[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.68,0:00:06.64,Default,,0000,0000,0000,,Functions often come defined\Nas quotients, so.
Dialogue: 0,0:00:06.81,0:00:11.98,Default,,0000,0000,0000,,Let's just write that word down\Nquotients so functions come
Dialogue: 0,0:00:11.98,0:00:17.15,Default,,0000,0000,0000,,defined. This quotients by,\Nwhich we mean we have one
Dialogue: 0,0:00:17.15,0:00:21.80,Default,,0000,0000,0000,,function cause X, let's say\Ndivided by another function,
Dialogue: 0,0:00:21.80,0:00:28.52,Default,,0000,0000,0000,,cause X divided by X squared.\NWhat we do is we identify that
Dialogue: 0,0:00:28.52,0:00:33.18,Default,,0000,0000,0000,,as one function you divided by\Nanother function V.
Dialogue: 0,0:00:34.06,0:00:37.75,Default,,0000,0000,0000,,This then gives us yet another
Dialogue: 0,0:00:37.75,0:00:42.49,Default,,0000,0000,0000,,result. Another formula that we\Nneed to be able to remember.
Dialogue: 0,0:00:43.50,0:00:46.84,Default,,0000,0000,0000,,This one goes VU
Dialogue: 0,0:00:46.84,0:00:53.23,Default,,0000,0000,0000,,by TX. Minus\Nyou DV by
Dialogue: 0,0:00:53.23,0:00:56.32,Default,,0000,0000,0000,,DX all over
Dialogue: 0,0:00:56.32,0:01:01.34,Default,,0000,0000,0000,,V squared. Now it looks very\Ncomplicated formula, but it's
Dialogue: 0,0:01:01.34,0:01:04.50,Default,,0000,0000,0000,,not really you just have to\Nremember the minus side and I
Dialogue: 0,0:01:04.50,0:01:07.92,Default,,0000,0000,0000,,must say the way that I always\Nremember it is if anything is
Dialogue: 0,0:01:07.92,0:01:11.34,Default,,0000,0000,0000,,going to go wrong with anything,\Nit's going to be what's in the
Dialogue: 0,0:01:11.34,0:01:14.23,Default,,0000,0000,0000,,denominator. So when we do the\Nderivative. We're going to have
Dialogue: 0,0:01:14.23,0:01:15.55,Default,,0000,0000,0000,,to have a minus sign.
Dialogue: 0,0:01:16.25,0:01:20.81,Default,,0000,0000,0000,,Let's have a look how this\Nformula is going to work with
Dialogue: 0,0:01:20.81,0:01:21.95,Default,,0000,0000,0000,,this particular example.
Dialogue: 0,0:01:22.50,0:01:29.32,Default,,0000,0000,0000,,So let's have a\Nlook at this example.
Dialogue: 0,0:01:29.32,0:01:36.15,Default,,0000,0000,0000,,Y equals cause X\Nover X squared, and
Dialogue: 0,0:01:36.15,0:01:42.97,Default,,0000,0000,0000,,we've identified this as\Nbeing you over VU
Dialogue: 0,0:01:42.97,0:01:45.53,Default,,0000,0000,0000,,divided by V.
Dialogue: 0,0:01:46.31,0:01:49.75,Default,,0000,0000,0000,,So you is
Dialogue: 0,0:01:49.75,0:01:57.09,Default,,0000,0000,0000,,cause X. And\Nthe is X squared. We
Dialogue: 0,0:01:57.09,0:02:03.73,Default,,0000,0000,0000,,can write down their\Nderivatives. Do you buy DX
Dialogue: 0,0:02:03.73,0:02:11.11,Default,,0000,0000,0000,,is minus sign X and\NDV by DX is 2
Dialogue: 0,0:02:11.11,0:02:14.89,Default,,0000,0000,0000,,X? Quote the
Dialogue: 0,0:02:14.89,0:02:18.63,Default,,0000,0000,0000,,formula. DY by the X
Dialogue: 0,0:02:18.63,0:02:24.31,Default,,0000,0000,0000,,is. VU\Nby DX
Dialogue: 0,0:02:24.31,0:02:30.19,Default,,0000,0000,0000,,minus UDVX\Nall over
Dialogue: 0,0:02:30.19,0:02:33.14,Default,,0000,0000,0000,,the squared.
Dialogue: 0,0:02:34.48,0:02:37.43,Default,,0000,0000,0000,,Again, we're quoting the formula\Nevery time, because that way we
Dialogue: 0,0:02:37.43,0:02:39.30,Default,,0000,0000,0000,,get to remember it. We get to
Dialogue: 0,0:02:39.30,0:02:45.42,Default,,0000,0000,0000,,know it. Equals and now we can\Nplug in the various bits that
Dialogue: 0,0:02:45.42,0:02:48.47,Default,,0000,0000,0000,,we've got. So V will be X
Dialogue: 0,0:02:48.47,0:02:55.74,Default,,0000,0000,0000,,squared. Times by DU by\NDX. So that's times by minus
Dialogue: 0,0:02:55.74,0:03:01.79,Default,,0000,0000,0000,,sign X. Minus because of the\Nminus that there is in the
Dialogue: 0,0:03:01.79,0:03:05.58,Default,,0000,0000,0000,,formula and then we want you\Nwhich is convex.
Dialogue: 0,0:03:06.16,0:03:12.51,Default,,0000,0000,0000,,Times by and we want DV by\NDX, which is 2 X.
Dialogue: 0,0:03:13.01,0:03:16.03,Default,,0000,0000,0000,,And then all over.
Dialogue: 0,0:03:16.88,0:03:22.90,Default,,0000,0000,0000,,The squared, and in this case V\Nis X squared, so that's X
Dialogue: 0,0:03:22.90,0:03:24.29,Default,,0000,0000,0000,,squared all squared.
Dialogue: 0,0:03:25.21,0:03:30.02,Default,,0000,0000,0000,,Now again, this doesn't look\Nvery nice. It needs tidying up.
Dialogue: 0,0:03:30.02,0:03:35.26,Default,,0000,0000,0000,,We need to gather things\Ntogether, so if I turn over the
Dialogue: 0,0:03:35.26,0:03:40.07,Default,,0000,0000,0000,,page, write this expression\Nagain at the top of the page.
Dialogue: 0,0:03:40.84,0:03:46.96,Default,,0000,0000,0000,,Writing that down\Nagain DYX is
Dialogue: 0,0:03:46.96,0:03:53.39,Default,,0000,0000,0000,,equal 2.\NX squared times
Dialogue: 0,0:03:53.39,0:04:00.42,Default,,0000,0000,0000,,minus sign X\Nminus cause X.
Dialogue: 0,0:04:00.96,0:04:08.71,Default,,0000,0000,0000,,Times 2X all\Nover X squared
Dialogue: 0,0:04:08.71,0:04:15.02,Default,,0000,0000,0000,,squared. OK, we need to\Nsimplify this look again for the
Dialogue: 0,0:04:15.02,0:04:20.82,Default,,0000,0000,0000,,common factor, and there's an X\Nthere in the X. Squared is a
Dialogue: 0,0:04:20.82,0:04:26.62,Default,,0000,0000,0000,,minus sign there a minus sign\Nthere and an X there. So from
Dialogue: 0,0:04:26.62,0:04:33.31,Default,,0000,0000,0000,,each term we can take out the\Nminus sign and the X and on top
Dialogue: 0,0:04:33.31,0:04:36.43,Default,,0000,0000,0000,,that will leave. As with X Sign
Dialogue: 0,0:04:36.43,0:04:43.10,Default,,0000,0000,0000,,X. Plus most be a plus\Nnow because we've taken the
Dialogue: 0,0:04:43.10,0:04:47.92,Default,,0000,0000,0000,,minus sign out plus and here\Nwe've got 2.
Dialogue: 0,0:04:47.96,0:04:50.22,Default,,0000,0000,0000,,Kohl's X left.
Dialogue: 0,0:04:50.89,0:04:57.75,Default,,0000,0000,0000,,All over and now X squared all\Nsquared is X to the power 4.
Dialogue: 0,0:04:58.59,0:05:04.13,Default,,0000,0000,0000,,Having done that, we can see\Nthat there's now a factor of X
Dialogue: 0,0:05:04.13,0:05:09.24,Default,,0000,0000,0000,,common to both the top, the\Nnumerator and to the bottom the
Dialogue: 0,0:05:09.24,0:05:14.78,Default,,0000,0000,0000,,denominator. So we can divide\Nthe top and the bottom by X in
Dialogue: 0,0:05:14.78,0:05:19.46,Default,,0000,0000,0000,,order to simplify it, so the\Nminus sign stays minus X.
Dialogue: 0,0:05:20.10,0:05:23.62,Default,,0000,0000,0000,,Sign X +2
Dialogue: 0,0:05:23.62,0:05:30.97,Default,,0000,0000,0000,,cause X. All\Nover X cubed and we can leave
Dialogue: 0,0:05:30.97,0:05:32.63,Default,,0000,0000,0000,,it like that.
Dialogue: 0,0:05:33.72,0:05:36.04,Default,,0000,0000,0000,,Simplified in that way.
Dialogue: 0,0:05:36.66,0:05:40.76,Default,,0000,0000,0000,,And that's useful, because now\Nif we wanted to, we could go on
Dialogue: 0,0:05:40.76,0:05:45.16,Default,,0000,0000,0000,,and put it equal to 0 and we\Ncould sort out Maxima and minima
Dialogue: 0,0:05:45.16,0:05:49.26,Default,,0000,0000,0000,,and all that kind of thing. So\Nit's always helpful to try and
Dialogue: 0,0:05:49.26,0:05:52.41,Default,,0000,0000,0000,,rearrange these expressions,\Nparticularly to get the top in a
Dialogue: 0,0:05:52.41,0:05:55.56,Default,,0000,0000,0000,,simplified form. Notice though,\Nthat we didn't cancel any of
Dialogue: 0,0:05:55.56,0:05:59.97,Default,,0000,0000,0000,,these axes in here. That part of\Nthe sine X. That part of the
Dialogue: 0,0:05:59.97,0:06:03.44,Default,,0000,0000,0000,,variables. You can't just go\Ncanceling them out. You can only
Dialogue: 0,0:06:03.44,0:06:05.01,Default,,0000,0000,0000,,cancel those out, which are
Dialogue: 0,0:06:05.01,0:06:10.98,Default,,0000,0000,0000,,common factors. Let's take a\Nsecond example. This time. Let's
Dialogue: 0,0:06:10.98,0:06:16.83,Default,,0000,0000,0000,,take it one that's just got\Npolynomial functions of accent,
Dialogue: 0,0:06:16.83,0:06:22.10,Default,,0000,0000,0000,,so we've got X squared +6\Nall over 2X.
Dialogue: 0,0:06:22.62,0:06:29.44,Default,,0000,0000,0000,,Minus 7, just polynomials now\Ncauses no signs etc, so this is
Dialogue: 0,0:06:29.44,0:06:32.84,Default,,0000,0000,0000,,a U over VA quotient again.
Dialogue: 0,0:06:34.24,0:06:42.20,Default,,0000,0000,0000,,Let's line this one up. You\Nequals X squared, and so do
Dialogue: 0,0:06:42.20,0:06:46.17,Default,,0000,0000,0000,,you find the X will be
Dialogue: 0,0:06:46.17,0:06:52.99,Default,,0000,0000,0000,,2X. The is 2X\Nminus Seven, and so DV
Dialogue: 0,0:06:52.99,0:06:56.44,Default,,0000,0000,0000,,by DX is just two.
Dialogue: 0,0:06:57.24,0:07:01.06,Default,,0000,0000,0000,,Quote the formula Y
Dialogue: 0,0:07:01.06,0:07:04.95,Default,,0000,0000,0000,,equals V. You buy
Dialogue: 0,0:07:04.95,0:07:11.50,Default,,0000,0000,0000,,TX. Minus\NUDVX
Dialogue: 0,0:07:12.54,0:07:15.56,Default,,0000,0000,0000,,All over the squared.
Dialogue: 0,0:07:17.02,0:07:22.14,Default,,0000,0000,0000,,Now we quoted the formula. We\Nnow in a position to be able to
Dialogue: 0,0:07:22.14,0:07:25.44,Default,,0000,0000,0000,,substitute in the various pieces\Nthat we need so.
Dialogue: 0,0:07:26.25,0:07:28.97,Default,,0000,0000,0000,,Why by
Dialogue: 0,0:07:28.97,0:07:34.24,Default,,0000,0000,0000,,the X?\NIs equal 2.
Dialogue: 0,0:07:34.98,0:07:41.48,Default,,0000,0000,0000,,Now this is VDU by DX\Nso that's 2X minus 7.
Dialogue: 0,0:07:42.51,0:07:49.35,Default,,0000,0000,0000,,Times by du by DX,\Nwhich is 2 X.
Dialogue: 0,0:07:49.35,0:07:56.59,Default,,0000,0000,0000,,Minus. U which was\NX squared plus six times divided
Dialogue: 0,0:07:56.59,0:08:03.100,Default,,0000,0000,0000,,by DX, which was just two\Nand this is all over the
Dialogue: 0,0:08:03.100,0:08:09.55,Default,,0000,0000,0000,,square so it's all over 2X\Nminus 7 squared.
Dialogue: 0,0:08:10.44,0:08:15.15,Default,,0000,0000,0000,,Now again we need to think about\Nthis one. We need to simplify
Dialogue: 0,0:08:15.15,0:08:19.85,Default,,0000,0000,0000,,it. We need to get together the\Nvarious terms and if we look,
Dialogue: 0,0:08:19.85,0:08:24.56,Default,,0000,0000,0000,,there's a common factor of two\Nthere and there so we can take
Dialogue: 0,0:08:24.56,0:08:29.99,Default,,0000,0000,0000,,that two out as a common factor\Nand put it at the front. So we
Dialogue: 0,0:08:29.99,0:08:36.59,Default,,0000,0000,0000,,have two. Then when we multiply\Nout with X times by two X, that
Dialogue: 0,0:08:36.59,0:08:38.74,Default,,0000,0000,0000,,gives us two X squared.
Dialogue: 0,0:08:40.05,0:08:47.03,Default,,0000,0000,0000,,X times by 7 gives us minus\NSeven X. Then we have minus
Dialogue: 0,0:08:47.03,0:08:52.94,Default,,0000,0000,0000,,this, so it's minus X squared\Nminus six. Close the bracket.
Dialogue: 0,0:08:53.68,0:09:00.45,Default,,0000,0000,0000,,All over 2X\Nminus Seven or
Dialogue: 0,0:09:00.45,0:09:06.95,Default,,0000,0000,0000,,square. Keep the two outside and\Nlet's simplify the terms inside.
Dialogue: 0,0:09:07.57,0:09:13.29,Default,,0000,0000,0000,,X squared Minus Seven\NX minus 6.
Dialogue: 0,0:09:13.98,0:09:17.78,Default,,0000,0000,0000,,All over 2X minus Seven\Nor squared and that's
Dialogue: 0,0:09:17.78,0:09:21.100,Default,,0000,0000,0000,,now informed. We need to\Ngo on and do something
Dialogue: 0,0:09:21.100,0:09:24.53,Default,,0000,0000,0000,,else with it we can do.
Dialogue: 0,0:09:25.82,0:09:30.03,Default,,0000,0000,0000,,3rd example that I'd like to\Ndo with you is one where
Dialogue: 0,0:09:30.03,0:09:33.89,Default,,0000,0000,0000,,we're going to use this\Nresult in order to help us
Dialogue: 0,0:09:33.89,0:09:34.95,Default,,0000,0000,0000,,establish something new.
Dialogue: 0,0:09:36.66,0:09:42.94,Default,,0000,0000,0000,,Now we're going to use this\Nresult to help us prove another
Dialogue: 0,0:09:42.94,0:09:48.41,Default,,0000,0000,0000,,result. So let's begin with Y\Nequals 10 X. It's a standard
Dialogue: 0,0:09:48.41,0:09:53.25,Default,,0000,0000,0000,,function, so we want to be able\Nto differentiate it in a
Dialogue: 0,0:09:53.25,0:09:58.49,Default,,0000,0000,0000,,standard way. We want to result.\NWe can use and just keep on
Dialogue: 0,0:09:58.49,0:10:04.13,Default,,0000,0000,0000,,using it. So we've got to begin\Nwith the definition of Tan X Tan
Dialogue: 0,0:10:04.13,0:10:09.37,Default,,0000,0000,0000,,X is defined as being sign X\Nover cause X, and of course
Dialogue: 0,0:10:09.37,0:10:14.20,Default,,0000,0000,0000,,that's now a quotient, isn't it?\NThat's now you over V, because
Dialogue: 0,0:10:14.20,0:10:20.99,Default,,0000,0000,0000,,we've been. Able to identify the\Nyou over V. Then we can have U
Dialogue: 0,0:10:20.99,0:10:27.89,Default,,0000,0000,0000,,equals sign X and so do you buy\Nthe X will be cause X.
Dialogue: 0,0:10:29.01,0:10:36.43,Default,,0000,0000,0000,,We've got the equals cause X\Nand so DV by the X
Dialogue: 0,0:10:36.43,0:10:38.28,Default,,0000,0000,0000,,will be minus.
Dialogue: 0,0:10:38.83,0:10:44.59,Default,,0000,0000,0000,,Sign X. Lots of causes and\Nsigns about, so we need to be
Dialogue: 0,0:10:44.59,0:10:46.35,Default,,0000,0000,0000,,very, very careful when we do
Dialogue: 0,0:10:46.35,0:10:53.69,Default,,0000,0000,0000,,the substitution. Close quote\Nthe formula DY by
Dialogue: 0,0:10:53.69,0:11:01.41,Default,,0000,0000,0000,,the X is VU\Nby DX minus UDVX
Dialogue: 0,0:11:01.41,0:11:05.28,Default,,0000,0000,0000,,all over V squared.
Dialogue: 0,0:11:06.40,0:11:11.13,Default,,0000,0000,0000,,And we're going to make this\Nsubstitution, so let's just work
Dialogue: 0,0:11:11.13,0:11:12.85,Default,,0000,0000,0000,,our way through that.
Dialogue: 0,0:11:14.08,0:11:17.37,Default,,0000,0000,0000,,The why by DX.
Dialogue: 0,0:11:17.99,0:11:25.67,Default,,0000,0000,0000,,Will be. Now it's VDU\Nby the X so V was
Dialogue: 0,0:11:25.67,0:11:26.97,Default,,0000,0000,0000,,cause X.
Dialogue: 0,0:11:28.15,0:11:35.91,Default,,0000,0000,0000,,You was sign X, so\Nits derivative is caused X.
Dialogue: 0,0:11:36.61,0:11:39.23,Default,,0000,0000,0000,,Minus from the formula.
Dialogue: 0,0:11:40.35,0:11:46.57,Default,,0000,0000,0000,,You divvy by DX. Now\Nyou was sign X.
Dialogue: 0,0:11:47.33,0:11:54.96,Default,,0000,0000,0000,,And DV by the X will V was cause\Nexo DV by the X is minus sign X.
Dialogue: 0,0:11:55.84,0:12:03.34,Default,,0000,0000,0000,,Oh, over.\NThe squared and V was cause X,
Dialogue: 0,0:12:03.34,0:12:06.56,Default,,0000,0000,0000,,so that's all over Cos squared
Dialogue: 0,0:12:06.56,0:12:13.73,Default,,0000,0000,0000,,X. We now need to simplify the\Ntop so we have cause X times by
Dialogue: 0,0:12:13.73,0:12:19.30,Default,,0000,0000,0000,,cause X, so that's cost squared\NX. We have a minus and minus
Dialogue: 0,0:12:19.30,0:12:25.74,Default,,0000,0000,0000,,sign, so that's going to give us\Na plus and we've sign X times by
Dialogue: 0,0:12:25.74,0:12:27.88,Default,,0000,0000,0000,,cynex, so we've signed squared
Dialogue: 0,0:12:27.88,0:12:31.54,Default,,0000,0000,0000,,X. All over cause
Dialogue: 0,0:12:31.54,0:12:34.90,Default,,0000,0000,0000,,squared X. Equals.
Dialogue: 0,0:12:35.77,0:12:40.04,Default,,0000,0000,0000,,Now this is a standard result,\Nwell known result cost squared
Dialogue: 0,0:12:40.04,0:12:44.69,Default,,0000,0000,0000,,plus sign squared is always\Nequal to 1 cost squared X plus
Dialogue: 0,0:12:44.69,0:12:49.35,Default,,0000,0000,0000,,sign squared. X is one, so\Nthat's one over cause squared X
Dialogue: 0,0:12:49.35,0:12:54.01,Default,,0000,0000,0000,,and of course we have another\Nway of writing one over cost
Dialogue: 0,0:12:54.01,0:13:00.32,Default,,0000,0000,0000,,squared. One over kozaks we\Nusually write as being sack X,
Dialogue: 0,0:13:00.32,0:13:07.35,Default,,0000,0000,0000,,so one over cost squared X. We\Nwould write as SEK squared X.
Dialogue: 0,0:13:07.37,0:13:10.68,Default,,0000,0000,0000,,And so that's how we\Ndifferentiate tab. And now we've
Dialogue: 0,0:13:10.68,0:13:14.32,Default,,0000,0000,0000,,got a standard result that the\Nderivative of tangent is sex
Dialogue: 0,0:13:14.32,0:13:18.62,Default,,0000,0000,0000,,squared X. We can simply quote\Nthat and use it anytime that we
Dialogue: 0,0:13:18.62,0:13:22.60,Default,,0000,0000,0000,,want to. We take one more\Nexample of using this in the
Dialogue: 0,0:13:22.60,0:13:23.92,Default,,0000,0000,0000,,same sort of way.
Dialogue: 0,0:13:24.45,0:13:29.99,Default,,0000,0000,0000,,Let's take the function Y\Nequals second X.
Dialogue: 0,0:13:31.21,0:13:37.95,Default,,0000,0000,0000,,Now we know the definition of\NPsychics. It's one over cause X.
Dialogue: 0,0:13:38.91,0:13:44.37,Default,,0000,0000,0000,,One of the ways of doing this\Nnow is to realize that this is a
Dialogue: 0,0:13:44.37,0:13:46.19,Default,,0000,0000,0000,,quotient. It's AU over V.
Dialogue: 0,0:13:47.07,0:13:51.50,Default,,0000,0000,0000,,Having identified those, we\Ncan say you equals 1, and so
Dialogue: 0,0:13:51.50,0:13:57.14,Default,,0000,0000,0000,,do you buy. The X will be\Nequal to. Now one is just a
Dialogue: 0,0:13:57.14,0:14:00.77,Default,,0000,0000,0000,,constant, so remember a\Nconstant is about rate of
Dialogue: 0,0:14:00.77,0:14:04.40,Default,,0000,0000,0000,,derivative is about rate of\Nchange. So if we
Dialogue: 0,0:14:04.40,0:14:07.22,Default,,0000,0000,0000,,differentiate something\Nwhich is constant rate of
Dialogue: 0,0:14:07.22,0:14:08.83,Default,,0000,0000,0000,,change must be 0.
Dialogue: 0,0:14:10.20,0:14:17.77,Default,,0000,0000,0000,,The is cause X and\Nso DV by the X
Dialogue: 0,0:14:17.77,0:14:21.56,Default,,0000,0000,0000,,will be minus sign X.
Dialogue: 0,0:14:22.11,0:14:26.79,Default,,0000,0000,0000,,And again, our\Nformula is DY by
Dialogue: 0,0:14:26.79,0:14:29.46,Default,,0000,0000,0000,,X is equal to.
Dialogue: 0,0:14:30.75,0:14:34.14,Default,,0000,0000,0000,,The DU by The
Dialogue: 0,0:14:34.14,0:14:40.93,Default,,0000,0000,0000,,X. Minus UDV\Nby the X all
Dialogue: 0,0:14:40.93,0:14:43.63,Default,,0000,0000,0000,,over V squared.
Dialogue: 0,0:14:44.52,0:14:46.41,Default,,0000,0000,0000,,So we're going to make the
Dialogue: 0,0:14:46.41,0:14:51.07,Default,,0000,0000,0000,,substitution now. This is going\Nto change things slightly if do
Dialogue: 0,0:14:51.07,0:14:55.62,Default,,0000,0000,0000,,you buy the Axis zero when we\Nstart multiplying by zero sum of
Dialogue: 0,0:14:55.62,0:14:57.02,Default,,0000,0000,0000,,things may happen so.
Dialogue: 0,0:14:57.85,0:15:04.92,Default,,0000,0000,0000,,DY by DX is VDU\Nby DX now remember V
Dialogue: 0,0:15:04.92,0:15:11.99,Default,,0000,0000,0000,,was cause X times do\Nyou buy DX that was
Dialogue: 0,0:15:11.99,0:15:15.52,Default,,0000,0000,0000,,zero because you was one.
Dialogue: 0,0:15:16.20,0:15:19.78,Default,,0000,0000,0000,,Minus U, that's one.
Dialogue: 0,0:15:20.41,0:15:27.19,Default,,0000,0000,0000,,Times by DV by DX now remember V\Nwas cause X and so its
Dialogue: 0,0:15:27.19,0:15:32.99,Default,,0000,0000,0000,,derivative is minus sign X and\Nthen all over V squared, which
Dialogue: 0,0:15:32.99,0:15:39.77,Default,,0000,0000,0000,,is cost squared X. So what we've\Ngot cause X times by zero is
Dialogue: 0,0:15:39.77,0:15:46.06,Default,,0000,0000,0000,,zero. Anything times by zero is\N0 minus. Minus gives us a plus,
Dialogue: 0,0:15:46.06,0:15:50.90,Default,,0000,0000,0000,,and so we've got sine X over\Ncause squared X.
Dialogue: 0,0:15:51.63,0:15:57.50,Default,,0000,0000,0000,,Looks looks like it might be\Nsomething else, and remember
Dialogue: 0,0:15:57.50,0:16:04.54,Default,,0000,0000,0000,,that we've just seen that tan\Nis sign over cars, so I
Dialogue: 0,0:16:04.54,0:16:11.59,Default,,0000,0000,0000,,can write this as one over\Ncause X times sign X over
Dialogue: 0,0:16:11.59,0:16:18.04,Default,,0000,0000,0000,,cause X here I've got one\Nover cause is sex X.
Dialogue: 0,0:16:19.27,0:16:25.86,Default,,0000,0000,0000,,Times sign over cause which is\N10 X so I end up with the
Dialogue: 0,0:16:25.86,0:16:30.10,Default,,0000,0000,0000,,result that the derivative of\Nsex is sex tinix.
Dialogue: 0,0:16:30.62,0:16:35.61,Default,,0000,0000,0000,,So. Anytime we want to use the\Nderivative of sex, we can do so.
Dialogue: 0,0:16:35.61,0:16:39.47,Default,,0000,0000,0000,,All we do is we just write it\Nstraight down sex tanks and
Dialogue: 0,0:16:39.47,0:16:42.74,Default,,0000,0000,0000,,that's it. We don't have to work\Nit all out again.
Dialogue: 0,0:16:43.29,0:16:45.42,Default,,0000,0000,0000,,And that's the end of Quotients.