WEBVTT 00:00:00.840 --> 00:00:04.090 So I've been requested to do the proof of the derivative of 00:00:04.090 --> 00:00:06.300 the square root of x, so I thought I would do a quick 00:00:06.300 --> 00:00:08.300 video on the proof of the derivative of the 00:00:08.300 --> 00:00:10.370 square root of x. 00:00:10.370 --> 00:00:13.680 So we know from the definition of a derivative that the 00:00:13.680 --> 00:00:22.280 derivative of the function square root of x, that is equal 00:00:22.280 --> 00:00:26.520 to-- let me switch colors, just for a variety-- that's equal to 00:00:26.520 --> 00:00:33.080 the limit as delta x approaches 0. 00:00:33.080 --> 00:00:35.595 And you know, some people say h approaches 0, 00:00:35.595 --> 00:00:36.360 or d approaches 0. 00:00:36.360 --> 00:00:37.450 I just use delta x. 00:00:37.450 --> 00:00:39.450 So the change in x over 0. 00:00:39.450 --> 00:00:41.830 And then we say f of x plus delta x, so in this 00:00:41.830 --> 00:00:42.910 case this is f of x. 00:00:42.910 --> 00:00:52.260 So it's the square root of x plus delta x minus f of x, 00:00:52.260 --> 00:00:54.640 which in this case it's square root of x. 00:00:54.640 --> 00:00:57.140 All of that over the change in x, over delta x. 00:01:00.040 --> 00:01:02.580 Right now when I look at that, there's not much simplification 00:01:02.580 --> 00:01:04.945 I can do to make this come out with something meaningful. 00:01:09.940 --> 00:01:12.540 I'm going to multiply the numerator and the denominator 00:01:12.540 --> 00:01:13.790 by the conjugate of the numerator is what 00:01:13.790 --> 00:01:14.200 I mean by that. 00:01:14.200 --> 00:01:15.480 Let me rewrite it. 00:01:15.480 --> 00:01:19.740 Limit is delta x approaching 0-- I'm just rewriting 00:01:19.740 --> 00:01:21.280 what I have here. 00:01:21.280 --> 00:01:26.650 So I said the square root of x plus delta x minus 00:01:26.650 --> 00:01:28.610 square root of x. 00:01:28.610 --> 00:01:31.200 All of that over delta x. 00:01:31.200 --> 00:01:34.490 And I'm going to multiply that-- after switching colors-- 00:01:34.490 --> 00:01:41.840 times square root of x plus delta x plus the square root of 00:01:41.840 --> 00:01:48.260 x, over the square root of x plus delta x plus the 00:01:48.260 --> 00:01:49.250 square root of x. 00:01:49.250 --> 00:01:53.420 This is just 1, so I could of course multiply that times-- if 00:01:53.420 --> 00:01:57.110 we assume that x and delta x aren't both 0, this is a 00:01:57.110 --> 00:01:59.090 defined number and this will be 1. 00:01:59.090 --> 00:02:00.010 And we can do that. 00:02:00.010 --> 00:02:02.130 This is 1/1, we're just multiplying it times this 00:02:02.130 --> 00:02:10.900 equation, and we get limit as delta x approaches 0. 00:02:10.900 --> 00:02:13.510 This is a minus b times a plus b. 00:02:13.510 --> 00:02:15.360 Let me do little aside here. 00:02:15.360 --> 00:02:20.880 Let me say a plus b times a minus b is equal to a 00:02:20.880 --> 00:02:23.150 squared minus b squared. 00:02:23.150 --> 00:02:26.600 So this is a plus b times a minus b. 00:02:26.600 --> 00:02:29.410 So it's going to be equal to a squared. 00:02:29.410 --> 00:02:32.010 So what's this quantity squared or this quantity squared, 00:02:32.010 --> 00:02:33.180 either one, these are my a's. 00:02:33.180 --> 00:02:35.450 Well it's just going to be x plus delta x. 00:02:35.450 --> 00:02:39.430 So we get x plus delta x. 00:02:39.430 --> 00:02:41.050 And then what's b squared? 00:02:41.050 --> 00:02:46.380 So minus square root of x is b in this analogy. 00:02:46.380 --> 00:02:50.640 So square root of x squared is just x. 00:02:50.640 --> 00:02:56.760 And all of that over delta x times square root of x 00:02:56.760 --> 00:03:04.210 plus delta x plus the square root of x. 00:03:04.210 --> 00:03:05.900 Let's see what simplification we can do. 00:03:05.900 --> 00:03:08.580 Well we have an x and then a minus x, so those 00:03:08.580 --> 00:03:11.480 cancel out. x minus x. 00:03:11.480 --> 00:03:13.460 And then we're left in the numerator and the denominator, 00:03:13.460 --> 00:03:15.690 all we have is a delta x here and a delta x here, so let's 00:03:15.690 --> 00:03:18.770 divide the numerator and the denominator by delta x. 00:03:18.770 --> 00:03:22.822 So this goes to 1, this goes to 1. 00:03:22.822 --> 00:03:26.350 And so this equals the limit-- I'll write smaller, because I'm 00:03:26.350 --> 00:03:34.920 running out of space-- limit as delta x approaches 0 of 1 over. 00:03:34.920 --> 00:03:37.780 And of course we can only do this assuming that delta-- 00:03:37.780 --> 00:03:40.220 well, we're dividing by delta x to begin with, so we know 00:03:40.220 --> 00:03:42.420 it's not 0, it's just approaching zero. 00:03:42.420 --> 00:03:50.320 So we get square root of x plus delta x plus 00:03:50.320 --> 00:03:51.860 the square root of x. 00:03:51.860 --> 00:03:53.550 And now we can just directly take the limit 00:03:53.550 --> 00:03:54.410 as it approaches 0. 00:03:54.410 --> 00:03:56.440 We can just set delta x as equal to 0. 00:03:56.440 --> 00:03:58.140 That's what it's approaching. 00:03:58.140 --> 00:04:04.260 So then that equals one over the square root of x. 00:04:04.260 --> 00:04:06.790 Right, delta x is 0, so we can ignore that. 00:04:06.790 --> 00:04:09.120 We could take the limit all the way to 0. 00:04:09.120 --> 00:04:13.000 And then this is of course just a square root of x here plus 00:04:13.000 --> 00:04:17.160 the square root of x, and that equals 1 over 00:04:17.160 --> 00:04:19.350 2 square root of x. 00:04:19.350 --> 00:04:24.890 And that equals 1/2x to the negative 1/2. 00:04:24.890 --> 00:04:28.900 So we just proved that x to the 1/2 power, the derivative of it 00:04:28.900 --> 00:04:35.220 is 1/2x to the negative 1/2, and so it is consistent with 00:04:35.220 --> 00:04:41.700 the general property that the derivative of-- oh I don't 00:04:41.700 --> 00:04:50.850 know-- the derivative of x to the n is equal to nx to the n 00:04:50.850 --> 00:04:55.150 minus 1, even in this case where the n was 1/2. 00:04:55.150 --> 00:04:56.100 Well hopefully that's satisfying. 00:04:56.100 --> 00:04:58.960 I didn't prove it for all fractions but this is a start. 00:04:58.960 --> 00:05:01.120 This is a common one you see, square root of x, and 00:05:01.120 --> 00:05:03.770 it's hopefully not too complicated for proof. 00:05:03.770 --> 00:05:05.180 I will see you in future videos.