clear how we would\Nfinish this off. We will take
Dialogue: 0,0:25:03.95,0:25:07.70,Default,,0000,0000,0000,,this factor here and divide both\Nsides by it.
Dialogue: 0,0:25:08.40,0:25:15.54,Default,,0000,0000,0000,,So we get DY by\Nthe X was equal 2.
Dialogue: 0,0:25:16.28,0:25:23.62,Default,,0000,0000,0000,,Just go back. We've got Y\Nminus three X squared to go
Dialogue: 0,0:25:23.62,0:25:27.30,Default,,0000,0000,0000,,in the numerator on the top.
Dialogue: 0,0:25:28.42,0:25:34.66,Default,,0000,0000,0000,,And this factor to go on the\Nbottom in the denominator two Y
Dialogue: 0,0:25:34.66,0:25:36.10,Default,,0000,0000,0000,,minus Sign Y.
Dialogue: 0,0:25:36.61,0:25:40.60,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:25:40.60,0:25:44.50,Default,,0000,0000,0000,,X. And then
Dialogue: 0,0:25:44.50,0:25:51.55,Default,,0000,0000,0000,,we have. Our derivative,\NLet's just take one
Dialogue: 0,0:25:51.55,0:25:55.100,Default,,0000,0000,0000,,more example.\NY
Dialogue: 0,0:25:55.100,0:26:01.40,Default,,0000,0000,0000,,cubed\Nminus
Dialogue: 0,0:26:01.40,0:26:04.86,Default,,0000,0000,0000,,X. Sign
Dialogue: 0,0:26:04.86,0:26:08.79,Default,,0000,0000,0000,,Y. Plus Y squared
Dialogue: 0,0:26:08.79,0:26:13.17,Default,,0000,0000,0000,,over X. Equals\N8.
Dialogue: 0,0:26:14.52,0:26:19.98,Default,,0000,0000,0000,,Again, all the axes and Wise\Nbundled up together if possible.
Dialogue: 0,0:26:19.98,0:26:27.42,Default,,0000,0000,0000,,We want to be able to do this\Ndirectly. We want to be able to
Dialogue: 0,0:26:27.42,0:26:31.88,Default,,0000,0000,0000,,differentiate it straight away\Nwithout going through the chain
Dialogue: 0,0:26:31.88,0:26:38.33,Default,,0000,0000,0000,,rule. So the derivative of Y\Ncubed with respect to Y times by
Dialogue: 0,0:26:38.33,0:26:44.78,Default,,0000,0000,0000,,DY by DX. So that's three Y\Nsquared times DY by X minus.
Dialogue: 0,0:26:44.84,0:26:51.17,Default,,0000,0000,0000,,We want the derivative of this.\NThis is a product so let's see
Dialogue: 0,0:26:51.17,0:26:58.48,Default,,0000,0000,0000,,if we can do it all in one\Ngo. Again you want X Times the
Dialogue: 0,0:26:58.48,0:27:03.83,Default,,0000,0000,0000,,derivative of sine Y with\Nrespect to X. That's X times
Dialogue: 0,0:27:03.83,0:27:05.78,Default,,0000,0000,0000,,cause YDY by X.
Dialogue: 0,0:27:06.31,0:27:12.79,Default,,0000,0000,0000,,And now we want sign Y times the\Nderivative of X so that sign Y
Dialogue: 0,0:27:12.79,0:27:15.81,Default,,0000,0000,0000,,and the derivative of X is just
Dialogue: 0,0:27:15.81,0:27:20.83,Default,,0000,0000,0000,,one. This one is a bit trickier.\NThis is a quotient Y squared
Dialogue: 0,0:27:20.83,0:27:25.78,Default,,0000,0000,0000,,over X, so we want plus. Now\Nlet's remember what we do with
Dialogue: 0,0:27:25.78,0:27:32.92,Default,,0000,0000,0000,,the quotient. It's V which is on\Nthe bottom that's X Times the
Dialogue: 0,0:27:32.92,0:27:39.28,Default,,0000,0000,0000,,derivative of what's on the top.\NThe derivative of Y squared with
Dialogue: 0,0:27:39.28,0:27:45.11,Default,,0000,0000,0000,,respect to X, which is the\Nderivative of Y squared with
Dialogue: 0,0:27:45.11,0:27:52.00,Default,,0000,0000,0000,,respect to Y2Y times DY by the\NX minus Y squared times the
Dialogue: 0,0:27:52.00,0:27:57.30,Default,,0000,0000,0000,,derivative of V, which in this\Ncase is just X.
Dialogue: 0,0:27:57.30,0:28:03.47,Default,,0000,0000,0000,,All over V squared, which is\NX squared equals 0.
Dialogue: 0,0:28:04.43,0:28:06.42,Default,,0000,0000,0000,,Now this needs a little bit of
Dialogue: 0,0:28:06.42,0:28:10.53,Default,,0000,0000,0000,,tidying up. We've got a\Ndenominator here that we can
Dialogue: 0,0:28:10.53,0:28:13.41,Default,,0000,0000,0000,,probably multiply out by, so\Nlet's do that.
Dialogue: 0,0:28:13.99,0:28:21.12,Default,,0000,0000,0000,,And do some tidying on the way,\Nso this will be three X squared
Dialogue: 0,0:28:21.12,0:28:24.17,Default,,0000,0000,0000,,Y squared DY by The X.
Dialogue: 0,0:28:25.07,0:28:32.97,Default,,0000,0000,0000,,Minus X cubed cause\NYDY by the X
Dialogue: 0,0:28:32.97,0:28:36.91,Default,,0000,0000,0000,,minus X squared Sign
Dialogue: 0,0:28:36.91,0:28:44.18,Default,,0000,0000,0000,,Y. +2 XYDY by\NX minus Y squared equals 0,
Dialogue: 0,0:28:44.18,0:28:50.76,Default,,0000,0000,0000,,so I've multiplied throughout by\Nthis X squared so the X
Dialogue: 0,0:28:50.76,0:28:55.54,Default,,0000,0000,0000,,squared is appeared there,\Nmultiplying that it's appeared
Dialogue: 0,0:28:55.54,0:29:00.93,Default,,0000,0000,0000,,there inside that X cubed,\Nmultiplying that it's appeared
Dialogue: 0,0:29:00.93,0:29:07.50,Default,,0000,0000,0000,,there, multiplying the sign Y,\Nand it's gone. From here, 'cause
Dialogue: 0,0:29:07.50,0:29:09.90,Default,,0000,0000,0000,,we've multiplied by so.
Dialogue: 0,0:29:09.93,0:29:15.11,Default,,0000,0000,0000,,Multiplying and dividing by, in\Neffect, leaving this on changed.
Dialogue: 0,0:29:15.11,0:29:21.84,Default,,0000,0000,0000,,Now let's get together all the\Nterms in DY by DX, so we
Dialogue: 0,0:29:21.84,0:29:28.58,Default,,0000,0000,0000,,have the why by DX times this\Nterm, three X squared Y squared.
Dialogue: 0,0:29:29.10,0:29:33.00,Default,,0000,0000,0000,,This term minus X cubed
Dialogue: 0,0:29:33.00,0:29:39.96,Default,,0000,0000,0000,,cause why? This\Nterm +2 XY.
Dialogue: 0,0:29:41.63,0:29:46.31,Default,,0000,0000,0000,,And here I've got minus X\Nsquared sign Y or I think I want
Dialogue: 0,0:29:46.31,0:29:50.98,Default,,0000,0000,0000,,to add that to the other side.\NSo that's plus X squared sign Y,
Dialogue: 0,0:29:50.98,0:29:54.99,Default,,0000,0000,0000,,and here I've got minus Y\Nsquared. Again, I think I want
Dialogue: 0,0:29:54.99,0:29:59.67,Default,,0000,0000,0000,,to add that to both sides, so I\Nget plus Y squared over there.
Dialogue: 0,0:30:00.34,0:30:08.06,Default,,0000,0000,0000,,Now why by X is\Nequal to X squared sign
Dialogue: 0,0:30:08.06,0:30:15.78,Default,,0000,0000,0000,,Y plus Y squared in\Nthe numerator and dividing by
Dialogue: 0,0:30:15.78,0:30:22.73,Default,,0000,0000,0000,,this expression as the\Ndenominator. Three X squared Y
Dialogue: 0,0:30:22.73,0:30:28.90,Default,,0000,0000,0000,,squared minus X cubed cause\NY, +2 XY.
Dialogue: 0,0:30:28.94,0:30:31.71,Default,,0000,0000,0000,,Notice how much shorter\Nautomating that
Dialogue: 0,0:30:31.71,0:30:35.39,Default,,0000,0000,0000,,Differentiation's made? What's\Nquite a complicated problem, and
Dialogue: 0,0:30:35.39,0:30:40.46,Default,,0000,0000,0000,,that's something you want to\Nwork at. Trying to automate your
Dialogue: 0,0:30:40.46,0:30:45.54,Default,,0000,0000,0000,,differentiation so you don't\Nhave to go through the rules and
Dialogue: 0,0:30:45.54,0:30:47.84,Default,,0000,0000,0000,,write them down every time.