[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.84,0:00:04.09,Default,,0000,0000,0000,,So I've been requested to do\Nthe proof of the derivative of
Dialogue: 0,0:00:04.09,0:00:06.30,Default,,0000,0000,0000,,the square root of x, so I\Nthought I would do a quick
Dialogue: 0,0:00:06.30,0:00:08.30,Default,,0000,0000,0000,,video on the proof of the\Nderivative of the
Dialogue: 0,0:00:08.30,0:00:10.37,Default,,0000,0000,0000,,square root of x.
Dialogue: 0,0:00:10.37,0:00:13.68,Default,,0000,0000,0000,,So we know from the definition\Nof a derivative that the
Dialogue: 0,0:00:13.68,0:00:22.28,Default,,0000,0000,0000,,derivative of the function\Nsquare root of x, that is equal
Dialogue: 0,0:00:22.28,0:00:26.52,Default,,0000,0000,0000,,to-- let me switch colors, just\Nfor a variety-- that's equal to
Dialogue: 0,0:00:26.52,0:00:33.08,Default,,0000,0000,0000,,the limit as delta\Nx approaches 0.
Dialogue: 0,0:00:33.08,0:00:35.60,Default,,0000,0000,0000,,And you know, some people\Nsay h approaches 0,
Dialogue: 0,0:00:35.60,0:00:36.36,Default,,0000,0000,0000,,or d approaches 0.
Dialogue: 0,0:00:36.36,0:00:37.45,Default,,0000,0000,0000,,I just use delta x.
Dialogue: 0,0:00:37.45,0:00:39.45,Default,,0000,0000,0000,,So the change in x over 0.
Dialogue: 0,0:00:39.45,0:00:41.83,Default,,0000,0000,0000,,And then we say f of x\Nplus delta x, so in this
Dialogue: 0,0:00:41.83,0:00:42.91,Default,,0000,0000,0000,,case this is f of x.
Dialogue: 0,0:00:42.91,0:00:52.26,Default,,0000,0000,0000,,So it's the square root of x\Nplus delta x minus f of x,
Dialogue: 0,0:00:52.26,0:00:54.64,Default,,0000,0000,0000,,which in this case it's\Nsquare root of x.
Dialogue: 0,0:00:54.64,0:00:57.14,Default,,0000,0000,0000,,All of that over the change\Nin x, over delta x.
Dialogue: 0,0:01:00.04,0:01:02.58,Default,,0000,0000,0000,,Right now when I look at that,\Nthere's not much simplification
Dialogue: 0,0:01:02.58,0:01:04.94,Default,,0000,0000,0000,,I can do to make this come out\Nwith something meaningful.
Dialogue: 0,0:01:09.94,0:01:12.54,Default,,0000,0000,0000,,I'm going to multiply the\Nnumerator and the denominator
Dialogue: 0,0:01:12.54,0:01:13.79,Default,,0000,0000,0000,,by the conjugate of the\Nnumerator is what
Dialogue: 0,0:01:13.79,0:01:14.20,Default,,0000,0000,0000,,I mean by that.
Dialogue: 0,0:01:14.20,0:01:15.48,Default,,0000,0000,0000,,Let me rewrite it.
Dialogue: 0,0:01:15.48,0:01:19.74,Default,,0000,0000,0000,,Limit is delta x approaching\N0-- I'm just rewriting
Dialogue: 0,0:01:19.74,0:01:21.28,Default,,0000,0000,0000,,what I have here.
Dialogue: 0,0:01:21.28,0:01:26.65,Default,,0000,0000,0000,,So I said the square root\Nof x plus delta x minus
Dialogue: 0,0:01:26.65,0:01:28.61,Default,,0000,0000,0000,,square root of x.
Dialogue: 0,0:01:28.61,0:01:31.20,Default,,0000,0000,0000,,All of that over delta x.
Dialogue: 0,0:01:31.20,0:01:34.49,Default,,0000,0000,0000,,And I'm going to multiply\Nthat-- after switching colors--
Dialogue: 0,0:01:34.49,0:01:41.84,Default,,0000,0000,0000,,times square root of x plus\Ndelta x plus the square root of
Dialogue: 0,0:01:41.84,0:01:48.26,Default,,0000,0000,0000,,x, over the square root of x\Nplus delta x plus the
Dialogue: 0,0:01:48.26,0:01:49.25,Default,,0000,0000,0000,,square root of x.
Dialogue: 0,0:01:49.25,0:01:53.42,Default,,0000,0000,0000,,This is just 1, so I could of\Ncourse multiply that times-- if
Dialogue: 0,0:01:53.42,0:01:57.11,Default,,0000,0000,0000,,we assume that x and delta x\Naren't both 0, this is a
Dialogue: 0,0:01:57.11,0:01:59.09,Default,,0000,0000,0000,,defined number and\Nthis will be 1.
Dialogue: 0,0:01:59.09,0:02:00.01,Default,,0000,0000,0000,,And we can do that.
Dialogue: 0,0:02:00.01,0:02:02.13,Default,,0000,0000,0000,,This is 1/1, we're just\Nmultiplying it times this
Dialogue: 0,0:02:02.13,0:02:10.90,Default,,0000,0000,0000,,equation, and we get limit\Nas delta x approaches 0.
Dialogue: 0,0:02:10.90,0:02:13.51,Default,,0000,0000,0000,,This is a minus b\Ntimes a plus b.
Dialogue: 0,0:02:13.51,0:02:15.36,Default,,0000,0000,0000,,Let me do little aside here.
Dialogue: 0,0:02:15.36,0:02:20.88,Default,,0000,0000,0000,,Let me say a plus b times\Na minus b is equal to a
Dialogue: 0,0:02:20.88,0:02:23.15,Default,,0000,0000,0000,,squared minus b squared.
Dialogue: 0,0:02:23.15,0:02:26.60,Default,,0000,0000,0000,,So this is a plus b\Ntimes a minus b.
Dialogue: 0,0:02:26.60,0:02:29.41,Default,,0000,0000,0000,,So it's going to be\Nequal to a squared.
Dialogue: 0,0:02:29.41,0:02:32.01,Default,,0000,0000,0000,,So what's this quantity squared\Nor this quantity squared,
Dialogue: 0,0:02:32.01,0:02:33.18,Default,,0000,0000,0000,,either one, these are my a's.
Dialogue: 0,0:02:33.18,0:02:35.45,Default,,0000,0000,0000,,Well it's just going\Nto be x plus delta x.
Dialogue: 0,0:02:35.45,0:02:39.43,Default,,0000,0000,0000,,So we get x plus delta x.
Dialogue: 0,0:02:39.43,0:02:41.05,Default,,0000,0000,0000,,And then what's b squared?
Dialogue: 0,0:02:41.05,0:02:46.38,Default,,0000,0000,0000,,So minus square root of\Nx is b in this analogy.
Dialogue: 0,0:02:46.38,0:02:50.64,Default,,0000,0000,0000,,So square root of x\Nsquared is just x.
Dialogue: 0,0:02:50.64,0:02:56.76,Default,,0000,0000,0000,,And all of that over delta\Nx times square root of x
Dialogue: 0,0:02:56.76,0:03:04.21,Default,,0000,0000,0000,,plus delta x plus the\Nsquare root of x.
Dialogue: 0,0:03:04.21,0:03:05.90,Default,,0000,0000,0000,,Let's see what\Nsimplification we can do.
Dialogue: 0,0:03:05.90,0:03:08.58,Default,,0000,0000,0000,,Well we have an x and\Nthen a minus x, so those
Dialogue: 0,0:03:08.58,0:03:11.48,Default,,0000,0000,0000,,cancel out. x minus x.
Dialogue: 0,0:03:11.48,0:03:13.46,Default,,0000,0000,0000,,And then we're left in the\Nnumerator and the denominator,
Dialogue: 0,0:03:13.46,0:03:15.69,Default,,0000,0000,0000,,all we have is a delta x here\Nand a delta x here, so let's
Dialogue: 0,0:03:15.69,0:03:18.77,Default,,0000,0000,0000,,divide the numerator and the\Ndenominator by delta x.
Dialogue: 0,0:03:18.77,0:03:22.82,Default,,0000,0000,0000,,So this goes to 1,\Nthis goes to 1.
Dialogue: 0,0:03:22.82,0:03:26.35,Default,,0000,0000,0000,,And so this equals the limit--\NI'll write smaller, because I'm
Dialogue: 0,0:03:26.35,0:03:34.92,Default,,0000,0000,0000,,running out of space-- limit as\Ndelta x approaches 0 of 1 over.
Dialogue: 0,0:03:34.92,0:03:37.78,Default,,0000,0000,0000,,And of course we can only do\Nthis assuming that delta--
Dialogue: 0,0:03:37.78,0:03:40.22,Default,,0000,0000,0000,,well, we're dividing by delta\Nx to begin with, so we know
Dialogue: 0,0:03:40.22,0:03:42.42,Default,,0000,0000,0000,,it's not 0, it's just\Napproaching zero.
Dialogue: 0,0:03:42.42,0:03:50.32,Default,,0000,0000,0000,,So we get square root\Nof x plus delta x plus
Dialogue: 0,0:03:50.32,0:03:51.86,Default,,0000,0000,0000,,the square root of x.
Dialogue: 0,0:03:51.86,0:03:53.55,Default,,0000,0000,0000,,And now we can just\Ndirectly take the limit
Dialogue: 0,0:03:53.55,0:03:54.41,Default,,0000,0000,0000,,as it approaches 0.
Dialogue: 0,0:03:54.41,0:03:56.44,Default,,0000,0000,0000,,We can just set delta\Nx as equal to 0.
Dialogue: 0,0:03:56.44,0:03:58.14,Default,,0000,0000,0000,,That's what it's approaching.
Dialogue: 0,0:03:58.14,0:04:04.26,Default,,0000,0000,0000,,So then that equals one\Nover the square root of x.
Dialogue: 0,0:04:04.26,0:04:06.79,Default,,0000,0000,0000,,Right, delta x is 0, so\Nwe can ignore that.
Dialogue: 0,0:04:06.79,0:04:09.12,Default,,0000,0000,0000,,We could take the limit\Nall the way to 0.
Dialogue: 0,0:04:09.12,0:04:13.00,Default,,0000,0000,0000,,And then this is of course just\Na square root of x here plus
Dialogue: 0,0:04:13.00,0:04:17.16,Default,,0000,0000,0000,,the square root of x,\Nand that equals 1 over
Dialogue: 0,0:04:17.16,0:04:19.35,Default,,0000,0000,0000,,2 square root of x.
Dialogue: 0,0:04:19.35,0:04:24.89,Default,,0000,0000,0000,,And that equals 1/2x\Nto the negative 1/2.
Dialogue: 0,0:04:24.89,0:04:28.90,Default,,0000,0000,0000,,So we just proved that x to the\N1/2 power, the derivative of it
Dialogue: 0,0:04:28.90,0:04:35.22,Default,,0000,0000,0000,,is 1/2x to the negative 1/2,\Nand so it is consistent with
Dialogue: 0,0:04:35.22,0:04:41.70,Default,,0000,0000,0000,,the general property that the\Nderivative of-- oh I don't
Dialogue: 0,0:04:41.70,0:04:50.85,Default,,0000,0000,0000,,know-- the derivative of x to\Nthe n is equal to nx to the n
Dialogue: 0,0:04:50.85,0:04:55.15,Default,,0000,0000,0000,,minus 1, even in this case\Nwhere the n was 1/2.
Dialogue: 0,0:04:55.15,0:04:56.10,Default,,0000,0000,0000,,Well hopefully\Nthat's satisfying.
Dialogue: 0,0:04:56.10,0:04:58.96,Default,,0000,0000,0000,,I didn't prove it for all\Nfractions but this is a start.
Dialogue: 0,0:04:58.96,0:05:01.12,Default,,0000,0000,0000,,This is a common one you\Nsee, square root of x, and
Dialogue: 0,0:05:01.12,0:05:03.77,Default,,0000,0000,0000,,it's hopefully not too\Ncomplicated for proof.
Dialogue: 0,0:05:03.77,0:05:05.18,Default,,0000,0000,0000,,I will see you in\Nfuture videos.