[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.57,0:00:04.92,Default,,0000,0000,0000,,The graph of Cynex looks like
Dialogue: 0,0:00:04.92,0:00:09.58,Default,,0000,0000,0000,,this. And we can see the\Ngradient of the tangents at
Dialogue: 0,0:00:09.58,0:00:11.32,Default,,0000,0000,0000,,certain points of the graph.
Dialogue: 0,0:00:12.34,0:00:16.52,Default,,0000,0000,0000,,For instance, when X equals\Nπ by two, the gradient of
Dialogue: 0,0:00:16.52,0:00:18.04,Default,,0000,0000,0000,,the tangent is 0.
Dialogue: 0,0:00:19.14,0:00:24.19,Default,,0000,0000,0000,,And it's also 0 - Π Pi 2, and at\Nthree π by 2.
Dialogue: 0,0:00:25.31,0:00:30.27,Default,,0000,0000,0000,,At other points, the symmetry of\Nthe sine curve can help.
Dialogue: 0,0:00:31.16,0:00:35.89,Default,,0000,0000,0000,,So the gradient of the tangent\Nat X equals 0 is positive.
Dialogue: 0,0:00:36.44,0:00:40.06,Default,,0000,0000,0000,,And it must be equal to\Nthe gradient of the
Dialogue: 0,0:00:40.06,0:00:41.87,Default,,0000,0000,0000,,tangent at X equals 2π.
Dialogue: 0,0:00:42.97,0:00:47.58,Default,,0000,0000,0000,,Also, the gradient of the\Ntangent at X equals minus. Pi
Dialogue: 0,0:00:47.58,0:00:53.03,Default,,0000,0000,0000,,must be equal to the gradient of\Nthe tangent at X equals π.
Dialogue: 0,0:00:54.41,0:00:58.41,Default,,0000,0000,0000,,They must also both have the\Nsame magnitude as the gradient
Dialogue: 0,0:00:58.41,0:00:59.87,Default,,0000,0000,0000,,at X equals 0.
Dialogue: 0,0:01:00.53,0:01:01.74,Default,,0000,0000,0000,,But be negative.
Dialogue: 0,0:01:03.85,0:01:07.69,Default,,0000,0000,0000,,Let's now plot the gradients\Nof these tangents on a graph
Dialogue: 0,0:01:07.69,0:01:09.43,Default,,0000,0000,0000,,directly underneath the sine\Ngraph.
Dialogue: 0,0:01:16.23,0:01:20.49,Default,,0000,0000,0000,,Now assume we can join up the\Npoints with a smooth curve.
Dialogue: 0,0:01:21.50,0:01:25.51,Default,,0000,0000,0000,,Once we don't know the values\Nof the peaks and the troughs,
Dialogue: 0,0:01:25.51,0:01:28.51,Default,,0000,0000,0000,,what we've drawn looks\Nremarkably like the graph of
Dialogue: 0,0:01:28.51,0:01:29.52,Default,,0000,0000,0000,,a cosine function.
Dialogue: 0,0:01:31.44,0:01:36.04,Default,,0000,0000,0000,,So now we've got an idea of what\Nwe're looking for. Let's
Dialogue: 0,0:01:36.04,0:01:39.10,Default,,0000,0000,0000,,differentiate cynex from first\Nprinciples. Now there's three
Dialogue: 0,0:01:39.10,0:01:41.40,Default,,0000,0000,0000,,things that we need to know.
Dialogue: 0,0:01:42.04,0:01:48.96,Default,,0000,0000,0000,,The first is our definition of a\Nderivative Cy by DX equals the
Dialogue: 0,0:01:48.96,0:01:55.87,Default,,0000,0000,0000,,limit as Delta X approaches zero\Nof F of X Plus Delta X
Dialogue: 0,0:01:55.87,0:01:58.00,Default,,0000,0000,0000,,minus F of X.
Dialogue: 0,0:01:58.62,0:02:00.92,Default,,0000,0000,0000,,Or divided by its Delta X?
Dialogue: 0,0:02:01.86,0:02:06.94,Default,,0000,0000,0000,,The second is one of our\Ntrigonometric identity's and
Dialogue: 0,0:02:06.94,0:02:10.32,Default,,0000,0000,0000,,that sign C minus sign D.
Dialogue: 0,0:02:11.13,0:02:18.34,Default,,0000,0000,0000,,Equals twice the cause of C\Nplus 3. / 2 multiplied by
Dialogue: 0,0:02:18.34,0:02:23.75,Default,,0000,0000,0000,,the sign of C minus D\Ndivided by two.
Dialogue: 0,0:02:24.77,0:02:28.31,Default,,0000,0000,0000,,And the third is the limit.
Dialogue: 0,0:02:28.88,0:02:36.25,Default,,0000,0000,0000,,As theater approaches zero, that\Nsign Theta divided by theater
Dialogue: 0,0:02:36.25,0:02:42.13,Default,,0000,0000,0000,,equals 1. Now we can see\Nthis from a table of values.
Dialogue: 0,0:02:42.75,0:02:46.15,Default,,0000,0000,0000,,So if we have a look at theater.
Dialogue: 0,0:02:46.66,0:02:48.73,Default,,0000,0000,0000,,When it's in radians.
Dialogue: 0,0:02:50.06,0:02:51.54,Default,,0000,0000,0000,,And then calculate.
Dialogue: 0,0:02:52.15,0:02:58.64,Default,,0000,0000,0000,,Sign theater And then\Nsign theater divided by theater.
Dialogue: 0,0:02:59.57,0:03:02.38,Default,,0000,0000,0000,,We can have a look and see
Dialogue: 0,0:03:02.38,0:03:07.02,Default,,0000,0000,0000,,what's happening. So our theater\Nin radians. Let's look at one
Dialogue: 0,0:03:07.02,0:03:08.07,Default,,0000,0000,0000,,first of all.
Dialogue: 0,0:03:09.01,0:03:15.18,Default,,0000,0000,0000,,The sign of\None is 0.84147.
Dialogue: 0,0:03:16.04,0:03:21.55,Default,,0000,0000,0000,,So if we calculate sine\NTheta divided by Sita, we
Dialogue: 0,0:03:21.55,0:03:24.30,Default,,0000,0000,0000,,just get the same 0.84147.
Dialogue: 0,0:03:25.96,0:03:29.39,Default,,0000,0000,0000,,Now let's make theater smaller.\NLet's go to 0.1.
Dialogue: 0,0:03:29.96,0:03:37.29,Default,,0000,0000,0000,,The sign of 0.1\Nis 0.0998 three and
Dialogue: 0,0:03:37.29,0:03:39.12,Default,,0000,0000,0000,,so on.
Dialogue: 0,0:03:40.19,0:03:42.71,Default,,0000,0000,0000,,Just it continued\Non there, sorry.
Dialogue: 0,0:03:43.72,0:03:50.54,Default,,0000,0000,0000,,OK, if we do sign\Ntheater divided by Theta then
Dialogue: 0,0:03:50.54,0:03:53.95,Default,,0000,0000,0000,,we get 0.99833 and again.
Dialogue: 0,0:03:54.51,0:03:55.50,Default,,0000,0000,0000,,Continues on.
Dialogue: 0,0:03:57.24,0:04:01.65,Default,,0000,0000,0000,,Let's make seat even\Nsmaller now at 0.01.
Dialogue: 0,0:04:02.86,0:04:09.76,Default,,0000,0000,0000,,The sign of Theta\Nis going to be
Dialogue: 0,0:04:09.76,0:04:13.20,Default,,0000,0000,0000,,0.00999, and so on.
Dialogue: 0,0:04:14.82,0:04:22.40,Default,,0000,0000,0000,,So sine Theta divided\Nby Sita equals 0.9998
Dialogue: 0,0:04:22.40,0:04:26.20,Default,,0000,0000,0000,,three and so on.
Dialogue: 0,0:04:27.15,0:04:32.06,Default,,0000,0000,0000,,And you can see that as Theta is\Ngetting smaller and smaller and
Dialogue: 0,0:04:32.06,0:04:33.20,Default,,0000,0000,0000,,tending to 0.
Dialogue: 0,0:04:33.72,0:04:37.82,Default,,0000,0000,0000,,Then sign Theta divided by\Ntheater is tending to one.
Dialogue: 0,0:04:39.01,0:04:42.83,Default,,0000,0000,0000,,OK, let's carry on with
Dialogue: 0,0:04:42.83,0:04:50.07,Default,,0000,0000,0000,,the calculation. So we have\Nour Y equals sign
Dialogue: 0,0:04:50.07,0:04:56.52,Default,,0000,0000,0000,,X. And let's start by just\Nlooking at the top part of our
Dialogue: 0,0:04:56.52,0:05:02.04,Default,,0000,0000,0000,,formula. Our function of X Plus\NDelta X minus a function of X.
Dialogue: 0,0:05:03.58,0:05:09.90,Default,,0000,0000,0000,,Sign X is our function of X, so\Nour function of X Plus Delta X
Dialogue: 0,0:05:09.90,0:05:13.26,Default,,0000,0000,0000,,is the sign of X Plus Delta X.
Dialogue: 0,0:05:13.93,0:05:16.75,Default,,0000,0000,0000,,Minus R function of X, so minus
Dialogue: 0,0:05:16.75,0:05:22.01,Default,,0000,0000,0000,,sign X. And this is where we\Nhave our sign. See take away as
Dialogue: 0,0:05:22.01,0:05:28.52,Default,,0000,0000,0000,,signed D. So we can rewrite\Nthis as twice the cause.
Dialogue: 0,0:05:29.03,0:05:36.53,Default,,0000,0000,0000,,Of C Plus D divided by\Ntwo. So it's X Plus Delta
Dialogue: 0,0:05:36.53,0:05:40.28,Default,,0000,0000,0000,,X Plus X divided by two.
Dialogue: 0,0:05:40.91,0:05:47.34,Default,,0000,0000,0000,,Multiplied by the sign of C\Nminus D divided by two.
Dialogue: 0,0:05:48.06,0:05:51.18,Default,,0000,0000,0000,,So it's X Plus Delta X.
Dialogue: 0,0:05:51.86,0:05:54.87,Default,,0000,0000,0000,,Minus X divided by two.
Dialogue: 0,0:05:55.95,0:06:03.19,Default,,0000,0000,0000,,So this equals twice the cause.\NX Plus X is 2X Plus Delta
Dialogue: 0,0:06:03.19,0:06:10.99,Default,,0000,0000,0000,,X or divided by 2 multiplied by\Nthe sign of X minus X is
Dialogue: 0,0:06:10.99,0:06:16.13,Default,,0000,0000,0000,,0. So we just have Delta X\Ndivided by two.
Dialogue: 0,0:06:16.91,0:06:21.26,Default,,0000,0000,0000,,I'm just going to tidy this up a\Nlittle bit further, so we've got
Dialogue: 0,0:06:21.26,0:06:22.51,Default,,0000,0000,0000,,two times the cause.
Dialogue: 0,0:06:23.50,0:06:30.67,Default,,0000,0000,0000,,2 / 2 gives us just RX plus\NDelta X over 2 multiplied by the
Dialogue: 0,0:06:30.67,0:06:33.54,Default,,0000,0000,0000,,sign of Delta X over 2.
Dialogue: 0,0:06:34.28,0:06:39.69,Default,,0000,0000,0000,,So now if we go back to\Ndivide by DX.
Dialogue: 0,0:06:40.19,0:06:45.64,Default,,0000,0000,0000,,Equals the limit as Delta\NX approaches 0.
Dialogue: 0,0:06:46.63,0:06:50.15,Default,,0000,0000,0000,,Of F of X Plus Delta X\Nminus F of X.
Dialogue: 0,0:06:51.20,0:06:57.99,Default,,0000,0000,0000,,Which is this? So it's two calls\NX plus Delta X over 2 multiplied
Dialogue: 0,0:06:57.99,0:07:01.87,Default,,0000,0000,0000,,by the sign of Delta X over 2.
Dialogue: 0,0:07:02.54,0:07:05.41,Default,,0000,0000,0000,,All divided by Delta X.
Dialogue: 0,0:07:06.93,0:07:11.88,Default,,0000,0000,0000,,Now multiplying the numerator,\Nthe top part of the fraction by
Dialogue: 0,0:07:11.88,0:07:16.38,Default,,0000,0000,0000,,two is exactly the same as\Ndividing the denominator. The
Dialogue: 0,0:07:16.38,0:07:18.18,Default,,0000,0000,0000,,bottom part by two.
Dialogue: 0,0:07:18.69,0:07:20.75,Default,,0000,0000,0000,,So I'm going to rewrite this.
Dialogue: 0,0:07:21.41,0:07:24.34,Default,,0000,0000,0000,,Limit of Delta X tends to 0.
Dialogue: 0,0:07:25.13,0:07:28.38,Default,,0000,0000,0000,,Taking this too from here.
Dialogue: 0,0:07:28.43,0:07:36.02,Default,,0000,0000,0000,,And.\NPutting it as a division
Dialogue: 0,0:07:36.02,0:07:38.32,Default,,0000,0000,0000,,in the denominator.
Dialogue: 0,0:07:39.53,0:07:43.60,Default,,0000,0000,0000,,And I've done that so that you\Ncan see here. We've got the sign
Dialogue: 0,0:07:43.60,0:07:45.06,Default,,0000,0000,0000,,of Delta X over 2.
Dialogue: 0,0:07:45.57,0:07:50.36,Default,,0000,0000,0000,,Divided by Delta X over 2\Nand that was where I third
Dialogue: 0,0:07:50.36,0:07:54.35,Default,,0000,0000,0000,,fact came in that when we\Nhad signed theater divided
Dialogue: 0,0:07:54.35,0:07:58.74,Default,,0000,0000,0000,,by Sita when we took the\Nlimit of Delta extending to
Dialogue: 0,0:07:58.74,0:08:01.13,Default,,0000,0000,0000,,0, this will tend to one.
Dialogue: 0,0:08:02.66,0:08:06.91,Default,,0000,0000,0000,,So let's actually take the\Nlimit. Now is Delta X approaches
Dialogue: 0,0:08:06.91,0:08:11.54,Default,,0000,0000,0000,,0 sign Delta X over 2 divided by\NDelta X over 2?
Dialogue: 0,0:08:12.15,0:08:18.20,Default,,0000,0000,0000,,Is one and Delta Rex over 2. As\NDelta X approaches, zero will be
Dialogue: 0,0:08:18.20,0:08:23.42,Default,,0000,0000,0000,,0. So we're just left\Nwith our derivative of
Dialogue: 0,0:08:23.42,0:08:25.96,Default,,0000,0000,0000,,sine X thing Cos X.
Dialogue: 0,0:08:27.27,0:08:32.06,Default,,0000,0000,0000,,Let's have a look at the\Nderivative now from first
Dialogue: 0,0:08:32.06,0:08:33.98,Default,,0000,0000,0000,,principles of Cos X.
Dialogue: 0,0:08:35.15,0:08:37.69,Default,,0000,0000,0000,,The graph of Cos X looks\Nlike this.
Dialogue: 0,0:08:39.75,0:08:43.22,Default,,0000,0000,0000,,Again, we can draw the tangents\Nat certain points.
Dialogue: 0,0:08:45.52,0:08:49.97,Default,,0000,0000,0000,,And we can plot their values on\Na graph directly underneath the
Dialogue: 0,0:08:49.97,0:08:55.12,Default,,0000,0000,0000,,cosine graph. This time if we\Njoin the points with a smooth
Dialogue: 0,0:08:55.12,0:08:58.97,Default,,0000,0000,0000,,curve, we get a graph that looks\Nlike minus sign X.
Dialogue: 0,0:08:59.70,0:09:02.14,Default,,0000,0000,0000,,Now before we\Ndifferentiate cause X
Dialogue: 0,0:09:02.14,0:09:05.79,Default,,0000,0000,0000,,from first principles,\Nlet's just have a look at
Dialogue: 0,0:09:05.79,0:09:10.26,Default,,0000,0000,0000,,the graph of the sign of\NX plus Π by 2.
Dialogue: 0,0:09:11.96,0:09:17.13,Default,,0000,0000,0000,,The graph of sign of X plus Π by\N2 looks like this.
Dialogue: 0,0:09:17.78,0:09:21.76,Default,,0000,0000,0000,,It's just the graph of cynex\Nwith everything happening
Dialogue: 0,0:09:21.76,0:09:23.53,Default,,0000,0000,0000,,pie by two earlier.
Dialogue: 0,0:09:24.65,0:09:29.46,Default,,0000,0000,0000,,It's as though we've shifted the\Nsign curve back along the X axis
Dialogue: 0,0:09:29.46,0:09:30.94,Default,,0000,0000,0000,,by Π by 2.
Dialogue: 0,0:09:33.13,0:09:35.48,Default,,0000,0000,0000,,And that just gives us Cossacks.
Dialogue: 0,0:09:37.12,0:09:39.52,Default,,0000,0000,0000,,The two functions are the same.
Dialogue: 0,0:09:41.59,0:09:46.12,Default,,0000,0000,0000,,Now shifting the sign curve back\Nalong the Axis doesn't affect
Dialogue: 0,0:09:46.12,0:09:51.48,Default,,0000,0000,0000,,the shape of the curve, so it\Ndoesn't affect the shape of its
Dialogue: 0,0:09:51.48,0:09:54.77,Default,,0000,0000,0000,,gradient function. Only the\Nposition of the gradient
Dialogue: 0,0:09:54.77,0:09:56.83,Default,,0000,0000,0000,,function on the X axis.
Dialogue: 0,0:10:01.88,0:10:06.94,Default,,0000,0000,0000,,So the derivative of sign of X\Nplus Π by two is the cause of X
Dialogue: 0,0:10:06.94,0:10:08.20,Default,,0000,0000,0000,,plus Π by 2.
Dialogue: 0,0:10:09.63,0:10:14.94,Default,,0000,0000,0000,,But the graph of cause of X plus\NΠ by two is identical to the
Dialogue: 0,0:10:14.94,0:10:16.71,Default,,0000,0000,0000,,graph of minus sign X.
Dialogue: 0,0:10:17.75,0:10:20.48,Default,,0000,0000,0000,,So the derivative of Cos X.
Dialogue: 0,0:10:21.05,0:10:22.71,Default,,0000,0000,0000,,Is minus sign X?
Dialogue: 0,0:10:24.51,0:10:27.23,Default,,0000,0000,0000,,Let's have a look at the
Dialogue: 0,0:10:27.23,0:10:31.05,Default,,0000,0000,0000,,calculation now. Y equals Cos
Dialogue: 0,0:10:31.05,0:10:34.74,Default,,0000,0000,0000,,X. And we need to know three
Dialogue: 0,0:10:34.74,0:10:41.42,Default,,0000,0000,0000,,things again. To help us do the\Ncalculation, the first is our
Dialogue: 0,0:10:41.42,0:10:47.86,Default,,0000,0000,0000,,definition of our derivative DY\Nby DX equals the limit as Delta
Dialogue: 0,0:10:47.86,0:10:49.48,Default,,0000,0000,0000,,X approaches 0.
Dialogue: 0,0:10:50.48,0:10:55.40,Default,,0000,0000,0000,,About function of X Plus\NDelta X minus a function
Dialogue: 0,0:10:55.40,0:10:58.35,Default,,0000,0000,0000,,of X divided by Delta X.
Dialogue: 0,0:10:59.57,0:11:03.89,Default,,0000,0000,0000,,The second thing we need to know\Nis one of the trigonometric
Dialogue: 0,0:11:03.89,0:11:10.64,Default,,0000,0000,0000,,identity's. And that cause see\Nminus calls D equals minus two
Dialogue: 0,0:11:10.64,0:11:17.71,Default,,0000,0000,0000,,times the sign of C Plus D\Ndivided by 2 multiplied by the
Dialogue: 0,0:11:17.71,0:11:21.52,Default,,0000,0000,0000,,sign of C minus D divided by
Dialogue: 0,0:11:21.52,0:11:28.64,Default,,0000,0000,0000,,two. And the third thing\Nis the limit. As theater
Dialogue: 0,0:11:28.64,0:11:35.82,Default,,0000,0000,0000,,approaches 0. Sign Theta divided\Nby feature is equal to 1.
Dialogue: 0,0:11:37.37,0:11:39.38,Default,,0000,0000,0000,,So let's have a look.
Dialogue: 0,0:11:40.16,0:11:45.84,Default,,0000,0000,0000,,Add putting our function into\Nthis top part against RF of X
Dialogue: 0,0:11:45.84,0:11:49.15,Default,,0000,0000,0000,,Plus Delta X minus a function of
Dialogue: 0,0:11:49.15,0:11:52.18,Default,,0000,0000,0000,,X. Is equal to.
Dialogue: 0,0:11:53.42,0:11:57.14,Default,,0000,0000,0000,,I cause of X Plus
Dialogue: 0,0:11:57.14,0:12:02.92,Default,,0000,0000,0000,,Delta X. Minus a function of\NX which is called sex.
Dialogue: 0,0:12:03.76,0:12:08.21,Default,,0000,0000,0000,,So here we have our cause of C\Nminus icons of D.
Dialogue: 0,0:12:09.31,0:12:12.11,Default,,0000,0000,0000,,So let's do this\Nsubstitution now.
Dialogue: 0,0:12:13.14,0:12:20.48,Default,,0000,0000,0000,,So it's minus twice the sign of\NC Plus D over 2, so that's X
Dialogue: 0,0:12:20.48,0:12:21.94,Default,,0000,0000,0000,,Plus Delta X.
Dialogue: 0,0:12:22.62,0:12:25.70,Default,,0000,0000,0000,,Plus, X all divided by two.
Dialogue: 0,0:12:28.88,0:12:35.55,Default,,0000,0000,0000,,Multiplied by the sign of C\Nminus T over 2. So that's X
Dialogue: 0,0:12:35.55,0:12:37.09,Default,,0000,0000,0000,,Plus Delta X.
Dialogue: 0,0:12:37.59,0:12:40.70,Default,,0000,0000,0000,,Minus X divided by two.
Dialogue: 0,0:12:41.70,0:12:48.53,Default,,0000,0000,0000,,Which equals minus two times\Nthe sign of X plus
Dialogue: 0,0:12:48.53,0:12:54.68,Default,,0000,0000,0000,,X is 2X plus Delta\NX divided by two.
Dialogue: 0,0:12:54.68,0:13:01.44,Default,,0000,0000,0000,,Multiplied by the sign of X\Nminus X is 0, so we're just left
Dialogue: 0,0:13:01.44,0:13:04.34,Default,,0000,0000,0000,,with Delta X divided by two.
Dialogue: 0,0:13:04.34,0:13:07.33,Default,,0000,0000,0000,,And again, as we did before.
Dialogue: 0,0:13:07.33,0:13:14.05,Default,,0000,0000,0000,,We write this is minus 2 sign 2X\Ndivided by two leaves us with X
Dialogue: 0,0:13:14.05,0:13:19.43,Default,,0000,0000,0000,,Plus our Delta X over 2\Nmultiplied by our sign of Delta
Dialogue: 0,0:13:19.43,0:13:20.77,Default,,0000,0000,0000,,X over 2.
Dialogue: 0,0:13:21.89,0:13:24.49,Default,,0000,0000,0000,,Now we can turn the page and put
Dialogue: 0,0:13:24.49,0:13:31.41,Default,,0000,0000,0000,,this into our.\NFormula.
Dialogue: 0,0:13:32.44,0:13:39.81,Default,,0000,0000,0000,,So that we have the why\Nby DX equals the limit a
Dialogue: 0,0:13:39.81,0:13:47.18,Default,,0000,0000,0000,,Delta X approaches zero of minus\N2 signs of X Plus Delta
Dialogue: 0,0:13:47.18,0:13:49.02,Default,,0000,0000,0000,,X over 2.
Dialogue: 0,0:13:49.76,0:13:52.95,Default,,0000,0000,0000,,Multiplied by sign Delta X over
Dialogue: 0,0:13:52.95,0:13:56.52,Default,,0000,0000,0000,,2. All divided by Delta
Dialogue: 0,0:13:56.52,0:14:02.75,Default,,0000,0000,0000,,X. I'm not equals, and again\Nwe're going to do this change
Dialogue: 0,0:14:02.75,0:14:07.50,Default,,0000,0000,0000,,where instead of multiplying the\Nnumerator by two, I'm going to
Dialogue: 0,0:14:07.50,0:14:13.12,Default,,0000,0000,0000,,divide the denominator by two.\NSo we have the limit as Delta X
Dialogue: 0,0:14:13.12,0:14:19.13,Default,,0000,0000,0000,,approaches 0. Of minus\Nthe sign of X Plus Delta
Dialogue: 0,0:14:19.13,0:14:24.94,Default,,0000,0000,0000,,X divided by 2 multiplied\Nby the sign of Delta X
Dialogue: 0,0:14:24.94,0:14:29.17,Default,,0000,0000,0000,,divided by two all\Ndivided by Delta X
Dialogue: 0,0:14:29.17,0:14:30.75,Default,,0000,0000,0000,,divided by two.
Dialogue: 0,0:14:32.01,0:14:36.61,Default,,0000,0000,0000,,Now as we take the limit as\NDelta X approaches 0. Again,
Dialogue: 0,0:14:36.61,0:14:42.35,Default,,0000,0000,0000,,here we've got the sign of Delta\NX over 2 divided by Delta X over
Dialogue: 0,0:14:42.35,0:14:47.33,Default,,0000,0000,0000,,2, and we know that as Delta X\Napproaches zero, then this tends
Dialogue: 0,0:14:47.33,0:14:53.37,Default,,0000,0000,0000,,to one. And then as Delta X\Ndivided by two, here tends to 0.
Dialogue: 0,0:14:54.09,0:14:59.49,Default,,0000,0000,0000,,Then our derivative will be\Nminus the sign of X.
Dialogue: 0,0:15:00.94,0:15:06.31,Default,,0000,0000,0000,,So our derivative of Cos X is\Nminus sign X.