[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.03,0:00:06.23,Default,,0000,0000,0000,,We're now going to extend our\Ntable of derivatives by looking
Dialogue: 0,0:00:06.23,0:00:12.38,Default,,0000,0000,0000,,at functions such as tanks sex,\Narcsine X, or sign minus one X&A
Dialogue: 0,0:00:12.38,0:00:16.17,Default,,0000,0000,0000,,to the power X, where a is a
Dialogue: 0,0:00:16.17,0:00:23.41,Default,,0000,0000,0000,,positive constant. Let's start\Nwith the table with our
Dialogue: 0,0:00:23.41,0:00:25.23,Default,,0000,0000,0000,,function. F of X.
Dialogue: 0,0:00:25.99,0:00:27.92,Default,,0000,0000,0000,,And our derivative.
Dialogue: 0,0:00:30.26,0:00:37.12,Default,,0000,0000,0000,,Either referred to as DF, DX,\Nor F dashed of X.
Dialogue: 0,0:00:38.08,0:00:44.14,Default,,0000,0000,0000,,And let's\Nstart with
Dialogue: 0,0:00:44.14,0:00:45.65,Default,,0000,0000,0000,,tonics.
Dialogue: 0,0:00:46.97,0:00:49.34,Default,,0000,0000,0000,,Sex.
Dialogue: 0,0:00:50.38,0:00:56.62,Default,,0000,0000,0000,,Call Tex.\NAnd Cosec X.
Dialogue: 0,0:00:56.69,0:01:01.45,Default,,0000,0000,0000,,And the first thing they want to\Nwant to do is to actually write
Dialogue: 0,0:01:01.45,0:01:03.49,Default,,0000,0000,0000,,these in terms of sine and
Dialogue: 0,0:01:03.49,0:01:10.51,Default,,0000,0000,0000,,cosine. So tonics we can write\Na sign X divided by Cos X.
Dialogue: 0,0:01:11.72,0:01:15.60,Default,,0000,0000,0000,,Sex is 1 divided by Cos X.
Dialogue: 0,0:01:16.20,0:01:24.08,Default,,0000,0000,0000,,Call Tex is Arcos X divided\Nby sine X and cosec, X
Dialogue: 0,0:01:24.08,0:01:28.03,Default,,0000,0000,0000,,is 1 divided by sine X.
Dialogue: 0,0:01:28.85,0:01:33.47,Default,,0000,0000,0000,,So let's have a look now at\Nfinding the derivatives of some
Dialogue: 0,0:01:33.47,0:01:38.75,Default,,0000,0000,0000,,of these. Let's\Nstart with tonics
Dialogue: 0,0:01:38.75,0:01:42.55,Default,,0000,0000,0000,,Y equals TAN X.
Dialogue: 0,0:01:45.85,0:01:51.92,Default,,0000,0000,0000,,Now we've already written tonics\Nin terms of sine and cosine, so
Dialogue: 0,0:01:51.92,0:01:56.98,Default,,0000,0000,0000,,let's write it a sign X divided\Nby cosine X.
Dialogue: 0,0:01:57.51,0:02:04.79,Default,,0000,0000,0000,,And this is as if we had\NU&V where you is a function of
Dialogue: 0,0:02:04.79,0:02:07.91,Default,,0000,0000,0000,,X&V is a function of X.
Dialogue: 0,0:02:08.76,0:02:10.65,Default,,0000,0000,0000,,So we have a quotient.
Dialogue: 0,0:02:11.20,0:02:16.26,Default,,0000,0000,0000,,So we're going to use our rule\Nfor differentiating quotients.
Dialogue: 0,0:02:16.26,0:02:24.08,Default,,0000,0000,0000,,Do Y by ZX equals V\Ntimes? Do you buy the X
Dialogue: 0,0:02:24.08,0:02:27.100,Default,,0000,0000,0000,,minus use times DV Pi DX?
Dialogue: 0,0:02:28.56,0:02:31.22,Default,,0000,0000,0000,,All divided by V squared.
Dialogue: 0,0:02:32.28,0:02:36.00,Default,,0000,0000,0000,,So in this case, IQ equals
Dialogue: 0,0:02:36.00,0:02:39.91,Default,,0000,0000,0000,,sign X. So do you buy the
Dialogue: 0,0:02:39.91,0:02:43.40,Default,,0000,0000,0000,,X? It was cool sex.
Dialogue: 0,0:02:44.23,0:02:48.08,Default,,0000,0000,0000,,RV. It's called
Dialogue: 0,0:02:48.08,0:02:52.07,Default,,0000,0000,0000,,sex. So 2 feet by TX.
Dialogue: 0,0:02:52.74,0:02:55.18,Default,,0000,0000,0000,,Equals minus sign X.
Dialogue: 0,0:02:56.25,0:02:58.13,Default,,0000,0000,0000,,Let's substitute now.
Dialogue: 0,0:02:58.95,0:03:02.65,Default,,0000,0000,0000,,Into our derivative do I buy DX?
Dialogue: 0,0:03:03.28,0:03:07.22,Default,,0000,0000,0000,,Is it cool to V which is Cos X?
Dialogue: 0,0:03:08.24,0:03:13.27,Default,,0000,0000,0000,,Multiplied by do you buy DX,\Nwhich is cause X?
Dialogue: 0,0:03:14.65,0:03:15.56,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:03:16.82,0:03:19.33,Default,,0000,0000,0000,,You which is synex.
Dialogue: 0,0:03:19.87,0:03:27.28,Default,,0000,0000,0000,,Multiplied by TV by DX,\Nwhich is minus sign X.
Dialogue: 0,0:03:27.28,0:03:30.66,Default,,0000,0000,0000,,All divided by.
Dialogue: 0,0:03:31.36,0:03:37.45,Default,,0000,0000,0000,,V. Squared. Via skull sex so\Nsquared is called squared X.
Dialogue: 0,0:03:38.42,0:03:41.32,Default,,0000,0000,0000,,So we have cost squared X.
Dialogue: 0,0:03:42.07,0:03:49.15,Default,,0000,0000,0000,,Minus times minus gives us\Npositive sign squared X.
Dialogue: 0,0:03:49.94,0:03:53.31,Default,,0000,0000,0000,,Divided by Cos squared X.
Dialogue: 0,0:03:54.25,0:03:57.97,Default,,0000,0000,0000,,Now one of our trigonometric\Nidentity's is called squared X
Dialogue: 0,0:03:57.97,0:04:00.20,Default,,0000,0000,0000,,plus sign. Squared X equals 1.
Dialogue: 0,0:04:00.71,0:04:07.71,Default,,0000,0000,0000,,So we can substitute 1\Ndivided by Cos squared X.
Dialogue: 0,0:04:08.72,0:04:12.28,Default,,0000,0000,0000,,Which is the same as sex
Dialogue: 0,0:04:12.28,0:04:14.77,Default,,0000,0000,0000,,squared X. So our derivative of
Dialogue: 0,0:04:14.77,0:04:17.67,Default,,0000,0000,0000,,Tan X. Is sex squared X?
Dialogue: 0,0:04:19.23,0:04:26.01,Default,,0000,0000,0000,,Let's have a look at\Nanother example. This time Y
Dialogue: 0,0:04:26.01,0:04:32.83,Default,,0000,0000,0000,,equals sex. And again will\Nwrite sex in terms of sine and
Dialogue: 0,0:04:32.83,0:04:38.39,Default,,0000,0000,0000,,cosine, which is one over call\Nsex. And again we have our
Dialogue: 0,0:04:38.39,0:04:40.24,Default,,0000,0000,0000,,quotient you over V.
Dialogue: 0,0:04:40.94,0:04:47.97,Default,,0000,0000,0000,,So I see Y by ZX equals\NV times. Do you buy the X?
Dialogue: 0,0:04:48.59,0:04:52.54,Default,,0000,0000,0000,,Minus U times TV by ZX?
Dialogue: 0,0:04:53.16,0:04:55.54,Default,,0000,0000,0000,,Or divided by 3 squared.
Dialogue: 0,0:04:56.60,0:05:03.91,Default,,0000,0000,0000,,So in this case I you equals 1.\NSo I do you buy DX is 0.
Dialogue: 0,0:05:04.81,0:05:08.11,Default,,0000,0000,0000,,RV equals
Dialogue: 0,0:05:08.11,0:05:14.99,Default,,0000,0000,0000,,Cossacks. So I do feet\Nby DX equals minus sign X.
Dialogue: 0,0:05:16.02,0:05:23.72,Default,,0000,0000,0000,,So our derivative do why by DX\Nis equal to V, so it's cause
Dialogue: 0,0:05:23.72,0:05:29.61,Default,,0000,0000,0000,,X. Multiplied by do you buy\NDX which is 0?
Dialogue: 0,0:05:30.78,0:05:34.04,Default,,0000,0000,0000,,Minus you choose one.
Dialogue: 0,0:05:34.74,0:05:39.79,Default,,0000,0000,0000,,Times DV by The X which is\Nminus sign X.
Dialogue: 0,0:05:40.44,0:05:45.75,Default,,0000,0000,0000,,All divided by V squared, which\Nis called squared X.
Dialogue: 0,0:05:46.88,0:05:49.65,Default,,0000,0000,0000,,So we have 0.
Dialogue: 0,0:05:50.96,0:05:55.65,Default,,0000,0000,0000,,Minus minus that becomes\Npositive. One lot of Cynex
Dialogue: 0,0:05:55.65,0:06:00.34,Default,,0000,0000,0000,,so have sign X divided by\NCos squared X.
Dialogue: 0,0:06:01.50,0:06:03.82,Default,,0000,0000,0000,,And we can write this.
Dialogue: 0,0:06:05.85,0:06:13.39,Default,,0000,0000,0000,,As. One over Cos\NX multiplied by sign X over
Dialogue: 0,0:06:13.39,0:06:17.06,Default,,0000,0000,0000,,call sex. Which
Dialogue: 0,0:06:17.06,0:06:23.76,Default,,0000,0000,0000,,is. Sex tonics,\Nso a derivative of
Dialogue: 0,0:06:23.76,0:06:27.02,Default,,0000,0000,0000,,sex is sex tonics.
Dialogue: 0,0:06:28.25,0:06:30.94,Default,,0000,0000,0000,,Let's put those in our table.
Dialogue: 0,0:06:31.87,0:06:34.26,Default,,0000,0000,0000,,So our derivative of Tan X.
Dialogue: 0,0:06:35.01,0:06:36.87,Default,,0000,0000,0000,,Sex squared X.
Dialogue: 0,0:06:37.95,0:06:40.03,Default,,0000,0000,0000,,A derivative of sex.
Dialogue: 0,0:06:40.60,0:06:44.32,Default,,0000,0000,0000,,Is sex tonics?
Dialogue: 0,0:06:45.08,0:06:50.100,Default,,0000,0000,0000,,And in a similar way are\Nderivative of Cortex will be
Dialogue: 0,0:06:50.100,0:06:53.15,Default,,0000,0000,0000,,minus cosec squared X.
Dialogue: 0,0:06:53.68,0:06:59.75,Default,,0000,0000,0000,,And the derivative of\Ncosec X is minus Cosec
Dialogue: 0,0:06:59.75,0:07:01.09,Default,,0000,0000,0000,,X Cotex.
Dialogue: 0,0:07:03.29,0:07:09.19,Default,,0000,0000,0000,,Now we going to extend this\Nfurther to look at what happens
Dialogue: 0,0:07:09.19,0:07:12.15,Default,,0000,0000,0000,,when we have time of MX.
Dialogue: 0,0:07:12.93,0:07:15.42,Default,,0000,0000,0000,,And the sack of MX.
Dialogue: 0,0:07:16.24,0:07:21.30,Default,,0000,0000,0000,,The cast of air Max and the\Ncosec of MX.
Dialogue: 0,0:07:22.01,0:07:25.74,Default,,0000,0000,0000,,Let's have a look at the
Dialogue: 0,0:07:25.74,0:07:32.75,Default,,0000,0000,0000,,calculation. So this time\Nwe have Y equals the
Dialogue: 0,0:07:32.75,0:07:35.15,Default,,0000,0000,0000,,time of MX.
Dialogue: 0,0:07:36.19,0:07:39.66,Default,,0000,0000,0000,,And what we're going to do here\Nis to use a substitution.
Dialogue: 0,0:07:40.27,0:07:47.13,Default,,0000,0000,0000,,So you were going to substitute\Ninstead of MX, so RY is
Dialogue: 0,0:07:47.13,0:07:49.42,Default,,0000,0000,0000,,equal to tan you.
Dialogue: 0,0:07:50.53,0:07:53.73,Default,,0000,0000,0000,,So if we calculate, do you buy
Dialogue: 0,0:07:53.73,0:07:57.48,Default,,0000,0000,0000,,DX? We get M.
Dialogue: 0,0:07:57.99,0:08:03.64,Default,,0000,0000,0000,,And if we find the derivative of\NY with respect to you while
Dialogue: 0,0:08:03.64,0:08:08.86,Default,,0000,0000,0000,,we've just learned how to do\Nthat, the derivative of tan you
Dialogue: 0,0:08:08.86,0:08:10.60,Default,,0000,0000,0000,,is sex squared you.
Dialogue: 0,0:08:10.77,0:08:13.69,Default,,0000,0000,0000,,So I do why by DX.
Dialogue: 0,0:08:14.22,0:08:21.73,Default,,0000,0000,0000,,Is equal to do Y fi\NCU multiplied by to you by
Dialogue: 0,0:08:21.73,0:08:26.70,Default,,0000,0000,0000,,DMX. The Wi-Fi do you? Is\Nsex squared you?
Dialogue: 0,0:08:28.04,0:08:33.19,Default,,0000,0000,0000,,Multiplied by do you buy DX,\Nwhich is RM. Let's put the
Dialogue: 0,0:08:33.19,0:08:37.48,Default,,0000,0000,0000,,constant first M times this X\Nsquared. You now we've
Dialogue: 0,0:08:37.48,0:08:43.06,Default,,0000,0000,0000,,introduced EU and we don't want\Nthe you were trying to find the
Dialogue: 0,0:08:43.06,0:08:49.49,Default,,0000,0000,0000,,wide by DX we wanted in terms of\NX. So what we do is we
Dialogue: 0,0:08:49.49,0:08:55.07,Default,,0000,0000,0000,,substitute for the you so we\Nhave M times 6 squared of MX.
Dialogue: 0,0:08:56.64,0:09:00.67,Default,,0000,0000,0000,,So what we get when we\Ndifferentiate ton of MX?
Dialogue: 0,0:09:01.32,0:09:04.36,Default,,0000,0000,0000,,Is the sex squared MX multiplied
Dialogue: 0,0:09:04.36,0:09:10.16,Default,,0000,0000,0000,,by M? Let's have a\Nlook at another example.
Dialogue: 0,0:09:10.16,0:09:18.13,Default,,0000,0000,0000,,This time, let's have a look\Nat Y equals cosec of MX.
Dialogue: 0,0:09:19.30,0:09:25.05,Default,,0000,0000,0000,,Again, we're going to let you\Nequal MX as our substitution.
Dialogue: 0,0:09:25.65,0:09:28.33,Default,,0000,0000,0000,,And therefore Y is equal to
Dialogue: 0,0:09:28.33,0:09:30.31,Default,,0000,0000,0000,,Kosek. Of you.
Dialogue: 0,0:09:31.11,0:09:35.10,Default,,0000,0000,0000,,Do you buy the X is equal
Dialogue: 0,0:09:35.10,0:09:39.08,Default,,0000,0000,0000,,to end? And our DY bye see you.
Dialogue: 0,0:09:39.87,0:09:42.49,Default,,0000,0000,0000,,The derivative of cosec you.
Dialogue: 0,0:09:43.69,0:09:47.45,Default,,0000,0000,0000,,Is minus cosec you
Dialogue: 0,0:09:47.45,0:09:54.31,Default,,0000,0000,0000,,cut you? We\Nknow that divide by DX.
Dialogue: 0,0:09:54.94,0:10:00.81,Default,,0000,0000,0000,,It cause divide by do you\Nmultiplied by do you buy DX?
Dialogue: 0,0:10:00.85,0:10:04.10,Default,,0000,0000,0000,,RDY by do you?
Dialogue: 0,0:10:04.71,0:10:08.59,Default,,0000,0000,0000,,Is minus cosec you caught
Dialogue: 0,0:10:08.59,0:10:13.71,Default,,0000,0000,0000,,you? Multiplied by do you buy DX\Nwhich is M?
Dialogue: 0,0:10:14.25,0:10:17.02,Default,,0000,0000,0000,,So we have minus N.
Dialogue: 0,0:10:17.65,0:10:20.77,Default,,0000,0000,0000,,Cosec you caught you.
Dialogue: 0,0:10:21.58,0:10:23.58,Default,,0000,0000,0000,,Now we need to substitute for
Dialogue: 0,0:10:23.58,0:10:30.71,Default,,0000,0000,0000,,these use. So we\Nhave minus M, Cosec
Dialogue: 0,0:10:30.71,0:10:33.64,Default,,0000,0000,0000,,MX cult MX.
Dialogue: 0,0:10:34.61,0:10:40.55,Default,,0000,0000,0000,,So the derivative of\Ncosec MX is minus M.
Dialogue: 0,0:10:40.55,0:10:43.19,Default,,0000,0000,0000,,Cosec MX caught MX.
Dialogue: 0,0:10:44.46,0:10:48.56,Default,,0000,0000,0000,,Now let's go back to our table\Nand put those in.
Dialogue: 0,0:10:49.15,0:10:56.57,Default,,0000,0000,0000,,So we found that the\Nderivative of tan of MX.
Dialogue: 0,0:10:57.70,0:11:02.24,Default,,0000,0000,0000,,Is M6 squared\NMX?
Dialogue: 0,0:11:03.68,0:11:06.49,Default,,0000,0000,0000,,The derivative of SAC MX.
Dialogue: 0,0:11:07.43,0:11:14.69,Default,,0000,0000,0000,,Is M times the sack of\NMX times a ton of MX?
Dialogue: 0,0:11:15.71,0:11:22.05,Default,,0000,0000,0000,,And in a similar way we\Nwould find that the koptev
Dialogue: 0,0:11:22.05,0:11:27.23,Default,,0000,0000,0000,,MX is minus M Times the\Ncosec squared MX.
Dialogue: 0,0:11:28.31,0:11:34.85,Default,,0000,0000,0000,,And cosec MX. The\Nderivative is minus M
Dialogue: 0,0:11:34.85,0:11:38.93,Default,,0000,0000,0000,,times cosec MX cult\NMX.
Dialogue: 0,0:11:40.78,0:11:44.15,Default,,0000,0000,0000,,OK, we're going to extend our
Dialogue: 0,0:11:44.15,0:11:48.73,Default,,0000,0000,0000,,table further. Find a\Nclean page.
Dialogue: 0,0:11:50.81,0:11:57.28,Default,,0000,0000,0000,,Let's just rewrite our headings\Nare function F of X.
Dialogue: 0,0:11:57.80,0:12:00.82,Default,,0000,0000,0000,,That derivative
Dialogue: 0,0:12:01.58,0:12:08.05,Default,,0000,0000,0000,,DFDX\NOr F Dash Devex.
Dialogue: 0,0:12:09.16,0:12:15.14,Default,,0000,0000,0000,,And we're going to\Nlook now at.
Dialogue: 0,0:12:15.85,0:12:19.14,Default,,0000,0000,0000,,The inverse. Sign
Dialogue: 0,0:12:19.14,0:12:22.99,Default,,0000,0000,0000,,minus 1X. It's
Dialogue: 0,0:12:22.99,0:12:26.52,Default,,0000,0000,0000,,minus 1X. And tonics.
Dialogue: 0,0:12:27.19,0:12:30.11,Default,,0000,0000,0000,,Minus 1 - 1 X.
Dialogue: 0,0:12:31.55,0:12:36.97,Default,,0000,0000,0000,,And then finally will look at A\Nto the power X.
Dialogue: 0,0:12:38.19,0:12:43.73,Default,,0000,0000,0000,,Let's do the calculation now for\Nsign minus 1X.
Dialogue: 0,0:12:43.74,0:12:51.08,Default,,0000,0000,0000,,So this time Y\Nequals sine minus 1X.
Dialogue: 0,0:12:53.08,0:13:00.08,Default,,0000,0000,0000,,What we're going to do here is\Nright this as sign Y equals X.
Dialogue: 0,0:13:00.76,0:13:06.52,Default,,0000,0000,0000,,Now we want to find the\Nderivative of Y with respect to
Dialogue: 0,0:13:06.52,0:13:09.40,Default,,0000,0000,0000,,X. So we want to differentiate.
Dialogue: 0,0:13:09.41,0:13:11.83,Default,,0000,0000,0000,,Side Y with respect to X.
Dialogue: 0,0:13:12.47,0:13:19.19,Default,,0000,0000,0000,,And differentiate RX with\Nrespect to X.
Dialogue: 0,0:13:19.47,0:13:23.63,Default,,0000,0000,0000,,His right hand side is very easy\Nbecause the derivative of X is
Dialogue: 0,0:13:23.63,0:13:25.55,Default,,0000,0000,0000,,just one with respect to X.
Dialogue: 0,0:13:26.09,0:13:29.72,Default,,0000,0000,0000,,This side we're going to\Ndifferentiate implicitly.
Dialogue: 0,0:13:30.74,0:13:38.10,Default,,0000,0000,0000,,And when we do that, we get the\Nderivative of sine. Y is cause Y
Dialogue: 0,0:13:38.10,0:13:40.56,Default,,0000,0000,0000,,multiplied by divided by DX.
Dialogue: 0,0:13:41.79,0:13:49.23,Default,,0000,0000,0000,,So if we now divide both sides\Nby cause why we get DY by DX
Dialogue: 0,0:13:49.23,0:13:52.21,Default,,0000,0000,0000,,equals 1 divided by cause Y?
Dialogue: 0,0:13:53.17,0:13:57.57,Default,,0000,0000,0000,,Now that's fine, except for the\Nfact that we actually wanted in
Dialogue: 0,0:13:57.57,0:13:58.68,Default,,0000,0000,0000,,terms of X.
Dialogue: 0,0:13:59.81,0:14:02.63,Default,,0000,0000,0000,,So we need to use one of our
Dialogue: 0,0:14:02.63,0:14:08.00,Default,,0000,0000,0000,,trigonometric identity's. I'm\Nlooking to use call squared Y
Dialogue: 0,0:14:08.00,0:14:11.22,Default,,0000,0000,0000,,plus sign squared Y equals 1.
Dialogue: 0,0:14:12.04,0:14:17.22,Default,,0000,0000,0000,,And if we rearrange this, we\Nget cause Y.
Dialogue: 0,0:14:17.74,0:14:22.97,Default,,0000,0000,0000,,Equals 1 minus sign squared Y\Nand then we want 'cause why not
Dialogue: 0,0:14:22.97,0:14:26.99,Default,,0000,0000,0000,,cause squared Y? So we're going\Nto square root it.
Dialogue: 0,0:14:28.10,0:14:31.93,Default,,0000,0000,0000,,Now we introduce slight problem\Nhere because when we square root
Dialogue: 0,0:14:31.93,0:14:35.41,Default,,0000,0000,0000,,something we could have a\Npositive answer. Or we could
Dialogue: 0,0:14:35.41,0:14:40.28,Default,,0000,0000,0000,,have a negative answer. So we\Nneed to just look at that for a
Dialogue: 0,0:14:40.28,0:14:42.72,Default,,0000,0000,0000,,moment. Now if we look at the
Dialogue: 0,0:14:42.72,0:14:48.78,Default,,0000,0000,0000,,diagram. Of Y equals sine minus\NOne X, so here's a graph of Y
Dialogue: 0,0:14:48.78,0:14:54.32,Default,,0000,0000,0000,,equals sine minus One X will see\Nthat there are many values of Y.
Dialogue: 0,0:14:54.85,0:14:59.65,Default,,0000,0000,0000,,So our values of X. So for one\Nvalue of X we could have lots of
Dialogue: 0,0:14:59.65,0:15:00.85,Default,,0000,0000,0000,,different values of Y.
Dialogue: 0,0:15:01.62,0:15:06.62,Default,,0000,0000,0000,,So that's not a function, and\Nfor it to be a function, we
Dialogue: 0,0:15:06.62,0:15:12.40,Default,,0000,0000,0000,,just want one set of values of\NY for each set of values of X.
Dialogue: 0,0:15:12.40,0:15:16.64,Default,,0000,0000,0000,,So what we need to do is\Nactually restrict the section
Dialogue: 0,0:15:16.64,0:15:21.26,Default,,0000,0000,0000,,that we're looking at, and if\Nwe restrict it to between minus
Dialogue: 0,0:15:21.26,0:15:25.88,Default,,0000,0000,0000,,π by two and plus five by two,\Njust this section here.
Dialogue: 0,0:15:27.21,0:15:31.54,Default,,0000,0000,0000,,Then we do have a function\Nbecause we have just one value
Dialogue: 0,0:15:31.54,0:15:34.07,Default,,0000,0000,0000,,of Y for each value of X.
Dialogue: 0,0:15:35.35,0:15:39.72,Default,,0000,0000,0000,,And if we look at this section,\Nwe can see that the gradient.
Dialogue: 0,0:15:40.68,0:15:45.23,Default,,0000,0000,0000,,Is increasing and it's positive,\Nso we're actually going to take
Dialogue: 0,0:15:45.23,0:15:49.37,Default,,0000,0000,0000,,the positive square root when we\Ntake the square root.
Dialogue: 0,0:15:50.44,0:15:57.07,Default,,0000,0000,0000,,So if we continue down here one\Nover cause Y, we can actually
Dialogue: 0,0:15:57.07,0:16:03.70,Default,,0000,0000,0000,,write as one over the square\Nroot of 1 minus sign squared Y.
Dialogue: 0,0:16:04.39,0:16:06.86,Default,,0000,0000,0000,,Now we still wanted in terms of
Dialogue: 0,0:16:06.86,0:16:12.87,Default,,0000,0000,0000,,X. But if we look here, we\Nknow that sign Y is equal to
Dialogue: 0,0:16:12.87,0:16:17.50,Default,,0000,0000,0000,,X, so we can substitute in\Nfor sine squared Y here and
Dialogue: 0,0:16:17.50,0:16:22.52,Default,,0000,0000,0000,,put X squared. So we get one\Ndivided by the square root of
Dialogue: 0,0:16:22.52,0:16:24.06,Default,,0000,0000,0000,,1 minus X squared.
Dialogue: 0,0:16:26.35,0:16:32.10,Default,,0000,0000,0000,,White was cause minus One X can\Nbe done in a similar way and is
Dialogue: 0,0:16:32.10,0:16:37.07,Default,,0000,0000,0000,,left as an exercise. But note\Nthere that when you look at the
Dialogue: 0,0:16:37.07,0:16:41.29,Default,,0000,0000,0000,,square root should have to take\Na negative square root rather
Dialogue: 0,0:16:41.29,0:16:42.82,Default,,0000,0000,0000,,than a positive one.
Dialogue: 0,0:16:43.49,0:16:49.36,Default,,0000,0000,0000,,Let's have a look now at Y\Nequals 10 - 1 X.
Dialogue: 0,0:16:49.36,0:16:55.44,Default,,0000,0000,0000,,In the same\Nway, we're going
Dialogue: 0,0:16:55.44,0:17:01.53,Default,,0000,0000,0000,,to write Tan,\NY is equal
Dialogue: 0,0:17:01.53,0:17:08.39,Default,,0000,0000,0000,,to X. And\Nagain, take the derivative of
Dialogue: 0,0:17:08.39,0:17:11.94,Default,,0000,0000,0000,,each side with respect to
Dialogue: 0,0:17:11.94,0:17:18.76,Default,,0000,0000,0000,,X. I can right hand\Nside is easy, it's just
Dialogue: 0,0:17:18.76,0:17:24.55,Default,,0000,0000,0000,,one. And we differentiate this\Nside implicitly, which gives us.
Dialogue: 0,0:17:25.33,0:17:32.73,Default,,0000,0000,0000,,The derivative of tan Y is sex\Nsquared Y multiplied by DY by
Dialogue: 0,0:17:32.73,0:17:39.31,Default,,0000,0000,0000,,ZX. Sophie now divide\Nboth sides by sex squared
Dialogue: 0,0:17:39.31,0:17:44.36,Default,,0000,0000,0000,,Y we get 1 / 6\Nsquared Y.
Dialogue: 0,0:17:45.51,0:17:50.35,Default,,0000,0000,0000,,Now, in a similar way to before\Nwe need to use a trigonometric
Dialogue: 0,0:17:50.35,0:17:53.15,Default,,0000,0000,0000,,identity. And we need to use the
Dialogue: 0,0:17:53.15,0:17:58.68,Default,,0000,0000,0000,,one. What sex squared Y\Nequals 1 plus?
Dialogue: 0,0:17:59.23,0:18:00.62,Default,,0000,0000,0000,,10 squared, Y.
Dialogue: 0,0:18:02.21,0:18:05.22,Default,,0000,0000,0000,,So if we substitute that.
Dialogue: 0,0:18:05.73,0:18:12.13,Default,,0000,0000,0000,,In here we have 1 /\N1 + 10 squared Y.
Dialogue: 0,0:18:12.87,0:18:20.10,Default,,0000,0000,0000,,And since 10 Y is X, then that's\Njust one over 1 plus X squared.
Dialogue: 0,0:18:20.88,0:18:27.31,Default,,0000,0000,0000,,So the derivative of tan minus\NOne X is 1 / 1 plus X
Dialogue: 0,0:18:27.31,0:18:32.43,Default,,0000,0000,0000,,squared. Let's go back and put\Nthose in our table now.
Dialogue: 0,0:18:32.43,0:18:39.68,Default,,0000,0000,0000,,So the derivative of sine minus\NOne X is one over the square
Dialogue: 0,0:18:39.68,0:18:43.03,Default,,0000,0000,0000,,root of 1 minus X squared.
Dialogue: 0,0:18:44.02,0:18:50.35,Default,,0000,0000,0000,,A derivative of Cos minus One X\Nis minus one over the square
Dialogue: 0,0:18:50.35,0:18:56.68,Default,,0000,0000,0000,,root of 1 minus X squared and\Nour derivative of tan minus One
Dialogue: 0,0:18:56.68,0:19:00.58,Default,,0000,0000,0000,,X is one over 1 plus X squared.
Dialogue: 0,0:19:01.30,0:19:06.51,Default,,0000,0000,0000,,Let's have a look now at Y\Nequals 8 to the power X.
Dialogue: 0,0:19:07.16,0:19:12.72,Default,,0000,0000,0000,,Y equals A to the power X and\Nthis time we're going to start
Dialogue: 0,0:19:12.72,0:19:17.88,Default,,0000,0000,0000,,by taking natural logarithms of\Neach side. So log to the base E
Dialogue: 0,0:19:17.88,0:19:24.23,Default,,0000,0000,0000,,of Y is equal to log to the base\NE of A to the power X.
Dialogue: 0,0:19:24.83,0:19:28.21,Default,,0000,0000,0000,,Now we're going to use one of\Nthe laws of logarithms to bring
Dialogue: 0,0:19:28.21,0:19:33.74,Default,,0000,0000,0000,,down the power. So on the right\Nhand side we have X times log to
Dialogue: 0,0:19:33.74,0:19:35.40,Default,,0000,0000,0000,,the base E of A.
Dialogue: 0,0:19:36.43,0:19:41.66,Default,,0000,0000,0000,,Now we want to find the\Nderivative of this, so we want
Dialogue: 0,0:19:41.66,0:19:48.20,Default,,0000,0000,0000,,to differentiate log to the base\NE of Y with respect to X and we
Dialogue: 0,0:19:48.20,0:19:53.87,Default,,0000,0000,0000,,want to differentiate X times\Nlog to the base E of A with
Dialogue: 0,0:19:53.87,0:19:55.18,Default,,0000,0000,0000,,respect to X.
Dialogue: 0,0:19:55.84,0:20:01.38,Default,,0000,0000,0000,,Now if we look at the right hand\Nside, first log to the base E of
Dialogue: 0,0:20:01.38,0:20:04.84,Default,,0000,0000,0000,,a is a constant, so it's a\Nconstant times X.
Dialogue: 0,0:20:05.45,0:20:09.75,Default,,0000,0000,0000,,So when we differentiate with\Nrespect to X, we just get the
Dialogue: 0,0:20:09.75,0:20:12.61,Default,,0000,0000,0000,,constant log to the base E of A.
Dialogue: 0,0:20:13.26,0:20:18.83,Default,,0000,0000,0000,,If we differentiate log to the\Nbase E of Y with respect to X.
Dialogue: 0,0:20:19.40,0:20:25.08,Default,,0000,0000,0000,,We get one over Y multiplied by\NDY by DX.
Dialogue: 0,0:20:27.28,0:20:30.93,Default,,0000,0000,0000,,So do why by DX is going to
Dialogue: 0,0:20:30.93,0:20:37.23,Default,,0000,0000,0000,,be equal. To now, to get this\Nside is being divided by Y, so
Dialogue: 0,0:20:37.23,0:20:42.46,Default,,0000,0000,0000,,want to multiply both sides by\NY. So we've got Y multiplied by
Dialogue: 0,0:20:42.46,0:20:45.29,Default,,0000,0000,0000,,log to the base E of A.
Dialogue: 0,0:20:46.55,0:20:49.23,Default,,0000,0000,0000,,But we wanted in terms of X, not
Dialogue: 0,0:20:49.23,0:20:54.72,Default,,0000,0000,0000,,Y. But we know that Y\Nequals A to the power X,
Dialogue: 0,0:20:54.72,0:20:59.63,Default,,0000,0000,0000,,so I do why by DX becomes\NA to the power X
Dialogue: 0,0:20:59.63,0:21:03.31,Default,,0000,0000,0000,,multiplied by log to the\Nbase E of A.
Dialogue: 0,0:21:04.81,0:21:09.97,Default,,0000,0000,0000,,Now it's interesting to note\Nhere that actually if the
Dialogue: 0,0:21:09.97,0:21:12.55,Default,,0000,0000,0000,,constant a is equal to.
Dialogue: 0,0:21:13.08,0:21:18.80,Default,,0000,0000,0000,,2.7 one and so on. That's\Nthe value of E.
Dialogue: 0,0:21:19.64,0:21:26.81,Default,,0000,0000,0000,,Then what we have is Y equals\NE to the power X and out
Dialogue: 0,0:21:26.81,0:21:30.39,Default,,0000,0000,0000,,the why by DX is equal to.
Dialogue: 0,0:21:31.46,0:21:37.13,Default,,0000,0000,0000,,A to the power X? Well, that's\NE. To the power X multiplied by
Dialogue: 0,0:21:37.13,0:21:44.02,Default,,0000,0000,0000,,log to the base E of a, which in\Nthis case is E and log to the
Dialogue: 0,0:21:44.02,0:21:47.66,Default,,0000,0000,0000,,base E of E is one, so we get.
Dialogue: 0,0:21:48.21,0:21:52.59,Default,,0000,0000,0000,,Our derivative is E to the power\NX, which is what we expected.
Dialogue: 0,0:21:53.88,0:21:59.23,Default,,0000,0000,0000,,Let's go and complete our table\Nnow, then with the derivative of
Dialogue: 0,0:21:59.23,0:22:05.92,Default,,0000,0000,0000,,A to the X as A to the\Npower X multiplied by log to the
Dialogue: 0,0:22:05.92,0:22:07.71,Default,,0000,0000,0000,,base E of A.