[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.02,0:00:07.06,Default,,0000,0000,0000,,Sometimes we have functions that\Nlook like this.
Dialogue: 0,0:00:07.06,0:00:10.81,Default,,0000,0000,0000,,Cause of X squared.
Dialogue: 0,0:00:11.77,0:00:18.21,Default,,0000,0000,0000,,Now there are immediate things\Nthat are different about this
Dialogue: 0,0:00:18.21,0:00:24.01,Default,,0000,0000,0000,,than the straightforward cosine\Nfunction. It's cause of X
Dialogue: 0,0:00:24.01,0:00:31.73,Default,,0000,0000,0000,,squared, not just cause of X,\Nand so we call this function
Dialogue: 0,0:00:31.73,0:00:38.17,Default,,0000,0000,0000,,of a function Y equals a\Nfunction of another function.
Dialogue: 0,0:00:39.03,0:00:40.65,Default,,0000,0000,0000,,Now in this case.
Dialogue: 0,0:00:41.25,0:00:48.25,Default,,0000,0000,0000,,The function F is the cosine\Nfunction, and the function G is
Dialogue: 0,0:00:48.25,0:00:54.66,Default,,0000,0000,0000,,the square function. Or we could\Nidentify them, perhaps a bit
Dialogue: 0,0:00:54.66,0:01:00.49,Default,,0000,0000,0000,,more mathematically by saying\Nthat F of X is cause
Dialogue: 0,0:01:00.49,0:01:07.48,Default,,0000,0000,0000,,X&G of X is X squared.\NNow let's have a look at
Dialogue: 0,0:01:07.48,0:01:09.82,Default,,0000,0000,0000,,another example of this.
Dialogue: 0,0:01:10.57,0:01:15.54,Default,,0000,0000,0000,,This time we'll turn it around,\Nso to speak. Let's have a look
Dialogue: 0,0:01:15.54,0:01:17.06,Default,,0000,0000,0000,,at cause squared X.
Dialogue: 0,0:01:17.76,0:01:23.76,Default,,0000,0000,0000,,Now, how is this a function of a\Nfunction? Let's remember what
Dialogue: 0,0:01:23.76,0:01:29.76,Default,,0000,0000,0000,,cost squared of X means. It\Nmeans cause X squared cause X
Dialogue: 0,0:01:29.76,0:01:36.26,Default,,0000,0000,0000,,multiplied by itself. So now if\Nwe look at our function of a
Dialogue: 0,0:01:36.26,0:01:42.52,Default,,0000,0000,0000,,function. Let's see what we can\Ndo to identify which is F and
Dialogue: 0,0:01:42.52,0:01:49.12,Default,,0000,0000,0000,,which is G of X. So here we are\Nsquaring, so it's the F is the
Dialogue: 0,0:01:49.12,0:01:53.24,Default,,0000,0000,0000,,square function, and inside the\Nsquare function, the thing that
Dialogue: 0,0:01:53.24,0:01:59.42,Default,,0000,0000,0000,,we are actually squaring is G of\NX, namely cause X. So here the F
Dialogue: 0,0:01:59.42,0:02:04.77,Default,,0000,0000,0000,,is the square function and the G\Nis the cosine function. Or write
Dialogue: 0,0:02:04.77,0:02:07.24,Default,,0000,0000,0000,,it down more mathematically F of
Dialogue: 0,0:02:07.24,0:02:13.82,Default,,0000,0000,0000,,X. Is X squared? Angie of\NX is cause X?
Dialogue: 0,0:02:14.58,0:02:18.71,Default,,0000,0000,0000,,Now how do we differentiate\Na function like Cos squared
Dialogue: 0,0:02:18.71,0:02:23.67,Default,,0000,0000,0000,,X or a function like the\Ncause of X squared? What we
Dialogue: 0,0:02:23.67,0:02:27.80,Default,,0000,0000,0000,,need to do to be able to\Ndifferentiate something that
Dialogue: 0,0:02:27.80,0:02:30.27,Default,,0000,0000,0000,,is a function of a function?
Dialogue: 0,0:02:31.62,0:02:38.46,Default,,0000,0000,0000,,To do that, we need\Nto do two things. One,
Dialogue: 0,0:02:38.46,0:02:41.20,Default,,0000,0000,0000,,we need to substitute.
Dialogue: 0,0:02:42.87,0:02:50.67,Default,,0000,0000,0000,,You equals G of X. This would\Nthen give us Y equals F of
Dialogue: 0,0:02:50.67,0:02:57.91,Default,,0000,0000,0000,,you, which of course is much\Nsimpler than F of G of X.
Dialogue: 0,0:02:57.91,0:03:05.71,Default,,0000,0000,0000,,Next we need to use a rule\Nor a formula that's known as the
Dialogue: 0,0:03:05.71,0:03:10.61,Default,,0000,0000,0000,,chain rule. China\Nrules quite simple,
Dialogue: 0,0:03:10.61,0:03:17.60,Default,,0000,0000,0000,,says DY by DX is\Nequal to DY by EU
Dialogue: 0,0:03:17.60,0:03:21.09,Default,,0000,0000,0000,,Times du by The X.
Dialogue: 0,0:03:22.20,0:03:26.61,Default,,0000,0000,0000,,Notice it looks as though the D\Nuse cancel out. If these were
Dialogue: 0,0:03:26.61,0:03:29.100,Default,,0000,0000,0000,,fractions, which they're not, it\Nlooks as though they might
Dialogue: 0,0:03:29.100,0:03:32.03,Default,,0000,0000,0000,,cancel out. That's a way of
Dialogue: 0,0:03:32.03,0:03:36.29,Default,,0000,0000,0000,,remembering it. So this is how\Nwe're going to approach it.
Dialogue: 0,0:03:36.29,0:03:38.14,Default,,0000,0000,0000,,Substitute you equals G of X.
Dialogue: 0,0:03:38.66,0:03:43.51,Default,,0000,0000,0000,,And then apply this rule called\Nthe chain rule in order to find
Dialogue: 0,0:03:43.51,0:03:47.98,Default,,0000,0000,0000,,the derivative. So let's take a\Nnumber of examples and the first
Dialogue: 0,0:03:47.98,0:03:52.46,Default,,0000,0000,0000,,one will take is the very first\Nexample that we looked at.
Dialogue: 0,0:03:53.19,0:03:59.87,Default,,0000,0000,0000,,And that was why equals\Ncause of X squared.
Dialogue: 0,0:04:00.38,0:04:07.52,Default,,0000,0000,0000,,So our first step was to\Nput you equals X squared.
Dialogue: 0,0:04:08.08,0:04:14.65,Default,,0000,0000,0000,,And then Y is\Nequal to cause you.
Dialogue: 0,0:04:15.84,0:04:23.56,Default,,0000,0000,0000,,Now the chain rule says DY\Nby the X is equal to
Dialogue: 0,0:04:23.56,0:04:30.63,Default,,0000,0000,0000,,DY by du times du by\Nthe X. So we need.
Dialogue: 0,0:04:31.26,0:04:32.65,Default,,0000,0000,0000,,Do you buy the X?
Dialogue: 0,0:04:33.33,0:04:37.71,Default,,0000,0000,0000,,Well, that's 2X. We\Ndifferentiate this. We multiply
Dialogue: 0,0:04:37.71,0:04:45.39,Default,,0000,0000,0000,,by the two and take one off\Nthe index. That leaves us 2X and
Dialogue: 0,0:04:45.39,0:04:48.67,Default,,0000,0000,0000,,we need the why by DU.
Dialogue: 0,0:04:49.57,0:04:55.09,Default,,0000,0000,0000,,And the derivative of Cos you is\Nminus sign you.
Dialogue: 0,0:04:55.99,0:05:02.37,Default,,0000,0000,0000,,So now we need to put these\Ntogether DY by the X is equal
Dialogue: 0,0:05:02.37,0:05:09.72,Default,,0000,0000,0000,,to. Divide by do you minus\Nsign U times do you buy DX
Dialogue: 0,0:05:09.72,0:05:11.83,Default,,0000,0000,0000,,which is 2 X?
Dialogue: 0,0:05:12.53,0:05:17.83,Default,,0000,0000,0000,,Now this is all very well, but\Nreally we'd like to have
Dialogue: 0,0:05:17.83,0:05:23.58,Default,,0000,0000,0000,,everything in terms of X. And\Nhere we've got you, so we need
Dialogue: 0,0:05:23.58,0:05:29.33,Default,,0000,0000,0000,,to undo this substitution. If we\Nput you equals X squared, we now
Dialogue: 0,0:05:29.33,0:05:35.51,Default,,0000,0000,0000,,need to replace EU by X squared,\Nand so we write minus 2X and
Dialogue: 0,0:05:35.51,0:05:40.38,Default,,0000,0000,0000,,bringing the two X to the front\Nsign of X squared.
Dialogue: 0,0:05:40.89,0:05:46.58,Default,,0000,0000,0000,,Now they're all done like that.\NWhat I'm going to do with all
Dialogue: 0,0:05:46.58,0:05:51.84,Default,,0000,0000,0000,,the next examples that I do is\NI'm going to put this
Dialogue: 0,0:05:51.84,0:05:56.66,Default,,0000,0000,0000,,differentiation of the two bits\Nup here with these two bits.
Dialogue: 0,0:05:58.19,0:06:05.14,Default,,0000,0000,0000,,Let's have a look at the\Nother one. We had. Y equals
Dialogue: 0,0:06:05.14,0:06:09.77,Default,,0000,0000,0000,,cause squared X. Let's\Nremember that meant the
Dialogue: 0,0:06:09.77,0:06:12.66,Default,,0000,0000,0000,,cause of X all squared.
Dialogue: 0,0:06:13.89,0:06:21.70,Default,,0000,0000,0000,,So this is now the G\Nof X and so we will
Dialogue: 0,0:06:21.70,0:06:25.61,Default,,0000,0000,0000,,put you equals cause of X.
Dialogue: 0,0:06:26.34,0:06:32.55,Default,,0000,0000,0000,,Then Y is equal\Nto you squared.
Dialogue: 0,0:06:33.53,0:06:40.46,Default,,0000,0000,0000,,I can calculate do you\Nbuy DX that's minus sign
Dialogue: 0,0:06:40.46,0:06:47.39,Default,,0000,0000,0000,,X and I can calculate\NDY by du. That's two
Dialogue: 0,0:06:47.39,0:06:55.18,Default,,0000,0000,0000,,you. Write down the chain\Nrule the why by DX is
Dialogue: 0,0:06:55.18,0:06:59.03,Default,,0000,0000,0000,,DY by du times DU by
Dialogue: 0,0:06:59.03,0:07:05.11,Default,,0000,0000,0000,,The X. And now we can substitute\Nin the bits that we've already
Dialogue: 0,0:07:05.11,0:07:07.70,Default,,0000,0000,0000,,calculated. The why by do you is
Dialogue: 0,0:07:07.70,0:07:14.43,Default,,0000,0000,0000,,to you. Times do you buy DX\Nwhich is minus sign X equals.
Dialogue: 0,0:07:14.43,0:07:21.84,Default,,0000,0000,0000,,Now again we want this all in\Nterms of X. So what we have to
Dialogue: 0,0:07:21.84,0:07:28.27,Default,,0000,0000,0000,,do is reverse the substitution\Nwe put you equals to cause X and
Dialogue: 0,0:07:28.27,0:07:34.69,Default,,0000,0000,0000,,now we need to undo that by\Nreplacing EU with cause X. So
Dialogue: 0,0:07:34.69,0:07:39.13,Default,,0000,0000,0000,,bringing the minus sign to the\Nfront minus 2.
Dialogue: 0,0:07:39.15,0:07:41.96,Default,,0000,0000,0000,,Kohl's X
Dialogue: 0,0:07:41.96,0:07:48.45,Default,,0000,0000,0000,,Sign X.\NLet's take
Dialogue: 0,0:07:48.45,0:07:52.00,Default,,0000,0000,0000,,another example.
Dialogue: 0,0:07:53.24,0:08:00.62,Default,,0000,0000,0000,,Y equals 2X minus five\Nall raised to the power
Dialogue: 0,0:08:00.62,0:08:06.73,Default,,0000,0000,0000,,10. Now, it might be tempting to\Nsay, well surely we could just
Dialogue: 0,0:08:06.73,0:08:11.38,Default,,0000,0000,0000,,multiply out the brackets, but\Nthis is to the power 10 to
Dialogue: 0,0:08:11.38,0:08:15.26,Default,,0000,0000,0000,,multiply out. Those brackets\Nwould take his ages, and there's
Dialogue: 0,0:08:15.26,0:08:20.31,Default,,0000,0000,0000,,all those mistakes that could be\Nmade in doing it. Plus when we
Dialogue: 0,0:08:20.31,0:08:24.57,Default,,0000,0000,0000,,differentiate it, we may not\Nhave the best form for future
Dialogue: 0,0:08:24.57,0:08:29.62,Default,,0000,0000,0000,,work, so let's use function of a\Nfunction. So here we will put
Dialogue: 0,0:08:29.62,0:08:31.95,Default,,0000,0000,0000,,you equals this bit here inside
Dialogue: 0,0:08:31.95,0:08:35.39,Default,,0000,0000,0000,,the bracket. 2X minus five
Dialogue: 0,0:08:35.39,0:08:42.74,Default,,0000,0000,0000,,and then. Why is\Nequal to U to the power
Dialogue: 0,0:08:42.74,0:08:43.38,Default,,0000,0000,0000,,10?
Dialogue: 0,0:08:44.55,0:08:50.40,Default,,0000,0000,0000,,I can now do the differentiation\Nof the little bits. Do you buy
Dialogue: 0,0:08:50.40,0:08:57.15,Default,,0000,0000,0000,,the X is just two the derivative\Nof two X just giving us two and
Dialogue: 0,0:08:57.15,0:09:04.35,Default,,0000,0000,0000,,E? Why by DU is 10 we multiply\Nby the index U to the power 9.
Dialogue: 0,0:09:04.35,0:09:10.20,Default,,0000,0000,0000,,We take one away from the index.\NNow we can put this together
Dialogue: 0,0:09:10.20,0:09:15.15,Default,,0000,0000,0000,,using the chain rule so divided\Nby DX is equal to.
Dialogue: 0,0:09:15.88,0:09:23.39,Default,,0000,0000,0000,,The why by DU times\NDU by X, substituting our
Dialogue: 0,0:09:23.39,0:09:30.90,Default,,0000,0000,0000,,little bits. Here's the why\Nby DU10U to the ninth
Dialogue: 0,0:09:30.90,0:09:34.66,Default,,0000,0000,0000,,times two do you buy
Dialogue: 0,0:09:34.66,0:09:41.40,Default,,0000,0000,0000,,DX? 2 * 10 is 20 and I\Nwant you to the power 9 and I
Dialogue: 0,0:09:41.40,0:09:47.16,Default,,0000,0000,0000,,need to get this all in terms of\NX, so I need to replace the you
Dialogue: 0,0:09:47.16,0:09:53.63,Default,,0000,0000,0000,,here. By the two X minus five.\NSo that's 2X minus five all to
Dialogue: 0,0:09:53.63,0:09:57.61,Default,,0000,0000,0000,,the power 9, and that's a\Ncompact expression for the
Dialogue: 0,0:09:57.61,0:10:02.39,Default,,0000,0000,0000,,derivative. Think what it would\Nhave been like if I had to
Dialogue: 0,0:10:02.39,0:10:05.18,Default,,0000,0000,0000,,expand the brackets and\Ndifferentiate each term.
Dialogue: 0,0:10:06.52,0:10:13.31,Default,,0000,0000,0000,,I want now to take another trick\Nexample and then develope that
Dialogue: 0,0:10:13.31,0:10:19.54,Default,,0000,0000,0000,,trig example a little bit\Nfurther to a more general case.
Dialogue: 0,0:10:21.02,0:10:27.24,Default,,0000,0000,0000,,So we'll take\NY equals the
Dialogue: 0,0:10:27.24,0:10:30.35,Default,,0000,0000,0000,,sign of 5X.
Dialogue: 0,0:10:31.09,0:10:37.95,Default,,0000,0000,0000,,Very easy here to identify the\NG of X. It's this bit
Dialogue: 0,0:10:37.95,0:10:43.10,Default,,0000,0000,0000,,here the 5X so will put\Nyou equals 5X.
Dialogue: 0,0:10:43.79,0:10:49.86,Default,,0000,0000,0000,,And then why will be\Nequal to sign you?
Dialogue: 0,0:10:50.89,0:10:57.81,Default,,0000,0000,0000,,Differentiating do you buy the\NX is equal to 5.
Dialogue: 0,0:10:58.54,0:11:04.89,Default,,0000,0000,0000,,And DY bite U is\Nequal to cause you.
Dialogue: 0,0:11:05.56,0:11:12.64,Default,,0000,0000,0000,,Put the two bits together with\Nthe chain rule DY by the
Dialogue: 0,0:11:12.64,0:11:19.72,Default,,0000,0000,0000,,X is equal to DY by\NEU Times DU by The X.
Dialogue: 0,0:11:20.71,0:11:22.95,Default,,0000,0000,0000,,DIY Bindu is cause you.
Dialogue: 0,0:11:23.46,0:11:29.98,Default,,0000,0000,0000,,Times by and you by the\Nex here is 5.
Dialogue: 0,0:11:30.90,0:11:38.11,Default,,0000,0000,0000,,So let's bring the five to\Nthe front 5 cause of, and
Dialogue: 0,0:11:38.11,0:11:43.52,Default,,0000,0000,0000,,now let's reverse the\Nsubstitution. You is equal to
Dialogue: 0,0:11:43.52,0:11:47.13,Default,,0000,0000,0000,,5X, so will replace EU by
Dialogue: 0,0:11:47.13,0:11:52.84,Default,,0000,0000,0000,,5X. Now, notice how that five\Nhere and here is apparently
Dialogue: 0,0:11:52.84,0:11:58.97,Default,,0000,0000,0000,,appeared there, and it did so\Nbecause the derivative of 5X was
Dialogue: 0,0:11:58.97,0:12:05.62,Default,,0000,0000,0000,,five. So the question is, could\Nwe do this with any number that
Dialogue: 0,0:12:05.62,0:12:12.77,Default,,0000,0000,0000,,appeared there in front of the X\Nbit five or six? Or 1/2 or
Dialogue: 0,0:12:12.77,0:12:16.35,Default,,0000,0000,0000,,not .5? Or for that matter, an
Dialogue: 0,0:12:16.35,0:12:23.37,Default,,0000,0000,0000,,so? Have a look at\NY equal sign NX.
Dialogue: 0,0:12:25.13,0:12:28.77,Default,,0000,0000,0000,,You equals an X that
Dialogue: 0,0:12:28.77,0:12:35.83,Default,,0000,0000,0000,,sour substitution. And then Y\Nis equal to sign you we can
Dialogue: 0,0:12:35.83,0:12:41.94,Default,,0000,0000,0000,,differentiate EU with respect to\NX and the derivative of NX is
Dialogue: 0,0:12:41.94,0:12:45.50,Default,,0000,0000,0000,,just N because N is a constant
Dialogue: 0,0:12:45.50,0:12:52.96,Default,,0000,0000,0000,,and number. And the why by\NDU is equal to cause you.
Dialogue: 0,0:12:52.96,0:12:56.43,Default,,0000,0000,0000,,We can now put this back
Dialogue: 0,0:12:56.43,0:13:03.63,Default,,0000,0000,0000,,together again. My by the X\Nis DY by EU Times
Dialogue: 0,0:13:03.63,0:13:06.31,Default,,0000,0000,0000,,DU by The X.
Dialogue: 0,0:13:07.11,0:13:13.86,Default,,0000,0000,0000,,Equals. DIY\NBindu is cause you.
Dialogue: 0,0:13:14.47,0:13:17.65,Default,,0000,0000,0000,,Times by du by DX, which is just
Dialogue: 0,0:13:17.65,0:13:25.40,Default,,0000,0000,0000,,N. So let's move the\Nend to the front. That's
Dialogue: 0,0:13:25.40,0:13:27.66,Default,,0000,0000,0000,,N cause NX.
Dialogue: 0,0:13:27.67,0:13:31.44,Default,,0000,0000,0000,,So the ends of behaved in\Nexactly the same way that the
Dialogue: 0,0:13:31.44,0:13:34.58,Default,,0000,0000,0000,,fives behaved in the previous\Nquestion. And of course this
Dialogue: 0,0:13:34.58,0:13:38.66,Default,,0000,0000,0000,,means. Now we're in a position\Nto be able to do any question
Dialogue: 0,0:13:38.66,0:13:40.54,Default,,0000,0000,0000,,like this simply by writing down
Dialogue: 0,0:13:40.54,0:13:46.62,Default,,0000,0000,0000,,the answer. So if we're just\Ngoing to write down the answer,
Dialogue: 0,0:13:46.62,0:13:53.68,Default,,0000,0000,0000,,let's take. Why is\Nsign of 6X?
Dialogue: 0,0:13:54.24,0:14:02.22,Default,,0000,0000,0000,,And then divide by the X is\Njust six cause 6X just by using
Dialogue: 0,0:14:02.22,0:14:05.64,Default,,0000,0000,0000,,the standard result that we had
Dialogue: 0,0:14:05.64,0:14:12.68,Default,,0000,0000,0000,,before. Similarly, if we had\NY was equal to cause 1/2, X
Dialogue: 0,0:14:12.68,0:14:18.38,Default,,0000,0000,0000,,will just because it's cause\Nit's not going to be any
Dialogue: 0,0:14:18.38,0:14:24.59,Default,,0000,0000,0000,,different to sign really and\Nso D. Why by DX is equal
Dialogue: 0,0:14:24.59,0:14:30.29,Default,,0000,0000,0000,,to 1/2 and the derivative of\Ncauses minus sign 1/2 X.
Dialogue: 0,0:14:31.33,0:14:38.32,Default,,0000,0000,0000,,Now let's take one more example.\NIn this particular style, so
Dialogue: 0,0:14:38.32,0:14:45.94,Default,,0000,0000,0000,,will take Y equals E to\Nthe X cubed. And here this
Dialogue: 0,0:14:45.94,0:14:53.56,Default,,0000,0000,0000,,X cubed is our G of\NX. So we will put you
Dialogue: 0,0:14:53.56,0:14:55.46,Default,,0000,0000,0000,,equals X cubed.
Dialogue: 0,0:14:55.46,0:15:02.47,Default,,0000,0000,0000,,And then why will be equal 2\NE to the power you we can
Dialogue: 0,0:15:02.47,0:15:08.49,Default,,0000,0000,0000,,differentiate. So do you find\Nthe X is equal to three X
Dialogue: 0,0:15:08.49,0:15:14.50,Default,,0000,0000,0000,,squared? We multiply by the\Npower and take one off it. And
Dialogue: 0,0:15:14.50,0:15:20.51,Default,,0000,0000,0000,,why by DU is equal to E\Nto the power you we
Dialogue: 0,0:15:20.51,0:15:25.02,Default,,0000,0000,0000,,differentiate with respect to\Nyou. The exponential function is
Dialogue: 0,0:15:25.02,0:15:29.13,Default,,0000,0000,0000,,it's. Own derivative, and so it\Nstays the same.
Dialogue: 0,0:15:30.13,0:15:35.86,Default,,0000,0000,0000,,Bringing those two bits together\Nto give us the why by DX through
Dialogue: 0,0:15:35.86,0:15:42.92,Default,,0000,0000,0000,,the chain rule DY by DX is DY by\Ndu times by DU by The X.
Dialogue: 0,0:15:43.52,0:15:50.74,Default,,0000,0000,0000,,The why Bindu is E to the power\Nyou times by du by DX which is
Dialogue: 0,0:15:50.74,0:15:52.09,Default,,0000,0000,0000,,3 X squared.
Dialogue: 0,0:15:52.89,0:15:56.89,Default,,0000,0000,0000,,Move the three X squared through\Nto the front.
Dialogue: 0,0:15:57.91,0:16:02.75,Default,,0000,0000,0000,,E to the power. And now we made\Nthe substitution that gave us
Dialogue: 0,0:16:02.75,0:16:07.21,Default,,0000,0000,0000,,you we put you equals X cubed\Nand now we need to.
Dialogue: 0,0:16:07.76,0:16:14.64,Default,,0000,0000,0000,,Reverse that substitution and\Nreplace U by X cubed.
Dialogue: 0,0:16:16.04,0:16:21.27,Default,,0000,0000,0000,,So let's have a look at what\Nwe've got here.
Dialogue: 0,0:16:21.91,0:16:28.88,Default,,0000,0000,0000,,E to the X cubed. Now if we\Nthink of this as E to EU then E
Dialogue: 0,0:16:28.88,0:16:34.21,Default,,0000,0000,0000,,to EU is just the derivative of\Nthe F with respect to you.
Dialogue: 0,0:16:34.21,0:16:38.31,Default,,0000,0000,0000,,Because the exponential function\Nis own derivative and here the
Dialogue: 0,0:16:38.31,0:16:43.23,Default,,0000,0000,0000,,three X squared is simply the\Nderivative of U with respect to
Dialogue: 0,0:16:43.23,0:16:48.56,Default,,0000,0000,0000,,X. In other words, it's the\Nderivative of G of X. Now let's
Dialogue: 0,0:16:48.56,0:16:53.07,Default,,0000,0000,0000,,see if we can put that together\Nfor a general case.
Dialogue: 0,0:16:54.06,0:17:01.47,Default,,0000,0000,0000,,So we've got Y equals\NF of G of X.
Dialogue: 0,0:17:02.73,0:17:09.83,Default,,0000,0000,0000,,And we're putting you equals G\Nof X and our chain rule tells
Dialogue: 0,0:17:09.83,0:17:15.29,Default,,0000,0000,0000,,us that the why by the X\Nis equal to.
Dialogue: 0,0:17:15.85,0:17:23.18,Default,,0000,0000,0000,,DY by EU Times\NDU by The X.
Dialogue: 0,0:17:23.75,0:17:28.72,Default,,0000,0000,0000,,Equals let's start here. You\Nremember that in most of the
Dialogue: 0,0:17:28.72,0:17:34.15,Default,,0000,0000,0000,,examples that I've done, I've\Nbeen pushing the result of du by
Dialogue: 0,0:17:34.15,0:17:37.31,Default,,0000,0000,0000,,DX forward to the front of the
Dialogue: 0,0:17:37.31,0:17:43.60,Default,,0000,0000,0000,,expression. So let's do it to\Nbegin with. Instead, at the end
Dialogue: 0,0:17:43.60,0:17:49.16,Default,,0000,0000,0000,,when we've been simplifying, so\Ndo you buy DX? This is the
Dialogue: 0,0:17:49.16,0:17:55.64,Default,,0000,0000,0000,,derivative of G of X, so that's\NDG by D of X by DX.
Dialogue: 0,0:17:56.55,0:18:03.40,Default,,0000,0000,0000,,Times by and this is the why\Nby DU. So that's the derivative
Dialogue: 0,0:18:03.40,0:18:08.67,Default,,0000,0000,0000,,of F of G of X with\Nrespect to you.
Dialogue: 0,0:18:09.85,0:18:16.19,Default,,0000,0000,0000,,Now can we use this? Let's have\Na look at an example why equals
Dialogue: 0,0:18:16.19,0:18:22.53,Default,,0000,0000,0000,,the tan of X squared and so DY\Nby the X is equal to.
Dialogue: 0,0:18:23.17,0:18:28.57,Default,,0000,0000,0000,,Well, this bit here is the G of\NX, so we need its derivative
Dialogue: 0,0:18:28.57,0:18:33.21,Default,,0000,0000,0000,,and we're going to write it\Ndown at the front. That's easy,
Dialogue: 0,0:18:33.21,0:18:37.45,Default,,0000,0000,0000,,that's 2X. And now we need to\Ndifferentiate tan as though
Dialogue: 0,0:18:37.45,0:18:42.08,Default,,0000,0000,0000,,this was tamu with respect to\Nyou and the derivative of tan
Dialogue: 0,0:18:42.08,0:18:47.87,Default,,0000,0000,0000,,is just sex squared. And then\Nwe need to put in the G of X
Dialogue: 0,0:18:47.87,0:18:49.42,Default,,0000,0000,0000,,sex squared X squared.
Dialogue: 0,0:18:50.53,0:18:55.17,Default,,0000,0000,0000,,Notice how short this is. 2\Nlines. Let's just look back at
Dialogue: 0,0:18:55.17,0:19:00.20,Default,,0000,0000,0000,,the example that we did before\Nand it took us all of this.
Dialogue: 0,0:19:00.80,0:19:05.43,Default,,0000,0000,0000,,What we've done is we've\Ncontracted this whole process to
Dialogue: 0,0:19:05.43,0:19:08.21,Default,,0000,0000,0000,,one that takes place in our
Dialogue: 0,0:19:08.21,0:19:14.59,Default,,0000,0000,0000,,head. Quicker. You can still do\Nit this way if you like, but if
Dialogue: 0,0:19:14.59,0:19:19.12,Default,,0000,0000,0000,,you can get into the habit of\Ndoing this, it's much quicker.
Dialogue: 0,0:19:19.12,0:19:24.02,Default,,0000,0000,0000,,So let's have a look at some\Nexamples and try and keep track
Dialogue: 0,0:19:24.02,0:19:29.30,Default,,0000,0000,0000,,of what we're doing. So let's\Nbegin with Y equals E to the one
Dialogue: 0,0:19:29.30,0:19:30.43,Default,,0000,0000,0000,,plus X squared.
Dialogue: 0,0:19:30.44,0:19:33.56,Default,,0000,0000,0000,,Sunday why by DX is equal to.
Dialogue: 0,0:19:34.08,0:19:39.72,Default,,0000,0000,0000,,Well, we can identify the G of\NX. It's this up here. The power
Dialogue: 0,0:19:39.72,0:19:44.16,Default,,0000,0000,0000,,of the exponential function, and\Nso if we differentiate that it's
Dialogue: 0,0:19:44.16,0:19:48.18,Default,,0000,0000,0000,,2X. And now we want the\Nderivative of the exponential
Dialogue: 0,0:19:48.18,0:19:53.50,Default,,0000,0000,0000,,function. It is its own\Nderivative, so that's E to the
Dialogue: 0,0:19:53.50,0:19:55.20,Default,,0000,0000,0000,,one plus X squared.
Dialogue: 0,0:19:56.42,0:19:58.69,Default,,0000,0000,0000,,Able to write it down straight
Dialogue: 0,0:19:58.69,0:20:06.09,Default,,0000,0000,0000,,away. Let's take Y\Nequals the sign of X
Dialogue: 0,0:20:06.09,0:20:10.06,Default,,0000,0000,0000,,Plus E to the X.
Dialogue: 0,0:20:10.10,0:20:13.84,Default,,0000,0000,0000,,The why by DX is equal to
Dialogue: 0,0:20:13.84,0:20:19.80,Default,,0000,0000,0000,,well here. We can identify the G\Nof X, so we want the derivative
Dialogue: 0,0:20:19.80,0:20:24.09,Default,,0000,0000,0000,,of this to write down the\Nderivative of X is one. The
Dialogue: 0,0:20:24.09,0:20:28.37,Default,,0000,0000,0000,,derivative of E to the X is just\NE to the X.
Dialogue: 0,0:20:28.89,0:20:32.96,Default,,0000,0000,0000,,Times by and we want the\Nderivative of this.
Dialogue: 0,0:20:33.86,0:20:39.83,Default,,0000,0000,0000,,As though this were you and the\Nderivative of sine is cause, and
Dialogue: 0,0:20:39.83,0:20:43.50,Default,,0000,0000,0000,,then X Plus E to the power X.
Dialogue: 0,0:20:44.03,0:20:49.79,Default,,0000,0000,0000,,We take\NY equals
Dialogue: 0,0:20:49.79,0:20:56.54,Default,,0000,0000,0000,,the tan.\NAll X squared
Dialogue: 0,0:20:56.54,0:20:59.86,Default,,0000,0000,0000,,plus sign X.
Dialogue: 0,0:20:59.86,0:21:06.96,Default,,0000,0000,0000,,Why buy the X is equal\Nto identify the G of X?
Dialogue: 0,0:21:06.96,0:21:12.88,Default,,0000,0000,0000,,That's this lump here, the X\Nsquared plus sign X
Dialogue: 0,0:21:12.88,0:21:15.84,Default,,0000,0000,0000,,differentiate it 2X plus cause
Dialogue: 0,0:21:15.84,0:21:22.31,Default,,0000,0000,0000,,X. And now the derivative of\Ntan as though it were tan of
Dialogue: 0,0:21:22.31,0:21:27.04,Default,,0000,0000,0000,,U. Well that will be sex\Nsquared. But instead of EU
Dialogue: 0,0:21:27.04,0:21:33.06,Default,,0000,0000,0000,,we want you is G of X which\Nis X squared plus sign X.
Dialogue: 0,0:21:36.49,0:21:42.51,Default,,0000,0000,0000,,Y equals this time will have a\Nlook at one that's got brackets
Dialogue: 0,0:21:42.51,0:21:49.45,Default,,0000,0000,0000,,and powers in it. 2 minus X to\Nthe 5th, all raised to the 9th
Dialogue: 0,0:21:49.45,0:21:56.45,Default,,0000,0000,0000,,power. The why by DX is got\Nto identify the G of X first and
Dialogue: 0,0:21:56.45,0:22:02.84,Default,,0000,0000,0000,,the G of X is this bit inside\Nthe bracket the two minus X to
Dialogue: 0,0:22:02.84,0:22:07.53,Default,,0000,0000,0000,,the power 5. So we need to\Ndifferentiate that first, so
Dialogue: 0,0:22:07.53,0:22:11.79,Default,,0000,0000,0000,,that's going to be minus 5X to\Nthe power 4.
Dialogue: 0,0:22:12.65,0:22:16.97,Default,,0000,0000,0000,,Remember the derivative of two,\Nwhich is a constant is 0 and
Dialogue: 0,0:22:16.97,0:22:21.65,Default,,0000,0000,0000,,then multiply by the Five and\Ntake one off the index of minus
Dialogue: 0,0:22:21.65,0:22:23.45,Default,,0000,0000,0000,,5X to the power 4.
Dialogue: 0,0:22:24.08,0:22:29.27,Default,,0000,0000,0000,,Now we've got to differentiate\Nto the power 9 as though it was
Dialogue: 0,0:22:29.27,0:22:34.85,Default,,0000,0000,0000,,you to the power 9 will that\Nwould be times by 9, and then
Dialogue: 0,0:22:34.85,0:22:36.85,Default,,0000,0000,0000,,instead of EU we want.
Dialogue: 0,0:22:37.36,0:22:42.83,Default,,0000,0000,0000,,The two minus X to the 5th and\Nwe take one away from the
Dialogue: 0,0:22:42.83,0:22:47.53,Default,,0000,0000,0000,,power. We get that little bit\Nof tidying up to do here,
Dialogue: 0,0:22:47.53,0:22:51.04,Default,,0000,0000,0000,,'cause we've got a\Nmultiplication that we can do
Dialogue: 0,0:22:51.04,0:22:57.30,Default,,0000,0000,0000,,minus five times by 9 - 45 X\Nto the power four 2 minus X to
Dialogue: 0,0:22:57.30,0:23:00.04,Default,,0000,0000,0000,,the fifth, raised to the power\N8.
Dialogue: 0,0:23:01.46,0:23:04.76,Default,,0000,0000,0000,,Take 1 final example.
Dialogue: 0,0:23:05.54,0:23:13.01,Default,,0000,0000,0000,,Let's take Y equals the\Nlog of X plus sign
Dialogue: 0,0:23:13.01,0:23:17.91,Default,,0000,0000,0000,,X. We can identify this as our\NG of X.
Dialogue: 0,0:23:19.08,0:23:25.93,Default,,0000,0000,0000,,So D why by DX is equal to\Nwe can write down the derivative
Dialogue: 0,0:23:25.93,0:23:32.28,Default,,0000,0000,0000,,of this one plus cause X times\Nby and now we want the
Dialogue: 0,0:23:32.28,0:23:33.75,Default,,0000,0000,0000,,derivative of log.
Dialogue: 0,0:23:34.44,0:23:40.12,Default,,0000,0000,0000,,Well, the derivative of log of\Nyou would be one over you, so
Dialogue: 0,0:23:40.12,0:23:47.55,Default,,0000,0000,0000,,this is one over and this is EU\Nor the G of X one over X plus
Dialogue: 0,0:23:47.55,0:23:54.10,Default,,0000,0000,0000,,sign X and it be untidy to leave\Nit like that. So we move that
Dialogue: 0,0:23:54.10,0:23:58.48,Default,,0000,0000,0000,,numerator one plus cause X. So\Nit's more obviously the
Dialogue: 0,0:23:58.48,0:24:03.72,Default,,0000,0000,0000,,numerator of the fraction. So\Nwhat we've seen there is the use
Dialogue: 0,0:24:03.72,0:24:05.47,Default,,0000,0000,0000,,of the chain rule.
Dialogue: 0,0:24:06.07,0:24:10.32,Default,,0000,0000,0000,,But then modifying it slightly\Nso we get used to writing down
Dialogue: 0,0:24:10.32,0:24:13.86,Default,,0000,0000,0000,,the answer more or less\Nstraight away. You can still
Dialogue: 0,0:24:13.86,0:24:18.46,Default,,0000,0000,0000,,use the chain rule. You can\Nstill do it at full length, but
Dialogue: 0,0:24:18.46,0:24:23.42,Default,,0000,0000,0000,,if you can get into the habit\Nof doing it like this, you find
Dialogue: 0,0:24:23.42,0:24:27.31,Default,,0000,0000,0000,,it quicker and easier to be\Nable to manage and more
Dialogue: 0,0:24:27.31,0:24:31.20,Default,,0000,0000,0000,,complicated problems will be\Neasier to do if you can short
Dialogue: 0,0:24:31.20,0:24:33.68,Default,,0000,0000,0000,,cut short circuit some of the\Nprocesses.