[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.62,0:00:06.27,Default,,0000,0000,0000,,In previous tutorials, you've\Ndifferentiated from first
Dialogue: 0,0:00:06.27,0:00:12.74,Default,,0000,0000,0000,,principles functions such as\Nc, x, 2x, x to the power n,
Dialogue: 0,0:00:12.74,0:00:19.60,Default,,0000,0000,0000,,where n is any power, sine x,\Ncosine x, e to the power x
Dialogue: 0,0:00:19.60,0:00:22.45,Default,,0000,0000,0000,,and log to the base e of x.
Dialogue: 0,0:00:23.72,0:00:28.02,Default,,0000,0000,0000,,Now what we're going to do in\Nthis tutorial is to construct
Dialogue: 0,0:00:28.02,0:00:31.24,Default,,0000,0000,0000,,a table of these standard\Nderivatives, and when we've done
Dialogue: 0,0:00:31.24,0:00:34.46,Default,,0000,0000,0000,,that, we'll look at some\Nvariations of those standard
Dialogue: 0,0:00:34.46,0:00:36.97,Default,,0000,0000,0000,,derivatives and add those to\Nour table.
Dialogue: 0,0:00:38.77,0:00:42.19,Default,,0000,0000,0000,,So let's start with a function.
Dialogue: 0,0:00:43.65,0:00:44.73,Default,,0000,0000,0000,,f of x.
Dialogue: 0,0:00:46.24,0:00:47.65,Default,,0000,0000,0000,,And add derivative.
Dialogue: 0,0:00:49.80,0:00:55.96,Default,,0000,0000,0000,,Which may be referred to as,\Ndf by dx or f dashed of x.
Dialogue: 0,0:01:01.57,0:01:07.35,Default,,0000,0000,0000,,So let's start with c, well c is a\Nconstant. It's a straight line
Dialogue: 0,0:01:07.35,0:01:11.07,Default,,0000,0000,0000,,and it's horizontal. So I\Ngradient function is 0.
Dialogue: 0,0:01:12.98,0:01:16.96,Default,,0000,0000,0000,,x, the derivative of x is 1.
Dialogue: 0,0:01:18.71,0:01:21.84,Default,,0000,0000,0000,,2x, the derivative is 2.
Dialogue: 0,0:01:23.46,0:01:25.59,Default,,0000,0000,0000,,If we have x to the power n.
Dialogue: 0,0:01:26.74,0:01:32.93,Default,,0000,0000,0000,,Then the derivative is n times x\Nto the power of n - 1,
Dialogue: 0,0:01:32.93,0:01:35.58,Default,,0000,0000,0000,,where n is a real number.
Dialogue: 0,0:01:38.86,0:01:43.28,Default,,0000,0000,0000,,sine x, the derivative is cos x.
Dialogue: 0,0:01:45.37,0:01:50.76,Default,,0000,0000,0000,,cos x, the derivative is -sine x.
Dialogue: 0,0:01:52.83,0:01:57.85,Default,,0000,0000,0000,,e to the power x, that's the one that\Nstays the same, e to the power x.
Dialogue: 0,0:01:58.83,0:02:01.85,Default,,0000,0000,0000,,And log to the base e of x.
Dialogue: 0,0:02:02.56,0:02:05.91,Default,,0000,0000,0000,,Where the derivative is 1 over x.
Dialogue: 0,0:02:08.30,0:02:13.83,Default,,0000,0000,0000,,Let's look now at a variation\Nof some of these, we'll come
Dialogue: 0,0:02:13.83,0:02:16.61,Default,,0000,0000,0000,,back to the table in just a minute.
Dialogue: 0,0:02:18.85,0:02:25.34,Default,,0000,0000,0000,,OK, what if we have f of x\Nequals, instead of just sine x,
Dialogue: 0,0:02:25.34,0:02:27.77,Default,,0000,0000,0000,,we have 2 sine x.
Dialogue: 0,0:02:28.98,0:02:32.13,Default,,0000,0000,0000,,Well, this is just the same as
Dialogue: 0,0:02:32.13,0:02:35.08,Default,,0000,0000,0000,,taking sine x and taking two of them.
Dialogue: 0,0:02:35.08,0:02:39.34,Default,,0000,0000,0000,,So when we find the derivative,\Nwe actually find the
Dialogue: 0,0:02:39.34,0:02:43.06,Default,,0000,0000,0000,,derivative of sine x and take\Ntwo of those.
Dialogue: 0,0:02:43.31,0:02:46.76,Default,,0000,0000,0000,,Well, the derivative of sine x\Nis cos x.
Dialogue: 0,0:02:47.60,0:02:51.45,Default,,0000,0000,0000,,So the derivative of 2 sine x\Nis 2 cos x.
Dialogue: 0,0:02:52.92,0:03:02.81,Default,,0000,0000,0000,,Similarly, if we take f of x is\N-5 sine x, then our derivative
Dialogue: 0,0:03:02.81,0:03:07.64,Default,,0000,0000,0000,,f dashed of x is -5 cos x.
Dialogue: 0,0:03:08.91,0:03:10.76,Default,,0000,0000,0000,,And this in fact is the case for
Dialogue: 0,0:03:10.76,0:03:15.52,Default,,0000,0000,0000,,any constant multiplied by the function.
Dialogue: 0,0:03:16.36,0:03:21.34,Default,,0000,0000,0000,,So if we have f of x equals\NC times sine x.
Dialogue: 0,0:03:22.35,0:03:28.27,Default,,0000,0000,0000,,Then f dashed of x is going to\Nbe C times cos x.
Dialogue: 0,0:03:29.47,0:03:36.63,Default,,0000,0000,0000,,And this is called the\Nconstant multiplier rule.
Dialogue: 0,0:03:40.56,0:03:41.89,Default,,0000,0000,0000,,So if we have...
Dialogue: 0,0:03:43.47,0:03:46.39,Default,,0000,0000,0000,,C times our function of x.
Dialogue: 0,0:03:46.94,0:03:50.90,Default,,0000,0000,0000,,And we want to differentiate it.\NSo we want to find the
Dialogue: 0,0:03:50.90,0:03:52.55,Default,,0000,0000,0000,,derivative with respect to x.
Dialogue: 0,0:03:53.80,0:04:00.06,Default,,0000,0000,0000,,It's actually just C times the\Nderivative df by dx or
Dialogue: 0,0:04:00.06,0:04:04.90,Default,,0000,0000,0000,,C times our f dashed of x.
Dialogue: 0,0:04:07.59,0:04:10.89,Default,,0000,0000,0000,,Now let's prove this from\Nfirst principles.
Dialogue: 0,0:04:10.89,0:04:15.17,Default,,0000,0000,0000,,If we take our function of x\Nto be g of x and
Dialogue: 0,0:04:15.17,0:04:18.58,Default,,0000,0000,0000,,it's equal to some constant times f of x.
Dialogue: 0,0:04:19.86,0:04:22.60,Default,,0000,0000,0000,,Now, our definition of our derivative
Dialogue: 0,0:04:22.60,0:04:32.46,Default,,0000,0000,0000,,is g dashed x equals the limit \Nas delta x approaches 0 of
Dialogue: 0,0:04:32.46,0:04:42.64,Default,,0000,0000,0000,,g of x plus delta x, minus g of x,\Nall divided by delta x.
Dialogue: 0,0:04:42.93,0:04:46.99,Default,,0000,0000,0000,,Which equals, again our limit\Nas delta x approaches 0.
Dialogue: 0,0:04:47.58,0:04:55.46,Default,,0000,0000,0000,,And in this case our g of x plus delta x \Nis C of f of x plus delta x.
Dialogue: 0,0:04:55.46,0:04:58.52,Default,,0000,0000,0000,,So we substitute that in.
Dialogue: 0,0:04:59.54,0:05:05.38,Default,,0000,0000,0000,,Minus our g of x which is C f of x.
Dialogue: 0,0:05:05.38,0:05:08.59,Default,,0000,0000,0000,,All divided by delta x.
Dialogue: 0,0:05:09.18,0:05:13.92,Default,,0000,0000,0000,,And here I'm going to take the\NC outside the bracket so we
Dialogue: 0,0:05:13.92,0:05:17.76,Default,,0000,0000,0000,,have the limit as delta x\Napproaches zero of
Dialogue: 0,0:05:17.76,0:05:24.35,Default,,0000,0000,0000,,C and then our function of x plus delta x\Ntakeaway our function of x...
Dialogue: 0,0:05:25.38,0:05:27.58,Default,,0000,0000,0000,,divided by delta x.
Dialogue: 0,0:05:27.96,0:05:31.95,Default,,0000,0000,0000,,And because our constant is\Nnothing whatsoever to do with
Dialogue: 0,0:05:31.95,0:05:36.86,Default,,0000,0000,0000,,our limit is delta x approaches\Nzero we can actually take the
Dialogue: 0,0:05:36.86,0:05:43.02,Default,,0000,0000,0000,,constant outside of the limiting\Nsign, so the limit delta x approaches 0
Dialogue: 0,0:05:43.02,0:05:49.07,Default,,0000,0000,0000,,and we have f of x plus delta x, \Nminus our function of x...
Dialogue: 0,0:05:49.07,0:05:51.100,Default,,0000,0000,0000,,divided by delta x.
Dialogue: 0,0:05:52.62,0:05:58.74,Default,,0000,0000,0000,,So this is C and this...\Nis our derivative
Dialogue: 0,0:05:58.74,0:06:02.93,Default,,0000,0000,0000,,so it's actually our f dashed of x.
Dialogue: 0,0:06:03.28,0:06:06.71,Default,,0000,0000,0000,,So our g dashed of x, our derivative,
Dialogue: 0,0:06:07.44,0:06:12.25,Default,,0000,0000,0000,,is equal to C times our f dashed of x.
Dialogue: 0,0:06:17.20,0:06:21.03,Default,,0000,0000,0000,,Now let's have a look at what \Nwe're going to do when
Dialogue: 0,0:06:21.03,0:06:24.56,Default,,0000,0000,0000,,we have two functions that\Nare added together.
Dialogue: 0,0:06:24.56,0:06:28.42,Default,,0000,0000,0000,,So let's say we have \Nf of x, plus, g of x.
Dialogue: 0,0:06:28.78,0:06:32.30,Default,,0000,0000,0000,,And we want to differentiate\Nthem with respect to x.
Dialogue: 0,0:06:33.05,0:06:39.22,Default,,0000,0000,0000,,So we want the derivative of\Nthat function with respect to x.
Dialogue: 0,0:06:39.83,0:06:44.12,Default,,0000,0000,0000,,Well, quite simply, what we do\Nis we differentiate each part
Dialogue: 0,0:06:44.12,0:06:46.76,Default,,0000,0000,0000,,separately and add them together.
Dialogue: 0,0:06:46.76,0:06:54.99,Default,,0000,0000,0000,,So that's the same as \Ndf dx plus dg dx.
Dialogue: 0,0:06:56.58,0:07:02.17,Default,,0000,0000,0000,,Similarly, what happens if we\Nwant two functions that are
Dialogue: 0,0:07:02.17,0:07:06.33,Default,,0000,0000,0000,,subtracted and we want to\Ndifferentiate those?
Dialogue: 0,0:07:06.33,0:07:13.48,Default,,0000,0000,0000,,Well, again, the derivative of those\Nfunctions subtracted...
Dialogue: 0,0:07:14.98,0:07:16.94,Default,,0000,0000,0000,,It's very straightforward.
Dialogue: 0,0:07:17.05,0:07:19.63,Default,,0000,0000,0000,,Because what we've got here.
Dialogue: 0,0:07:20.26,0:07:23.32,Default,,0000,0000,0000,,Is just the same as we've got\Nhere. But with this second
Dialogue: 0,0:07:23.32,0:07:25.86,Default,,0000,0000,0000,,function multiplied by minus one.
Dialogue: 0,0:07:25.86,0:07:28.40,Default,,0000,0000,0000,,And there we are using the
Dialogue: 0,0:07:28.40,0:07:32.65,Default,,0000,0000,0000,,constant multiplier rule. So\Nthis is just the same as the
Dialogue: 0,0:07:32.65,0:07:39.94,Default,,0000,0000,0000,,derivative of d of x of f of x plus
Dialogue: 0,0:07:39.94,0:07:50.09,Default,,0000,0000,0000,,minus one times the derivative of g of x.
Dialogue: 0,0:07:53.26,0:07:58.34,Default,,0000,0000,0000,,So that's our df dx plus
Dialogue: 0,0:07:58.34,0:08:03.27,Default,,0000,0000,0000,,our minus one times our \Nderivative there which is
Dialogue: 0,0:08:03.27,0:08:08.34,Default,,0000,0000,0000,,plus minus one times dg dx.
Dialogue: 0,0:08:09.75,0:08:17.98,Default,,0000,0000,0000,,Which is just the same as df dx \Nminus dg dx.
Dialogue: 0,0:08:20.15,0:08:22.20,Default,,0000,0000,0000,,Let's just return to our table.
Dialogue: 0,0:08:23.87,0:08:25.37,Default,,0000,0000,0000,,And we can write those in.
Dialogue: 0,0:08:28.07,0:08:34.81,Default,,0000,0000,0000,,So we have our constant multiplier\Nrule: c times our function of x,
Dialogue: 0,0:08:34.81,0:08:38.85,Default,,0000,0000,0000,,is c times df dx.
Dialogue: 0,0:08:40.84,0:08:44.87,Default,,0000,0000,0000,,And if we have f of x plus g of x.
Dialogue: 0,0:08:45.92,0:08:52.01,Default,,0000,0000,0000,,Our derivative is df dx plus dg dx.
Dialogue: 0,0:08:53.50,0:08:57.86,Default,,0000,0000,0000,,And if they're subtracted f of x\Nminus g of x.
Dialogue: 0,0:08:58.47,0:09:04.17,Default,,0000,0000,0000,,Then the derivative\Nis df dx minus dg dx.
Dialogue: 0,0:09:06.02,0:09:08.20,Default,,0000,0000,0000,,Now let's have a look at\Nan example.
Dialogue: 0,0:09:10.54,0:09:15.13,Default,,0000,0000,0000,,2 x cubed minus 6 cos x
Dialogue: 0,0:09:15.13,0:09:20.96,Default,,0000,0000,0000,,And we want to find the\Nderivative with respect to x.
Dialogue: 0,0:09:22.83,0:09:27.14,Default,,0000,0000,0000,,And that equals the derivative of
Dialogue: 0,0:09:27.14,0:09:31.95,Default,,0000,0000,0000,,2 x cubed with respect to x, minus
Dialogue: 0,0:09:31.95,0:09:39.44,Default,,0000,0000,0000,,the derivative of 6 cos x \Nwith respect to x.
Dialogue: 0,0:09:40.53,0:09:42.91,Default,,0000,0000,0000,,Now what we have here is the 2, \Nwhich is,
Dialogue: 0,0:09:42.91,0:09:47.79,Default,,0000,0000,0000,,we can use the constant multiplier rule. \NSo that we've got actually:
Dialogue: 0,0:09:47.79,0:09:55.65,Default,,0000,0000,0000,,Twice the derivative of x cubed\Nwith respect to x, minus,
Dialogue: 0,0:09:55.65,0:09:58.82,Default,,0000,0000,0000,,and again here using the constant\Nmultiplier rule, we can
Dialogue: 0,0:09:58.82,0:10:00.34,Default,,0000,0000,0000,,take the six outside.
Dialogue: 0,0:10:01.05,0:10:07.75,Default,,0000,0000,0000,,So it's six times the derivative\Nof cos x with respect to x.
Dialogue: 0,0:10:09.72,0:10:13.14,Default,,0000,0000,0000,,So we have twice, now the\Nderivative of x cubed
Dialogue: 0,0:10:13.14,0:10:14.89,Default,,0000,0000,0000,,with respect to x,
Dialogue: 0,0:10:14.89,0:10:22.22,Default,,0000,0000,0000,,is 3 x squared minus, 6 times
Dialogue: 0,0:10:22.22,0:10:27.82,Default,,0000,0000,0000,,the derivative of cos x, which is\Nminus sine x.
Dialogue: 0,0:10:29.87,0:10:36.07,Default,,0000,0000,0000,,So we have two threes, 6 x squared,\Nminus times minus is
Dialogue: 0,0:10:36.07,0:10:40.51,Default,,0000,0000,0000,,positive times 6 sine x.
Dialogue: 0,0:10:45.83,0:10:48.61,Default,,0000,0000,0000,,Let's look now at extending\Nthe table further.
Dialogue: 0,0:10:50.43,0:10:55.94,Default,,0000,0000,0000,,So let's start with our function\Nf of x again.
Dialogue: 0,0:10:56.97,0:10:58.40,Default,,0000,0000,0000,,And our derivative.
Dialogue: 0,0:11:02.06,0:11:06.39,Default,,0000,0000,0000,,df dx or f dash of x.
Dialogue: 0,0:11:11.25,0:11:17.58,Default,,0000,0000,0000,,And this time let's look at sine\Nof mx where m is
Dialogue: 0,0:11:17.58,0:11:19.40,Default,,0000,0000,0000,,any constant number
Dialogue: 0,0:11:19.40,0:11:21.90,Default,,0000,0000,0000,,and the cos of mx.
Dialogue: 0,0:11:23.64,0:11:27.85,Default,,0000,0000,0000,,Then we'll have a look at e to\Nthe power of mx.
Dialogue: 0,0:11:29.60,0:11:33.70,Default,,0000,0000,0000,,And then log to the\Nbase e of mx.
Dialogue: 0,0:11:38.37,0:11:44.98,Default,,0000,0000,0000,,Let's look then at y equals sine\Nmx. Now here we're going to do a
Dialogue: 0,0:11:44.98,0:11:49.12,Default,,0000,0000,0000,,substitution and instead of mx,\Nwe're going to write
Dialogue: 0,0:11:49.12,0:11:51.42,Default,,0000,0000,0000,,u is equal to mx.
Dialogue: 0,0:11:52.52,0:11:55.92,Default,,0000,0000,0000,,So therefore our y is\Nequal to sine u.
Dialogue: 0,0:11:57.64,0:12:02.68,Default,,0000,0000,0000,,And we're going to differentiate\Nu with respect to x.
Dialogue: 0,0:12:02.68,0:12:05.34,Default,,0000,0000,0000,,So du by dx is equal to m.
Dialogue: 0,0:12:06.44,0:12:09.74,Default,,0000,0000,0000,,And then we're going to\Ndifferentiate y with respect to
Dialogue: 0,0:12:09.74,0:12:12.96,Default,,0000,0000,0000,,u. So dy by du equals
Dialogue: 0,0:12:12.96,0:12:17.10,Default,,0000,0000,0000,,the derivative of sine u,\Nwhich is cos u.
Dialogue: 0,0:12:18.46,0:12:23.94,Default,,0000,0000,0000,,Now dy by dx, and you'll\Nlearn this in the future,
Dialogue: 0,0:12:23.94,0:12:27.02,Default,,0000,0000,0000,,dy by dx is equal to
Dialogue: 0,0:12:27.02,0:12:33.78,Default,,0000,0000,0000,,equal to dy by du multiplied by\Ndu by dx.
Dialogue: 0,0:12:34.78,0:12:38.29,Default,,0000,0000,0000,,Now this is called differentiating\Na function of a function
Dialogue: 0,0:12:38.29,0:12:43.98,Default,,0000,0000,0000,,or differentiating using the chain rule,\Nand it's the subject of another tutorial.
Dialogue: 0,0:12:44.49,0:12:47.67,Default,,0000,0000,0000,,So we're not going to go\Nthrough the details of that now.
Dialogue: 0,0:12:47.67,0:12:51.97,Default,,0000,0000,0000,,What I want you to do\Nis just use it as a formula.
Dialogue: 0,0:12:53.98,0:12:57.89,Default,,0000,0000,0000,,Let's now substitute then\Ndy by du is equal to
Dialogue: 0,0:12:57.89,0:13:04.35,Default,,0000,0000,0000,,cos u multiplied by du by dx,\Nwhich equals m.
Dialogue: 0,0:13:05.44,0:13:09.35,Default,,0000,0000,0000,,Now we usually write the\Nconstant first, so
Dialogue: 0,0:13:09.35,0:13:12.48,Default,,0000,0000,0000,,it'll be m times the cos of u.
Dialogue: 0,0:13:13.18,0:13:19.64,Default,,0000,0000,0000,,But we introduced the u and we\Nwant actually dy in terms of dx
Dialogue: 0,0:13:19.64,0:13:24.28,Default,,0000,0000,0000,,in terms of x, dy by dx.\NSo what we need to do is to
Dialogue: 0,0:13:24.28,0:13:29.33,Default,,0000,0000,0000,,substitute back and instead of\Nwriting u we want to write mx.
Dialogue: 0,0:13:29.33,0:13:37.26,Default,,0000,0000,0000,,So we have as dy by dx equals m\Ntimes the cos of mx.
Dialogue: 0,0:13:38.45,0:13:42.30,Default,,0000,0000,0000,,So in effect, what's happened\Nwhen we found the derivative is
Dialogue: 0,0:13:42.30,0:13:43.97,Default,,0000,0000,0000,,we've multiplied by m.
Dialogue: 0,0:13:44.98,0:13:47.51,Default,,0000,0000,0000,,So let's go back and\Nput that in our table.
Dialogue: 0,0:13:48.59,0:13:54.63,Default,,0000,0000,0000,,The derivative of sine mx\Nis m cos mx.
Dialogue: 0,0:13:55.25,0:14:00.24,Default,,0000,0000,0000,,And in a similar way, the\Nderivative of cos mx will be
Dialogue: 0,0:14:00.24,0:14:06.86,Default,,0000,0000,0000,,minus m sine mx again, just like\Nmultiplying by the m.
Dialogue: 0,0:14:08.29,0:14:12.02,Default,,0000,0000,0000,,Let's look now at e\Nto the power of mx.
Dialogue: 0,0:14:17.77,0:14:21.92,Default,,0000,0000,0000,,So our y equals x to the \Npower of mx.
Dialogue: 0,0:14:21.92,0:14:26.31,Default,,0000,0000,0000,,Again, we're going to do a substitution
Dialogue: 0,0:14:26.31,0:14:29.46,Default,,0000,0000,0000,,and let u equal mx.
Dialogue: 0,0:14:31.13,0:14:34.52,Default,,0000,0000,0000,,So that our y is equal to e\Nto the power u.
Dialogue: 0,0:14:36.06,0:14:40.32,Default,,0000,0000,0000,,Again, we're going to\Ndifferentiate u with respect to x.
Dialogue: 0,0:14:40.32,0:14:46.19,Default,,0000,0000,0000,,I'm gonna get m and will differentiate\Ny with respect to u.
Dialogue: 0,0:14:47.16,0:14:48.49,Default,,0000,0000,0000,,And we get e to the u.
Dialogue: 0,0:14:50.37,0:15:00.12,Default,,0000,0000,0000,,Our dy by dx is equal to\Ndy by du multiplied by du by dx.
Dialogue: 0,0:15:01.37,0:15:06.74,Default,,0000,0000,0000,,And if we substitute in, dy by du is\Ne to the u.
Dialogue: 0,0:15:06.74,0:15:08.59,Default,,0000,0000,0000,,du by dx is m.
Dialogue: 0,0:15:09.32,0:15:12.61,Default,,0000,0000,0000,,Again, let's write our\Nconstant terms first, so
Dialogue: 0,0:15:12.61,0:15:17.13,Default,,0000,0000,0000,,we get m e to the power u.\NNow we need to
Dialogue: 0,0:15:17.13,0:15:24.21,Default,,0000,0000,0000,,substitute back for our u so\Nwe get m e to the power of mx.
Dialogue: 0,0:15:24.57,0:15:30.81,Default,,0000,0000,0000,,So again, our derivative we've\Nended up multiplying by m.
Dialogue: 0,0:15:32.03,0:15:33.72,Default,,0000,0000,0000,,Let's put that back in our table.
Dialogue: 0,0:15:36.32,0:15:40.73,Default,,0000,0000,0000,,So e to the power of mx.\NThe derivative is
Dialogue: 0,0:15:40.73,0:15:43.38,Default,,0000,0000,0000,,m e to the power of mx.
Dialogue: 0,0:15:45.24,0:15:49.38,Default,,0000,0000,0000,,Let's look now at log to the\Nbase e of mx.
Dialogue: 0,0:15:53.77,0:15:58.53,Default,,0000,0000,0000,,So y equals log to the base e\Nof mx.
Dialogue: 0,0:15:58.53,0:16:04.34,Default,,0000,0000,0000,,Same process again.\NLet's let u equal mx.
Dialogue: 0,0:16:05.31,0:16:09.71,Default,,0000,0000,0000,,So our y equals log to\Nthe base e of u.
Dialogue: 0,0:16:11.28,0:16:15.41,Default,,0000,0000,0000,,Our du by dx equals m.
Dialogue: 0,0:16:16.92,0:16:24.65,Default,,0000,0000,0000,,dy by du, the differential of\Nlog to the base e of u is 1 over u.
Dialogue: 0,0:16:26.54,0:16:30.88,Default,,0000,0000,0000,,So our dy by dx equals
Dialogue: 0,0:16:30.88,0:16:36.01,Default,,0000,0000,0000,,dy by du multiplied by\Ndu by dx.
Dialogue: 0,0:16:37.35,0:16:43.76,Default,,0000,0000,0000,,Our dy by du is 1 over u.\NOur du by dx is m.
Dialogue: 0,0:16:45.76,0:16:52.87,Default,,0000,0000,0000,,So we have m divided by u,\Nwhich is mx and very
Dialogue: 0,0:16:52.87,0:16:57.86,Default,,0000,0000,0000,,conveniently here, our m's cancel out
Dialogue: 0,0:16:57.86,0:17:06.81,Default,,0000,0000,0000,,and we end up with 1 over x.\NSo dy by dx is equal to 1 over x.
Dialogue: 0,0:17:07.72,0:17:09.72,Default,,0000,0000,0000,,So let's put that now in our
Dialogue: 0,0:17:09.72,0:17:15.74,Default,,0000,0000,0000,,table. So log to the base E of\NMX is our function, so our
Dialogue: 0,0:17:15.74,0:17:20.61,Default,,0000,0000,0000,,derivative is one over X and of\Ncourse this is only the case
Dialogue: 0,0:17:20.61,0:17:24.74,Default,,0000,0000,0000,,where M is greater than zero.\NSince we can't take the
Dialogue: 0,0:17:24.74,0:17:26.61,Default,,0000,0000,0000,,logarithm of a negative number.
Dialogue: 0,0:17:28.20,0:17:31.44,Default,,0000,0000,0000,,Let's just go back to the\Ncalculation that we've just
Dialogue: 0,0:17:31.44,0:17:35.00,Default,,0000,0000,0000,,done, because there is another\Nway that we could have looked
Dialogue: 0,0:17:35.00,0:17:39.54,Default,,0000,0000,0000,,at. Differentiating Y equals log\Nto the base E of MX and that was
Dialogue: 0,0:17:39.54,0:17:41.48,Default,,0000,0000,0000,,by using the laws of logarithms.
Dialogue: 0,0:17:42.18,0:17:47.03,Default,,0000,0000,0000,,What we could have done was\Nsaid that Y equals log to
Dialogue: 0,0:17:47.03,0:17:52.28,Default,,0000,0000,0000,,the base E of M plus log to\Nthe base E of X.
Dialogue: 0,0:17:53.43,0:17:55.37,Default,,0000,0000,0000,,And then differentiated them.
Dialogue: 0,0:17:56.41,0:18:03.63,Default,,0000,0000,0000,,So I DY by DX is equal to\Nwhat log to the base E of M.
Dialogue: 0,0:18:04.29,0:18:09.85,Default,,0000,0000,0000,,Is just a constant. So when we\Ndifferentiate that we get 0
Dialogue: 0,0:18:09.85,0:18:14.40,Default,,0000,0000,0000,,plus. Log to the base E of X\Nwhere we know the derivative
Dialogue: 0,0:18:14.40,0:18:15.58,Default,,0000,0000,0000,,is one over X.
Dialogue: 0,0:18:16.81,0:18:22.16,Default,,0000,0000,0000,,So we end up with just the same\Nas before divided by DX is equal
Dialogue: 0,0:18:22.16,0:18:23.59,Default,,0000,0000,0000,,to one over X.
Dialogue: 0,0:18:27.77,0:18:32.44,Default,,0000,0000,0000,,Not one final one to\Nhave a look at is log to
Dialogue: 0,0:18:32.44,0:18:35.16,Default,,0000,0000,0000,,the base E of AX plus B.
Dialogue: 0,0:18:37.29,0:18:39.56,Default,,0000,0000,0000,,So let's just have\Na look at that one.
Dialogue: 0,0:18:48.93,0:18:52.27,Default,,0000,0000,0000,,So Y equals log to the base.
Dialogue: 0,0:18:52.95,0:18:55.54,Default,,0000,0000,0000,,A of a X plus B.
Dialogue: 0,0:18:57.44,0:19:00.26,Default,,0000,0000,0000,,Again, we're going to\Nuse the substitution.
Dialogue: 0,0:19:00.26,0:19:02.68,Default,,0000,0000,0000,,You equals a X plus B.
Dialogue: 0,0:19:04.12,0:19:06.94,Default,,0000,0000,0000,,So why? Why is it cool to log to
Dialogue: 0,0:19:06.94,0:19:09.31,Default,,0000,0000,0000,,the base E? Of you.
Dialogue: 0,0:19:12.28,0:19:16.27,Default,,0000,0000,0000,,We differentiate you\Nwith respect to X. Do
Dialogue: 0,0:19:16.27,0:19:18.77,Default,,0000,0000,0000,,you buy DX equals a?
Dialogue: 0,0:19:20.43,0:19:24.80,Default,,0000,0000,0000,,At the Y fi do you\Nequals one over you?
Dialogue: 0,0:19:26.49,0:19:33.31,Default,,0000,0000,0000,,Again, I DYIDX is equal\Nto DY by du multiplied
Dialogue: 0,0:19:33.31,0:19:36.72,Default,,0000,0000,0000,,by to you by DMX.
Dialogue: 0,0:19:38.07,0:19:40.63,Default,,0000,0000,0000,,The Wi-Fi do you\Nis one over you.
Dialogue: 0,0:19:42.25,0:19:46.67,Default,,0000,0000,0000,,Multiplied by do you buy DX,\Nwhich is a?
Dialogue: 0,0:19:48.03,0:19:55.39,Default,,0000,0000,0000,,So we have a divided by\NIU, which is a X plus
Dialogue: 0,0:19:55.39,0:19:55.100,Default,,0000,0000,0000,,B.
Dialogue: 0,0:19:57.67,0:19:59.76,Default,,0000,0000,0000,,So do why by DX.
Dialogue: 0,0:20:00.18,0:20:03.79,Default,,0000,0000,0000,,Sequel to a divided by a\NX plus B.
Dialogue: 0,0:20:04.81,0:20:07.43,Default,,0000,0000,0000,,So finally, let's add that\Nto our table.
Dialogue: 0,0:20:10.03,0:20:16.23,Default,,0000,0000,0000,,The derivative of locked the\Nbase E of X Plus B is a divided
Dialogue: 0,0:20:16.23,0:20:18.00,Default,,0000,0000,0000,,by 8X Plus B.