[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.52,0:00:05.14,Default,,0000,0000,0000,,In this tutorial, we're going to\Nlook at differentiating x to the
Dialogue: 0,0:00:05.14,0:00:07.38,Default,,0000,0000,0000,,power n from first principles.
Dialogue: 0,0:00:07.38,0:00:12.24,Default,,0000,0000,0000,,Now n could be a positive\Ninteger, n could be a fraction.
Dialogue: 0,0:00:12.24,0:00:15.54,Default,,0000,0000,0000,,It could be negative, or it\Ncould be 0.
Dialogue: 0,0:00:15.54,0:00:20.24,Default,,0000,0000,0000,,So we're going to start by taking\Nthe case where n is a positive integer.
Dialogue: 0,0:00:20.24,0:00:25.13,Default,,0000,0000,0000,,So we'll be looking at things\Nlike x squared, x to the power 7.
Dialogue: 0,0:00:25.13,0:00:27.89,Default,,0000,0000,0000,,Even x to the power 1.
Dialogue: 0,0:00:27.89,0:00:33.25,Default,,0000,0000,0000,,So we have y equals x\Nto the power n.
Dialogue: 0,0:00:33.87,0:00:40.48,Default,,0000,0000,0000,,Our definition of our derivative\Nfunction is dy by dx equals
Dialogue: 0,0:00:40.48,0:00:47.52,Default,,0000,0000,0000,,the limit as delta x approaches\N0, of f of x + delta x,
Dialogue: 0,0:00:47.52,0:00:53.54,Default,,0000,0000,0000,,minus our function of x, \Ndivided by delta x.
Dialogue: 0,0:00:54.23,0:01:01.45,Default,,0000,0000,0000,,So let's just look at this part\Nfirst of all. Our f of x + delta x
Dialogue: 0,0:01:01.45,0:01:08.06,Default,,0000,0000,0000,,is going to equal x + delta x,\Nall to the power n.
Dialogue: 0,0:01:08.86,0:01:10.40,Default,,0000,0000,0000,,And this is a binomial.
Dialogue: 0,0:01:10.94,0:01:15.94,Default,,0000,0000,0000,,So what we're going to start by\Ndoing is actually just expanding
Dialogue: 0,0:01:16.85,0:01:20.31,Default,,0000,0000,0000,,a + b to the power n.
Dialogue: 0,0:01:22.04,0:01:24.56,Default,,0000,0000,0000,,And that is a to the power n,
Dialogue: 0,0:01:25.07,0:01:32.15,Default,,0000,0000,0000,,plus n, times a to the power of n - 1, \Nmultiplied by b,
Dialogue: 0,0:01:32.15,0:01:39.31,Default,,0000,0000,0000,,plus and there are lots of other terms in\Nbetween both containing powers of a and b
Dialogue: 0,0:01:39.31,0:01:43.88,Default,,0000,0000,0000,,along to our final term, which is\Nb to the power n.
Dialogue: 0,0:01:44.55,0:01:47.40,Default,,0000,0000,0000,,Well, now let's look at what we've got.
Dialogue: 0,0:01:47.40,0:01:52.32,Default,,0000,0000,0000,,Instead of our a, we've got x, and\Ninstead of the b, we've got delta x.
Dialogue: 0,0:01:52.32,0:01:57.37,Default,,0000,0000,0000,,So, we've got x + delta x \Nto the power n.
Dialogue: 0,0:01:57.37,0:02:02.43,Default,,0000,0000,0000,,So our a, we've got x to the power n,
Dialogue: 0,0:02:02.43,0:02:09.84,Default,,0000,0000,0000,,plus n x to the power n - 1\Nand our b is delta x plus, and
Dialogue: 0,0:02:09.84,0:02:15.51,Default,,0000,0000,0000,,again, all these terms which\Nwill be in terms of x and delta x
Dialogue: 0,0:02:15.51,0:02:20.82,Default,,0000,0000,0000,,Up to our last term, which is\Ndelta x to the power n.
Dialogue: 0,0:02:21.65,0:02:27.28,Default,,0000,0000,0000,,Now let's substitute this in\Ninto our derivative here.
Dialogue: 0,0:02:27.48,0:02:34.55,Default,,0000,0000,0000,,So our dy by dx. is the limit as\Ndelta x approaches 0,
Dialogue: 0,0:02:35.51,0:02:38.36,Default,,0000,0000,0000,,of our function of x + delta x.
Dialogue: 0,0:02:39.37,0:02:45.85,Default,,0000,0000,0000,,Which is x to the power n, \Nplus, n x to the n - 1, delta x, plus,
Dialogue: 0,0:02:45.85,0:02:52.33,Default,,0000,0000,0000,,and again all these terms in both x\Nand delta x, plus delta x to the power n.
Dialogue: 0,0:02:52.33,0:02:58.90,Default,,0000,0000,0000,,Minus our function of x,\Nwhich is x to the power n.
Dialogue: 0,0:02:59.59,0:03:03.06,Default,,0000,0000,0000,,All divided by delta x.
Dialogue: 0,0:03:03.79,0:03:08.82,Default,,0000,0000,0000,,Oh, here we have x to the power n\Ntakeaway x to the power n.
Dialogue: 0,0:03:09.45,0:03:15.44,Default,,0000,0000,0000,,So our derivative is the limit\Nas delta x approaches 0.
Dialogue: 0,0:03:16.53,0:03:23.64,Default,,0000,0000,0000,,Of n x to the n - 1, delta x\Nplus all the terms in x and delta x.
Dialogue: 0,0:03:23.64,0:03:31.53,Default,,0000,0000,0000,,Plus delta x to the power n.\NAll divided by delta x.
Dialogue: 0,0:03:34.07,0:03:37.90,Default,,0000,0000,0000,,Now all these terms, have\Ndelta x in them.
Dialogue: 0,0:03:38.80,0:03:45.29,Default,,0000,0000,0000,,And we're dividing by delta x,\Nso we can actually cancel the delta x.
Dialogue: 0,0:03:45.29,0:03:49.60,Default,,0000,0000,0000,,So that we have the limit as\Ndelta x approaches 0.
Dialogue: 0,0:03:50.62,0:03:57.58,Default,,0000,0000,0000,,Of n x to the power of n -1,\Nplus, now all these terms here
Dialogue: 0,0:03:57.58,0:04:03.81,Default,,0000,0000,0000,,have delta x squared, delta x cubed\Nand higher powers all the way up
Dialogue: 0,0:04:03.81,0:04:08.58,Default,,0000,0000,0000,,to delta x to the power n.\NSo when we've divided by delta x,
Dialogue: 0,0:04:08.58,0:04:13.83,Default,,0000,0000,0000,,all of these have still got a\Ndelta x in them up to our
Dialogue: 0,0:04:13.83,0:04:18.16,Default,,0000,0000,0000,,final term of delta x to the\Npower of n -1.
Dialogue: 0,0:04:18.47,0:04:23.46,Default,,0000,0000,0000,,So when we actually take the\Nlimit of delta x approaches 0.
Dialogue: 0,0:04:24.15,0:04:29.07,Default,,0000,0000,0000,,All of these terms are going to\Napproach 0.
Dialogue: 0,0:04:29.07,0:04:32.88,Default,,0000,0000,0000,,They've all got delta x in them\Nso they'll all be 0.
Dialogue: 0,0:04:32.88,0:04:40.49,Default,,0000,0000,0000,,So our derivative is just n times\Nx to the power of n - 1.
Dialogue: 0,0:04:41.63,0:04:44.45,Default,,0000,0000,0000,,So at derivative of x to the power n,
Dialogue: 0,0:04:44.76,0:04:50.19,Default,,0000,0000,0000,,is n times x to the power of n - 1.
Dialogue: 0,0:04:50.20,0:04:53.59,Default,,0000,0000,0000,,Let's have a look at some examples.
Dialogue: 0,0:04:53.59,0:04:56.45,Default,,0000,0000,0000,,Let's say y equals x squared.
Dialogue: 0,0:04:56.45,0:05:02.42,Default,,0000,0000,0000,,So our dy by dx equals
Dialogue: 0,0:05:02.42,0:05:05.77,Default,,0000,0000,0000,,Well, the power comes down \Nin front, so it's
Dialogue: 0,0:05:05.77,0:05:11.34,Default,,0000,0000,0000,,2 times x to the power of 2\Ntake away 1 which gives us
Dialogue: 0,0:05:11.34,0:05:16.44,Default,,0000,0000,0000,,2 times x to the power 1,\Nwhich is just 2x.
Dialogue: 0,0:05:18.64,0:05:20.89,Default,,0000,0000,0000,,y equals x to the power 7.
Dialogue: 0,0:05:23.28,0:05:27.84,Default,,0000,0000,0000,,dy by dx equals... \Nthe power comes down
Dialogue: 0,0:05:27.84,0:05:36.41,Default,,0000,0000,0000,,7 times x to the power of 7 - 1,\Nso we get 7x to the power of 6.
Dialogue: 0,0:05:39.46,0:05:41.55,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:05:42.17,0:05:46.43,Default,,0000,0000,0000,,What happens when we\Nhave y equals just x?
Dialogue: 0,0:05:46.43,0:05:48.69,Default,,0000,0000,0000,,x to the power of 1?
Dialogue: 0,0:05:48.69,0:05:52.22,Default,,0000,0000,0000,,Well, we already know the\Nderivative of this, but let's
Dialogue: 0,0:05:52.22,0:05:54.69,Default,,0000,0000,0000,,see how it fits with the rule.
Dialogue: 0,0:05:55.88,0:06:02.85,Default,,0000,0000,0000,,Bring down the power 1 times x\Nto the power of 1 take away 1,
Dialogue: 0,0:06:02.85,0:06:06.75,Default,,0000,0000,0000,,so we have 1 times x to the\Npower of 0.
Dialogue: 0,0:06:07.93,0:06:13.06,Default,,0000,0000,0000,,Well, x to the power of 0\Nis 1, so it's 1 times 1, so
Dialogue: 0,0:06:13.06,0:06:15.83,Default,,0000,0000,0000,,we end up with the\Nderivative of 1, which is
Dialogue: 0,0:06:15.83,0:06:18.36,Default,,0000,0000,0000,,exactly what we expected\Nbecause we know the
Dialogue: 0,0:06:18.36,0:06:22.59,Default,,0000,0000,0000,,gradient function of y\Nequals x is 1.
Dialogue: 0,0:06:26.02,0:06:32.88,Default,,0000,0000,0000,,So that was when x was a positive\Ninteger. What happens when x is 0?
Dialogue: 0,0:06:32.88,0:06:37.96,Default,,0000,0000,0000,,Well, let's have a look: y\Nequals x to the power 0.
Dialogue: 0,0:06:39.09,0:06:45.22,Default,,0000,0000,0000,,Well, we just saw here that x to\Nthe power 0 is actually 1.
Dialogue: 0,0:06:47.59,0:06:54.74,Default,,0000,0000,0000,,And if we find the derivative of\Ny equals 1. Well, y equals 1 is a
Dialogue: 0,0:06:54.74,0:06:58.08,Default,,0000,0000,0000,,horizontal line, so the\Nderivative is 0.
Dialogue: 0,0:06:59.70,0:07:06.24,Default,,0000,0000,0000,,So y equals x to the power 0\Nwhen n is 0. The derivative is 0.
Dialogue: 0,0:07:11.11,0:07:14.77,Default,,0000,0000,0000,,Let's have a look at some more\Ncomplicated examples.
Dialogue: 0,0:07:16.46,0:07:25.70,Default,,0000,0000,0000,,Let's try y equals 6 x cubed, minus\N12 x to the power 4, plus 5.
Dialogue: 0,0:07:27.99,0:07:32.28,Default,,0000,0000,0000,,dy by dx is equal to...
Dialogue: 0,0:07:32.28,0:07:37.72,Default,,0000,0000,0000,,If that's 6 multiplied by 3 as we\Nbring the power down,
Dialogue: 0,0:07:37.72,0:07:41.42,Default,,0000,0000,0000,,x to the power, take one from the power,
Dialogue: 0,0:07:42.42,0:07:47.87,Default,,0000,0000,0000,,minus 12 times,\Nbring the power down, 4,
Dialogue: 0,0:07:48.53,0:07:54.86,Default,,0000,0000,0000,,x to power of 4 take away 1 and\Nour derivative of 5 is 0.
Dialogue: 0,0:07:56.85,0:07:59.21,Default,,0000,0000,0000,,(I'll put the plus 0 there.)
Dialogue: 0,0:07:59.21,0:08:05.45,Default,,0000,0000,0000,,So three 6s... 18 x to the power\N3 - 1 is 2
Dialogue: 0,0:08:06.16,0:08:14.73,Default,,0000,0000,0000,,minus four 12s... 48 x to the power\N4 - 1 is 3.
Dialogue: 0,0:08:15.52,0:08:18.35,Default,,0000,0000,0000,,So there's our derivative.
Dialogue: 0,0:08:19.52,0:08:20.78,Default,,0000,0000,0000,,Let's try another one.
Dialogue: 0,0:08:20.78,0:08:31.38,Default,,0000,0000,0000,,y equals x minus 5x to the power 5 \N+ 6 x to the power 7 + 25.
Dialogue: 0,0:08:32.46,0:08:38.95,Default,,0000,0000,0000,,So our derivative dy by dx\Nequals... Now this is x to the power 1.
Dialogue: 0,0:08:40.58,0:08:45.04,Default,,0000,0000,0000,,So, 1 times x to the power 1 - 1,
Dialogue: 0,0:08:46.29,0:08:53.47,Default,,0000,0000,0000,,minus 5 times, bring the power down,\N5 times x to the power of 5 - 1,
Dialogue: 0,0:08:56.46,0:09:03.72,Default,,0000,0000,0000,,plus 6 times, bring the power down,\N7 multiplied by x to power 7 - 1,
Dialogue: 0,0:09:03.72,0:09:10.08,Default,,0000,0000,0000,,plus the derivative of 25, again 0.
Dialogue: 0,0:09:11.00,0:09:18.09,Default,,0000,0000,0000,,So we have 1 times x to the power\Nof 0, which is just 1,
Dialogue: 0,0:09:18.89,0:09:25.69,Default,,0000,0000,0000,,minus, five 5s are 25, times x\Nto the power 5 - 1 is 4,
Dialogue: 0,0:09:26.39,0:09:33.83,Default,,0000,0000,0000,,plus six 7s, 42 times x to the\Npower 7 - 1 which is 6.
Dialogue: 0,0:09:37.45,0:09:41.74,Default,,0000,0000,0000,,Now we've proved this result\Nwhen n is a positive integer,
Dialogue: 0,0:09:41.74,0:09:45.30,Default,,0000,0000,0000,,but it actually works also when\Nn is a fraction or when it's a
Dialogue: 0,0:09:45.30,0:09:49.38,Default,,0000,0000,0000,,negative number. Now we're not\Ngoing to do the proof of this
Dialogue: 0,0:09:49.38,0:09:52.93,Default,,0000,0000,0000,,because it requires a more\Ncomplicated version of the
Dialogue: 0,0:09:52.93,0:09:56.88,Default,,0000,0000,0000,,binomial expansion, but we're\Nstill going to use the result,
Dialogue: 0,0:09:56.88,0:10:02.02,Default,,0000,0000,0000,,so let's have a look at y equals\Nx to the power a half.
Dialogue: 0,0:10:02.91,0:10:10.08,Default,,0000,0000,0000,,Our dy by dx is going to be a half,\Nas we bring down the power,
Dialogue: 0,0:10:10.08,0:10:16.22,Default,,0000,0000,0000,,x to the power of a half take away 1\Nwhich equals
Dialogue: 0,0:10:16.22,0:10:21.85,Default,,0000,0000,0000,,a half times x, and a half take away 1,\Nis minus a half.
Dialogue: 0,0:10:22.71,0:10:27.26,Default,,0000,0000,0000,,Let's have a look now, when n is\Na negative number, so
Dialogue: 0,0:10:27.26,0:10:35.76,Default,,0000,0000,0000,,y equals x to the power of minus 1. So dy\Nby dx equals, let's bring the power down,
Dialogue: 0,0:10:35.76,0:10:43.64,Default,,0000,0000,0000,,minus 1 times x to the power of\Nminus 1 take away 1, so we have...
Dialogue: 0,0:10:43.64,0:10:49.48,Default,,0000,0000,0000,,Minus x to the power -1 - 1 is -2.
Dialogue: 0,0:10:54.12,0:10:56.91,Default,,0000,0000,0000,,Now, before we finish, let's\Nlook at two more complicated
Dialogue: 0,0:10:56.91,0:11:00.28,Default,,0000,0000,0000,,examples where we need to do a\Nlittle bit of rewriting in
Dialogue: 0,0:11:00.28,0:11:04.28,Default,,0000,0000,0000,,index notation before we can\Ncarry out the differentiation.
Dialogue: 0,0:11:05.61,0:11:12.46,Default,,0000,0000,0000,,So let's have a look at\Ny equals 1 over x plus 6x
Dialogue: 0,0:11:13.34,0:11:17.68,Default,,0000,0000,0000,,minus 4x to the power of 3\Nover 2 plus 8.
Dialogue: 0,0:11:19.96,0:11:25.15,Default,,0000,0000,0000,,Now we need to rewrite this in\Nindex notation so that we can
Dialogue: 0,0:11:25.15,0:11:30.63,Default,,0000,0000,0000,,easily differentiate it. So\Nthat's x to the power of minus 1
Dialogue: 0,0:11:30.63,0:11:36.68,Default,,0000,0000,0000,,plus 6x minus 4x to the power 3\Nover 2 plus 8.
Dialogue: 0,0:11:37.27,0:11:41.64,Default,,0000,0000,0000,,So let's differentiate dy by dx equals...
Dialogue: 0,0:11:44.45,0:11:50.00,Default,,0000,0000,0000,,Bring the power down, minus 1 x to\Nthe power of minus 1 take away 1,
Dialogue: 0,0:11:51.48,0:12:02.23,Default,,0000,0000,0000,,plus derivative of 6x to the power 1,\Nthat's 6 times x to the power 1 - 1,
Dialogue: 0,0:12:03.39,0:12:09.42,Default,,0000,0000,0000,,minus... now 4 times 3 over 2,
Dialogue: 0,0:12:10.22,0:12:14.62,Default,,0000,0000,0000,,multiplied by x to the power of\N3 over 2 take away 1,
Dialogue: 0,0:12:15.76,0:12:19.36,Default,,0000,0000,0000,,plus, the derivative of 8,\Nwhich is 0.
Dialogue: 0,0:12:19.87,0:12:26.60,Default,,0000,0000,0000,,So here we have minus x,\N-1 - 1 is -2.
Dialogue: 0,0:12:27.98,0:12:35.86,Default,,0000,0000,0000,,Plus 6 x to the power 1 - 1\Nis 0 which is 1, so it's just plus 6.
Dialogue: 0,0:12:36.87,0:12:43.86,Default,,0000,0000,0000,,Minus... three 4s are 12 divided by 2\Ngives us 6,
Dialogue: 0,0:12:43.86,0:12:47.75,Default,,0000,0000,0000,,x to the power of 3/2 minus 1...
Dialogue: 0,0:12:47.75,0:12:52.49,Default,,0000,0000,0000,,So that's 1 and a half minus 1, so we end\Nup with x to the power a half.
Dialogue: 0,0:12:53.61,0:12:55.23,Default,,0000,0000,0000,,And there's our derivative.
Dialogue: 0,0:12:56.33,0:13:05.40,Default,,0000,0000,0000,,OK, one more example, y equals\N4x to the power of one third,
Dialogue: 0,0:13:05.40,0:13:11.95,Default,,0000,0000,0000,,minus 5x plus 6 divided by x cubed.
Dialogue: 0,0:13:12.84,0:13:17.54,Default,,0000,0000,0000,,Now again, we've got a mixture\Nof notations and to
Dialogue: 0,0:13:17.54,0:13:22.71,Default,,0000,0000,0000,,differentiate it we need to\Nwrite it all in index notation
Dialogue: 0,0:13:22.71,0:13:28.82,Default,,0000,0000,0000,,rather than having the division.\NSo this will be 6x to the power of -3,
Dialogue: 0,0:13:30.25,0:13:34.76,Default,,0000,0000,0000,,So, our dy by dx equals...
Dialogue: 0,0:13:34.76,0:13:39.73,Default,,0000,0000,0000,,4 multiplied by the power,\Nwhich is a third,
Dialogue: 0,0:13:40.51,0:13:45.97,Default,,0000,0000,0000,,x to the power of 1/3 take away 1,
Dialogue: 0,0:13:45.97,0:13:52.49,Default,,0000,0000,0000,,minus, this is 5 x to the power 1,\Nso it's 5 times 1,
Dialogue: 0,0:13:52.49,0:13:57.08,Default,,0000,0000,0000,,multiplied by x to the\Npower of 1 - 1,
Dialogue: 0,0:13:57.10,0:14:08.15,Default,,0000,0000,0000,,plus 6 multiplied by minus 3 times\Nx to the power of -3 - 1.
Dialogue: 0,0:14:09.08,0:14:12.07,Default,,0000,0000,0000,,So let's tidy it all up.
Dialogue: 0,0:14:12.07,0:14:20.26,Default,,0000,0000,0000,,We get 4 thirds x to the power\N1/3 take away 1 is minus 2/3,
Dialogue: 0,0:14:20.26,0:14:23.78,Default,,0000,0000,0000,,so it's x to the power of minus 2/3,
Dialogue: 0,0:14:25.39,0:14:31.42,Default,,0000,0000,0000,,minus, now here x the power of 1 - 1\Nis x to the power 0, which is 1,
Dialogue: 0,0:14:31.42,0:14:33.10,Default,,0000,0000,0000,,so it's just minus 5,
Dialogue: 0,0:14:34.66,0:14:43.83,Default,,0000,0000,0000,,Six times -3 is -18 and x to the power\N-3 - 1 is -4.
Dialogue: 0,0:14:43.83,0:14:46.60,Default,,0000,0000,0000,,So, there's our derivative.