WEBVTT 00:00:00.000 --> 00:00:01.984 align:middle line:90% [SQUEAKING] 00:00:01.984 --> 00:00:04.464 align:middle line:90% [RUSTLING] 00:00:04.464 --> 00:00:05.952 align:middle line:90% [CLICKING] 00:00:05.952 --> 00:00:09.740 align:middle line:90% 00:00:09.740 --> 00:00:12.540 align:middle line:90% PROFESSOR: So next, we will-- 00:00:12.540 --> 00:00:15.660 align:middle line:84% so we're switching away from metric spaces, 00:00:15.660 --> 00:00:18.660 align:middle line:84% and we talk about differentiation. 00:00:18.660 --> 00:00:30.230 align:middle line:84% So if we have a function on R, say, and then 00:00:30.230 --> 00:00:51.860 align:middle line:84% we say that x that f is differentiable at x0, 00:00:51.860 --> 00:01:11.130 align:middle line:84% if the limit here f of dx minus f of x0, divided by x minus x0, 00:01:11.130 --> 00:01:16.440 align:middle line:84% if this limit here exists as x goes to x0. 00:01:16.440 --> 00:01:22.420 align:middle line:90% 00:01:22.420 --> 00:01:25.310 align:middle line:84% This quotient here is usually called the difference quotient. 00:01:25.310 --> 00:01:32.340 align:middle line:90% 00:01:32.340 --> 00:01:35.460 align:middle line:84% And it's always, of course, when you form this difference 00:01:35.460 --> 00:01:38.820 align:middle line:84% quotient, it's always assumed, obviously, 00:01:38.820 --> 00:01:43.770 align:middle line:84% that x is not equal to x0 so that the denominator makes 00:01:43.770 --> 00:01:44.770 align:middle line:90% sense. 00:01:44.770 --> 00:01:49.440 align:middle line:84% So if this limit here, as x goes to x0, if this exists, 00:01:49.440 --> 00:01:53.700 align:middle line:84% then we say that the function f is differentiable at x0. 00:01:53.700 --> 00:01:59.170 align:middle line:84% And the limit is called the derivative of f at x0. 00:01:59.170 --> 00:02:03.070 align:middle line:90% And this is how it's denoted. 00:02:03.070 --> 00:02:12.880 align:middle line:84% So now the simplest possible example of this 00:02:12.880 --> 00:02:17.410 align:middle line:84% would be that suppose that you're looking 00:02:17.410 --> 00:02:19.940 align:middle line:90% at a function that is constant. 00:02:19.940 --> 00:02:32.620 align:middle line:84% So f is equal to f for some constant C. 00:02:32.620 --> 00:02:35.720 align:middle line:84% Then if you're looking at this difference quotient, 00:02:35.720 --> 00:02:45.160 align:middle line:84% so f of dx minus f of x0 x minus x0, well, the value at any point 00:02:45.160 --> 00:02:46.010 align:middle line:90% is C-- 00:02:46.010 --> 00:02:52.510 align:middle line:84% so this is C minus C divided by x minus y, x0-- 00:02:52.510 --> 00:02:56.370 align:middle line:84% and so the denominator is 0, so this whole thing is 0. 00:02:56.370 --> 00:02:58.350 align:middle line:84% And so of course, the limit-- this 00:02:58.350 --> 00:03:02.350 align:middle line:84% is 0 for each x and also each x0. 00:03:02.350 --> 00:03:05.470 align:middle line:84% So in particular, of course, the limit here exists. 00:03:05.470 --> 00:03:09.300 align:middle line:84% So the limit here of the difference quotient 00:03:09.300 --> 00:03:20.340 align:middle line:84% exists, and it's equal to 0 because it's 00:03:20.340 --> 00:03:24.900 align:middle line:90% 0 for all x for a given x0. 00:03:24.900 --> 00:03:28.600 align:middle line:84% So that would be the simplest possible example. 00:03:28.600 --> 00:03:32.250 align:middle line:90% The second simplest example. 00:03:32.250 --> 00:03:35.430 align:middle line:84% Second most simple example would be that you're 00:03:35.430 --> 00:03:37.060 align:middle line:90% looking at a linear function. 00:03:37.060 --> 00:03:42.690 align:middle line:84% So let's say that f of x is equal to x. 00:03:42.690 --> 00:03:47.765 align:middle line:84% In this case, the difference quotient here-- 00:03:47.765 --> 00:03:52.830 align:middle line:90% 00:03:52.830 --> 00:03:55.810 align:middle line:90% well, this here is x. 00:03:55.810 --> 00:03:57.343 align:middle line:90% This here is x0. 00:03:57.343 --> 00:04:00.490 align:middle line:90% 00:04:00.490 --> 00:04:03.800 align:middle line:84% So this here is for each x and x0. 00:04:03.800 --> 00:04:05.030 align:middle line:90% This is 1. 00:04:05.030 --> 00:04:08.025 align:middle line:84% So the limit of course, in this case, also exists. 00:04:08.025 --> 00:04:13.300 align:middle line:90% 00:04:13.300 --> 00:04:17.595 align:middle line:84% The limit here exists, and it's equal to 1. 00:04:17.595 --> 00:04:22.780 align:middle line:90% 00:04:22.780 --> 00:04:35.140 align:middle line:84% And so in this case, f is differentiable at all points. 00:04:35.140 --> 00:04:39.500 align:middle line:84% And the derivative at all points is equal to 0. 00:04:39.500 --> 00:04:43.280 align:middle line:84% In this case, f here is also differentiable. 00:04:43.280 --> 00:04:48.400 align:middle line:90% 00:04:48.400 --> 00:04:52.660 align:middle line:84% And the derivative at all points is equal to 1. 00:04:52.660 --> 00:04:55.640 align:middle line:90% 00:04:55.640 --> 00:04:59.690 align:middle line:84% So now, in general, how do we compute the-- so 00:04:59.690 --> 00:05:02.335 align:middle line:84% how do we compute the derivative of a function? 00:05:02.335 --> 00:05:10.490 align:middle line:90% 00:05:10.490 --> 00:05:17.580 align:middle line:84% How do we compute the derivative? 00:05:17.580 --> 00:05:26.390 align:middle line:90% 00:05:26.390 --> 00:05:30.530 align:middle line:84% And for that, there are some basic laws-- 00:05:30.530 --> 00:05:31.830 align:middle line:90% how to compute it. 00:05:31.830 --> 00:05:33.280 align:middle line:90% So there's these laws-- 00:05:33.280 --> 00:05:38.270 align:middle line:90% 00:05:38.270 --> 00:05:45.860 align:middle line:84% the sum rule, what's usually called 00:05:45.860 --> 00:06:02.040 align:middle line:84% the Leibniz rule, the quotient rule, and then 00:06:02.040 --> 00:06:03.395 align:middle line:90% also the chain rule. 00:06:03.395 --> 00:06:09.810 align:middle line:90% 00:06:09.810 --> 00:06:17.880 align:middle line:84% Now let me look at the first three rules first. 00:06:17.880 --> 00:06:26.640 align:middle line:84% So I have two functions, f and g. 00:06:26.640 --> 00:06:29.220 align:middle line:84% They are defined, say, on R, but it 00:06:29.220 --> 00:06:35.130 align:middle line:84% could have been a subset of R. And I'm assuming 00:06:35.130 --> 00:06:39.975 align:middle line:90% that they are both and both-- 00:06:39.975 --> 00:06:42.760 align:middle line:90% 00:06:42.760 --> 00:06:44.320 align:middle line:90% actually, we'll need one. 00:06:44.320 --> 00:07:02.410 align:middle line:84% So both f and g are differentiable at x0. 00:07:02.410 --> 00:07:04.670 align:middle line:90% Then we have the following. 00:07:04.670 --> 00:07:05.985 align:middle line:90% So we have the following lemma. 00:07:05.985 --> 00:07:26.020 align:middle line:90% 00:07:26.020 --> 00:07:28.250 align:middle line:90% One is we have that-- 00:07:28.250 --> 00:07:30.600 align:middle line:84% so 1, and this is really the sum rule. 00:07:30.600 --> 00:07:36.160 align:middle line:90% 00:07:36.160 --> 00:07:39.520 align:middle line:84% We have that the sum here f plus g-- 00:07:39.520 --> 00:07:42.400 align:middle line:84% so the sum of these two functions, this 00:07:42.400 --> 00:07:45.710 align:middle line:84% is the function that is just in x. 00:07:45.710 --> 00:07:50.600 align:middle line:84% It's just the sum of f of x and g of x We have 00:07:50.600 --> 00:08:04.670 align:middle line:84% that this function here, f plus g, is differentiable at x0. 00:08:04.670 --> 00:08:11.900 align:middle line:84% And the derivative, f plus g, the derivative of this at x0 00:08:11.900 --> 00:08:14.160 align:middle line:84% is just the sum of the derivative. 00:08:14.160 --> 00:08:27.590 align:middle line:90% 00:08:27.590 --> 00:08:31.615 align:middle line:84% Let me prove this one here, by the way. 00:08:31.615 --> 00:08:35.630 align:middle line:90% 00:08:35.630 --> 00:08:36.699 align:middle line:90% So let's prove this. 00:08:36.699 --> 00:08:42.169 align:middle line:90% 00:08:42.169 --> 00:08:44.130 align:middle line:90% So we have the sum-- 00:08:44.130 --> 00:08:49.690 align:middle line:84% so we are going to look at-- so we formed the difference 00:08:49.690 --> 00:09:13.070 align:middle line:84% quotient, so f plus g at x minus f plus g at x0 x minus x0. 00:09:13.070 --> 00:09:14.780 align:middle line:90% So now what is this? 00:09:14.780 --> 00:09:27.040 align:middle line:84% So this is just f of x plus g of x minus and then f of x0 00:09:27.040 --> 00:09:35.380 align:middle line:84% plus g of x0, and then divided by x minus x0. 00:09:35.380 --> 00:09:38.950 align:middle line:84% And so then you can split this up into two. 00:09:38.950 --> 00:09:41.680 align:middle line:84% You can write this difference quotient 00:09:41.680 --> 00:09:44.240 align:middle line:84% as if you split it up if you collect 00:09:44.240 --> 00:09:47.390 align:middle line:90% the f's in the numerator. 00:09:47.390 --> 00:09:49.330 align:middle line:90% So I just collect the f's. 00:09:49.330 --> 00:09:53.180 align:middle line:90% 00:09:53.180 --> 00:09:57.290 align:middle line:84% And then, afterwards, I collect and I put it on two different 00:09:57.290 --> 00:10:02.530 align:middle line:84% fractions, separate them out, and then I collect the g's. 00:10:02.530 --> 00:10:13.100 align:middle line:90% 00:10:13.100 --> 00:10:18.120 align:middle line:84% And now you see that because f is differentiable at x0, 00:10:18.120 --> 00:10:23.730 align:middle line:84% this thing here, as x goes to x0, this here goes to f prime. 00:10:23.730 --> 00:10:28.380 align:middle line:84% So this here goes to f prime of x0. 00:10:28.380 --> 00:10:30.870 align:middle line:84% This is what this difference quotient here. 00:10:30.870 --> 00:10:34.970 align:middle line:84% And this difference quotient goes to g prime of 0. 00:10:34.970 --> 00:10:38.400 align:middle line:84% So you have that indeed, it does converge. 00:10:38.400 --> 00:10:40.980 align:middle line:84% And this is what it converges to. 00:10:40.980 --> 00:10:44.370 align:middle line:84% And so this proves that the sum of these two functions 00:10:44.370 --> 00:10:47.280 align:middle line:84% is also differentiable, and their derivative 00:10:47.280 --> 00:10:49.000 align:middle line:90% is the sum of the derivative. 00:10:49.000 --> 00:10:52.480 align:middle line:84% The next thing we will prove is Leibniz rule. 00:10:52.480 --> 00:10:59.460 align:middle line:84% But before that, I need to step back and I want to-- 00:10:59.460 --> 00:11:03.240 align:middle line:84% so maybe I'll just state it and then we'll go back 00:11:03.240 --> 00:11:04.780 align:middle line:90% and we need something. 00:11:04.780 --> 00:11:06.700 align:middle line:84% The next is what's called Leibniz rule. 00:11:06.700 --> 00:11:11.460 align:middle line:90% 00:11:11.460 --> 00:11:14.140 align:middle line:84% So you could also call it a product rule, 00:11:14.140 --> 00:11:18.360 align:middle line:84% but it's really quite generally called the Leibniz rule. 00:11:18.360 --> 00:11:21.150 align:middle line:84% And so this is when you're looking 00:11:21.150 --> 00:11:23.200 align:middle line:84% at the products of these functions. 00:11:23.200 --> 00:11:28.500 align:middle line:84% So the product function is a function 00:11:28.500 --> 00:11:33.785 align:middle line:84% in x that is just the product of f of x and g of x. 00:11:33.785 --> 00:11:41.650 align:middle line:90% 00:11:41.650 --> 00:11:52.320 align:middle line:84% And Leibniz rule says the following, 00:11:52.320 --> 00:11:56.500 align:middle line:84% that if f and g are differentiable at x0, 00:11:56.500 --> 00:11:58.750 align:middle line:84% then the product of these two function 00:11:58.750 --> 00:12:04.060 align:middle line:84% here is also differentiable at x0, 00:12:04.060 --> 00:12:06.760 align:middle line:84% and the derivative of the product 00:12:06.760 --> 00:12:11.650 align:middle line:84% is the derivative of the first function at x0 times 00:12:11.650 --> 00:12:14.920 align:middle line:84% the value of the second function at x0 00:12:14.920 --> 00:12:20.600 align:middle line:84% plus the symmetric thing in f and g, 00:12:20.600 --> 00:12:28.490 align:middle line:84% which is the value of f at x0 times the derivative of g at x0. 00:12:28.490 --> 00:12:31.730 align:middle line:84% So the product is differentiable at x0 also. 00:12:31.730 --> 00:12:33.950 align:middle line:84% And this is what the derivative is. 00:12:33.950 --> 00:12:36.970 align:middle line:84% And in the notes, I just copied a little bit of thing 00:12:36.970 --> 00:12:40.040 align:middle line:84% just for fun from Wikipedia about Leibniz 00:12:40.040 --> 00:12:41.930 align:middle line:84% that was kind of an interesting figure 00:12:41.930 --> 00:12:46.920 align:middle line:84% and played an important role a long time ago. 00:12:46.920 --> 00:12:50.840 align:middle line:84% So before I want to prove this, let me step back 00:12:50.840 --> 00:12:59.570 align:middle line:84% and let me make an observation about what it means to be 00:12:59.570 --> 00:13:01.570 align:middle line:90% differentiable at a point. 00:13:01.570 --> 00:13:04.280 align:middle line:90% 00:13:04.280 --> 00:13:07.090 align:middle line:90% So this is the next lemma. 00:13:07.090 --> 00:13:12.560 align:middle line:90% 00:13:12.560 --> 00:13:25.530 align:middle line:84% So you have a function f on R into R. And you assume that f-- 00:13:25.530 --> 00:13:38.920 align:middle line:84% so if f here is differentiable at x0, 00:13:38.920 --> 00:13:52.900 align:middle line:84% then I claim that then f here is also continuous at x0. 00:13:52.900 --> 00:13:55.600 align:middle line:90% 00:13:55.600 --> 00:13:58.660 align:middle line:84% So in other words, being differentiable 00:13:58.660 --> 00:14:01.630 align:middle line:84% is stronger than being continuous. 00:14:01.630 --> 00:14:04.090 align:middle line:84% In a minute, we will see an example of something 00:14:04.090 --> 00:14:07.210 align:middle line:84% that's continuous but not differentiable. 00:14:07.210 --> 00:14:12.435 align:middle line:84% And so now let's try to prove that. 00:14:12.435 --> 00:14:25.490 align:middle line:90% 00:14:25.490 --> 00:14:27.770 align:middle line:84% So I'm assuming that it's differentiable. 00:14:27.770 --> 00:14:37.130 align:middle line:84% So this means that this difference quotient here 00:14:37.130 --> 00:14:38.220 align:middle line:90% has a limit. 00:14:38.220 --> 00:14:44.720 align:middle line:84% And that limit is the derivative as x goes to x0. 00:14:44.720 --> 00:14:59.750 align:middle line:84% So this means that there exists a delta greater than 0 such 00:14:59.750 --> 00:15:11.930 align:middle line:84% that if x here minus x0 is less than delta y-- 00:15:11.930 --> 00:15:17.360 align:middle line:84% of course, I'm assuming that it's not equal to 0. 00:15:17.360 --> 00:15:22.910 align:middle line:84% If it is less than, the difference between x and x0 00:15:22.910 --> 00:15:26.150 align:middle line:84% is less than delta, then this difference 00:15:26.150 --> 00:15:36.060 align:middle line:84% quotient here, then f of dx minus f of x here x 00:15:36.060 --> 00:15:42.150 align:middle line:84% minus x minus x0 minus f prime of x0-- 00:15:42.150 --> 00:15:47.050 align:middle line:84% this thing here have to be less than 1. 00:15:47.050 --> 00:15:50.700 align:middle line:84% I mean, this is just using that this limit exists, 00:15:50.700 --> 00:15:52.620 align:middle line:90% and I'm using-- 00:15:52.620 --> 00:15:55.145 align:middle line:90% in place of epsilon I'm using 1. 00:15:55.145 --> 00:16:06.330 align:middle line:90% 00:16:06.330 --> 00:16:13.050 align:middle line:84% Now so this means just multiplying over and multiplying 00:16:13.050 --> 00:16:13.840 align:middle line:90% over. 00:16:13.840 --> 00:16:21.600 align:middle line:84% You get that f of x minus f of x minus 0 00:16:21.600 --> 00:16:30.675 align:middle line:84% minus f prime of x0, this thing here-- 00:16:30.675 --> 00:16:33.670 align:middle line:90% 00:16:33.670 --> 00:16:47.530 align:middle line:84% and I'm multiplying both sides, so multiply both sides 00:16:47.530 --> 00:16:51.890 align:middle line:90% by x minus x0. 00:16:51.890 --> 00:16:53.800 align:middle line:90% This is a positive thing. 00:16:53.800 --> 00:16:56.540 align:middle line:84% So multiplying both things by that, 00:16:56.540 --> 00:17:01.540 align:middle line:84% so then I get this thing here, x minus x0. 00:17:01.540 --> 00:17:06.970 align:middle line:84% And then I get less or less strictly less than x minus x0 00:17:06.970 --> 00:17:09.640 align:middle line:84% because this is strictly positive 00:17:09.640 --> 00:17:11.770 align:middle line:90% because x is not equal to x0. 00:17:11.770 --> 00:17:13.480 align:middle line:90% So I have that. 00:17:13.480 --> 00:17:18.530 align:middle line:84% And so I get that-- so now I just split it up 00:17:18.530 --> 00:17:30.620 align:middle line:84% so I have that f of dx minus f of x0 is now less than x 00:17:30.620 --> 00:17:41.810 align:middle line:84% minus x0 plus f prime of x minus 0 times x minus x0. 00:17:41.810 --> 00:17:45.140 align:middle line:90% 00:17:45.140 --> 00:17:47.810 align:middle line:84% I mean, this is just the triangle inequality 00:17:47.810 --> 00:17:54.920 align:middle line:84% that we've used many times, just depend on R. 00:17:54.920 --> 00:18:00.080 align:middle line:84% And so now you see that if you now-- 00:18:00.080 --> 00:18:05.810 align:middle line:90% 00:18:05.810 --> 00:18:09.970 align:middle line:84% and this here, of course, I can also write like this. 00:18:09.970 --> 00:18:12.500 align:middle line:90% 00:18:12.500 --> 00:18:14.870 align:middle line:84% I can just factor this out, and I 00:18:14.870 --> 00:18:21.600 align:middle line:84% can write it as 1 plus f prime of x0 norm, like this. 00:18:21.600 --> 00:18:22.700 align:middle line:90% That's fine. 00:18:22.700 --> 00:18:25.760 align:middle line:84% And so now if I want this thing here-- 00:18:25.760 --> 00:18:32.530 align:middle line:84% so now you want to prove that f is continuous at x0. 00:18:32.530 --> 00:18:36.630 align:middle line:84% So you want this here to be less than some given epsilon. 00:18:36.630 --> 00:18:46.050 align:middle line:84% So given epsilon greater than 0, you 00:18:46.050 --> 00:18:54.910 align:middle line:90% let delta 1 be the minimum of-- 00:18:54.910 --> 00:18:59.730 align:middle line:84% if you set delta 1 to be the minimum of the delta 00:18:59.730 --> 00:19:04.080 align:middle line:84% that I already chose, so then I have this inequality, 00:19:04.080 --> 00:19:17.130 align:middle line:84% and then epsilon over 1 plus f prime of x0. 00:19:17.130 --> 00:19:28.430 align:middle line:84% So if I choose delta like this, then I have this inequality. 00:19:28.430 --> 00:19:30.630 align:middle line:90% I fill in that. 00:19:30.630 --> 00:19:38.420 align:middle line:84% So now I'm assuming that this thing here is less than delta 1. 00:19:38.420 --> 00:19:42.480 align:middle line:84% So again, I have this inequality because it's less than delta. 00:19:42.480 --> 00:19:49.160 align:middle line:84% So then when I fill in here delta 1 for this, 00:19:49.160 --> 00:19:56.360 align:middle line:84% then I see that this thing here times this is less than epsilon. 00:19:56.360 --> 00:20:09.060 align:middle line:84% You see this holds when x minus x0 is less than delta 1. 00:20:09.060 --> 00:20:13.650 align:middle line:84% Again, because if it's less than delta 1, it's less than delta, 00:20:13.650 --> 00:20:15.510 align:middle line:90% so I have this inequality. 00:20:15.510 --> 00:20:21.980 align:middle line:84% And then I can use that delta 1 since this is less than this. 00:20:21.980 --> 00:20:24.310 align:middle line:90% And then I get this inequality. 00:20:24.310 --> 00:20:28.800 align:middle line:90% 00:20:28.800 --> 00:20:31.840 align:middle line:84% So this is really what we want to prove. 00:20:31.840 --> 00:20:34.240 align:middle line:84% We want to prove that if it's differentiable, 00:20:34.240 --> 00:20:36.310 align:middle line:90% then it's also continuous. 00:20:36.310 --> 00:20:39.450 align:middle line:84% And the continuity here, I proved it 00:20:39.450 --> 00:20:44.590 align:middle line:84% with epsilon and delta that really should prove it. 00:20:44.590 --> 00:20:48.090 align:middle line:90% 00:20:48.090 --> 00:20:50.350 align:middle line:90% Now let's look at some examples. 00:20:50.350 --> 00:20:55.600 align:middle line:84% So this means that if a function is differentiable, 00:20:55.600 --> 00:20:56.906 align:middle line:90% then it's continuous. 00:20:56.906 --> 00:21:01.830 align:middle line:90% 00:21:01.830 --> 00:21:07.300 align:middle line:84% Let me give you an example of a function that is -- 00:21:07.300 --> 00:21:12.690 align:middle line:84% so let me give you three examples. 00:21:12.690 --> 00:21:18.010 align:middle line:84% The first example is maybe the simplest one. 00:21:18.010 --> 00:21:21.030 align:middle line:84% So this is the function f of x that 00:21:21.030 --> 00:21:24.470 align:middle line:84% is equal to the absolute value of x. 00:21:24.470 --> 00:21:30.970 align:middle line:84% So f here is defined on R into R. 00:21:30.970 --> 00:21:33.700 align:middle line:84% And it's just the absolute value. 00:21:33.700 --> 00:21:39.140 align:middle line:84% And so the absolute value here we've seen before. 00:21:39.140 --> 00:21:42.116 align:middle line:90% So f here is continuous. 00:21:42.116 --> 00:21:46.630 align:middle line:90% 00:21:46.630 --> 00:21:52.180 align:middle line:84% It's continuous at all points, in particular, of course, at 0. 00:21:52.180 --> 00:21:55.340 align:middle line:84% But if you now form the difference quotient. 00:21:55.340 --> 00:21:58.880 align:middle line:84% Suppose that x here is bigger than 0. 00:21:58.880 --> 00:22:01.700 align:middle line:84% And now look at the difference quotient. 00:22:01.700 --> 00:22:03.250 align:middle line:90% So the difference quotient-- 00:22:03.250 --> 00:22:09.700 align:middle line:90% 00:22:09.700 --> 00:22:18.400 align:middle line:84% quotient at x0 equal to 0 if you're looking 00:22:18.400 --> 00:22:21.680 align:middle line:90% at that difference quotient. 00:22:21.680 --> 00:22:24.330 align:middle line:84% And you first look at where x is positive. 00:22:24.330 --> 00:22:30.710 align:middle line:84% If x is positive, then f of x minus f of 0-- 00:22:30.710 --> 00:22:33.440 align:middle line:84% x minus 0, this difference quotient here. 00:22:33.440 --> 00:22:37.550 align:middle line:84% If x is positive, then this thing here is just x. 00:22:37.550 --> 00:22:41.940 align:middle line:90% So this is x minus 0, x minus 0. 00:22:41.940 --> 00:22:44.760 align:middle line:90% So this here is 1. 00:22:44.760 --> 00:22:46.530 align:middle line:90% This difference quotient is 1. 00:22:46.530 --> 00:22:48.780 align:middle line:90% This is when x is positive. 00:22:48.780 --> 00:22:55.250 align:middle line:84% When x is negative, then f of x-- 00:22:55.250 --> 00:23:01.730 align:middle line:84% if x is negative, then the absolute value is minus x. 00:23:01.730 --> 00:23:07.820 align:middle line:84% So the difference quotient here f of x minus f of 0, 00:23:07.820 --> 00:23:11.150 align:middle line:90% which is 0, x minus 0-- 00:23:11.150 --> 00:23:17.960 align:middle line:84% this here is just minus x divided by x. 00:23:17.960 --> 00:23:20.910 align:middle line:90% But this thing here is minus 1. 00:23:20.910 --> 00:23:29.760 align:middle line:84% So if you're looking at the difference quotient that 00:23:29.760 --> 00:23:33.180 align:middle line:84% comes from looking at the derivative is 0, 00:23:33.180 --> 00:23:37.920 align:middle line:84% then when you approach 0 from the right, 00:23:37.920 --> 00:23:40.930 align:middle line:84% then the difference quotient is 1. 00:23:40.930 --> 00:23:44.740 align:middle line:90% When you approach it from the-- 00:23:44.740 --> 00:23:46.740 align:middle line:84% and when you approach it from the left, 00:23:46.740 --> 00:23:49.360 align:middle line:84% then the difference quotient is minus 1. 00:23:49.360 --> 00:23:51.670 align:middle line:90% So it doesn't have a limit. 00:23:51.670 --> 00:23:54.250 align:middle line:84% It has a limit from the right from the left, 00:23:54.250 --> 00:23:57.180 align:middle line:84% but it doesn't have a limit if you don't restrict 00:23:57.180 --> 00:23:59.100 align:middle line:90% where it's coming from. 00:23:59.100 --> 00:24:03.850 align:middle line:84% So this function here is not differentiable. 00:24:03.850 --> 00:24:05.580 align:middle line:90% Yeah? 00:24:05.580 --> 00:24:08.460 align:middle line:84% AUDIENCE: Can you define what it means when they have 00:24:08.460 --> 00:24:11.090 align:middle line:90% a function of x [INAUDIBLE]? 00:24:11.090 --> 00:24:22.490 align:middle line:90% 00:24:22.490 --> 00:24:24.730 align:middle line:84% PROFESSOR: So we did this earlier in the class. 00:24:24.730 --> 00:24:28.260 align:middle line:90% 00:24:28.260 --> 00:24:34.610 align:middle line:84% So that's the difference quotient here 00:24:34.610 --> 00:24:45.380 align:middle line:84% that this thing here is converging to some number, 00:24:45.380 --> 00:24:47.330 align:middle line:90% say a. 00:24:47.330 --> 00:24:53.810 align:middle line:84% This here means that for all epsilon greater than 0, 00:24:53.810 --> 00:25:04.280 align:middle line:84% there exists a delta greater than 0 such that if x minus x0 00:25:04.280 --> 00:25:10.460 align:middle line:84% is less than delta and, again, x is assumed 00:25:10.460 --> 00:25:12.800 align:middle line:90% to be different from x0. 00:25:12.800 --> 00:25:17.070 align:middle line:84% If you have that, then this difference quotient 00:25:17.070 --> 00:25:31.260 align:middle line:84% here minus this a is supposed to be less than epsilon. 00:25:31.260 --> 00:25:32.950 align:middle line:90% And that's this. 00:25:32.950 --> 00:25:37.620 align:middle line:90% 00:25:37.620 --> 00:25:40.740 align:middle line:84% And so you see, again, in this example 00:25:40.740 --> 00:25:45.660 align:middle line:84% that we just looked at, where in the example that we just 00:25:45.660 --> 00:25:46.260 align:middle line:90% looked at-- 00:25:46.260 --> 00:25:49.440 align:middle line:90% 00:25:49.440 --> 00:25:54.030 align:middle line:84% so in the example where f of x was 00:25:54.030 --> 00:26:00.330 align:middle line:84% equal to the absolute value of x, then if you were graphing it, 00:26:00.330 --> 00:26:04.200 align:middle line:90% then this looks like this. 00:26:04.200 --> 00:26:07.590 align:middle line:84% And we saw that the difference quotient 00:26:07.590 --> 00:26:11.380 align:middle line:84% when you approach it over here, the difference quotient was 1. 00:26:11.380 --> 00:26:14.410 align:middle line:84% When you approach it over here, the difference quotient 00:26:14.410 --> 00:26:15.620 align:middle line:90% is minus 1. 00:26:15.620 --> 00:26:21.100 align:middle line:84% So if you take a tiny little interval around 0, 00:26:21.100 --> 00:26:27.740 align:middle line:84% then the difference quotient could be either minus 1 or 1. 00:26:27.740 --> 00:26:31.550 align:middle line:84% But however small you make this little interval, 00:26:31.550 --> 00:26:36.050 align:middle line:84% it's not like that it converges to some number because it has-- 00:26:36.050 --> 00:26:37.580 align:middle line:90% it is both. 00:26:37.580 --> 00:26:42.070 align:middle line:84% There are both cases where it's 1 and cases where it's minus 1. 00:26:42.070 --> 00:26:44.720 align:middle line:90% 00:26:44.720 --> 00:26:48.280 align:middle line:84% So that's like the first example. 00:26:48.280 --> 00:26:52.840 align:middle line:84% And again, you know of a function that is continuous, 00:26:52.840 --> 00:26:56.630 align:middle line:84% but it's not differentiable at 0. 00:26:56.630 --> 00:27:00.460 align:middle line:90% In this case, it was a 0. 00:27:00.460 --> 00:27:03.170 align:middle line:84% And maybe a little bit more interesting example, 00:27:03.170 --> 00:27:08.120 align:middle line:84% along the same line, is the following. 00:27:08.120 --> 00:27:30.920 align:middle line:90% 00:27:30.920 --> 00:27:35.960 align:middle line:84% So this is, again, a function on R into R, 00:27:35.960 --> 00:27:42.240 align:middle line:84% and I set f of x equal to x times sine of 1 over x. 00:27:42.240 --> 00:27:45.510 align:middle line:84% This here is not really defined at 0. 00:27:45.510 --> 00:27:47.850 align:middle line:84% So I need to define it also at 0. 00:27:47.850 --> 00:27:54.540 align:middle line:84% So when x is equal to 0, then I just set it equal to 0. 00:27:54.540 --> 00:28:02.840 align:middle line:84% And otherwise, I set it equal to x times sine of 1 over x. 00:28:02.840 --> 00:28:06.260 align:middle line:84% And so the thing is that this function here, so f here, 00:28:06.260 --> 00:28:07.476 align:middle line:90% is continuous. 00:28:07.476 --> 00:28:12.540 align:middle line:90% 00:28:12.540 --> 00:28:18.160 align:middle line:84% In particular, it's continuous at 0. 00:28:18.160 --> 00:28:19.750 align:middle line:84% It's actually continuous everywhere, 00:28:19.750 --> 00:28:23.830 align:middle line:84% but I'm mostly interested in it at 0. 00:28:23.830 --> 00:28:25.480 align:middle line:90% The graph of this-- 00:28:25.480 --> 00:28:35.730 align:middle line:84% so remember that sine function looks like this, That at 0, 00:28:35.730 --> 00:28:41.860 align:middle line:84% at pi, at 2 pi, at 3 pi, and so on. 00:28:41.860 --> 00:28:45.180 align:middle line:90% And all of these-- 00:28:45.180 --> 00:28:49.870 align:middle line:84% or at all integer multiples, also, for the negative ones, 00:28:49.870 --> 00:28:56.910 align:middle line:84% at all integer multiple of pi, sine function vanishes. 00:28:56.910 --> 00:29:01.110 align:middle line:84% And in between here, it is at pi over 2, 00:29:01.110 --> 00:29:04.540 align:middle line:84% it achieves its maximum, which is 1. 00:29:04.540 --> 00:29:08.820 align:middle line:84% And so the function looks like this. 00:29:08.820 --> 00:29:09.840 align:middle line:90% So this here is 1. 00:29:09.840 --> 00:29:13.690 align:middle line:90% 00:29:13.690 --> 00:29:17.230 align:middle line:84% That's how a sine function looks like. 00:29:17.230 --> 00:29:20.320 align:middle line:84% And so now if you're looking at sine of 1 00:29:20.320 --> 00:29:25.990 align:middle line:84% over x, that's going to be a function as you're going down. 00:29:25.990 --> 00:29:28.990 align:middle line:84% So if you're looking at this-- here's a sine of x. 00:29:28.990 --> 00:29:34.060 align:middle line:84% But if you're looking at sine of 1 over x, as x goes to 0, 00:29:34.060 --> 00:29:36.620 align:middle line:90% this becomes very large. 00:29:36.620 --> 00:29:39.490 align:middle line:84% So this means that as you're going to 0, 00:29:39.490 --> 00:29:45.790 align:middle line:84% it goes very rapidly through these periods, where 00:29:45.790 --> 00:29:48.620 align:middle line:84% it oscillates between negative 1 and 1. 00:29:48.620 --> 00:29:50.790 align:middle line:90% So the value here is minus 1. 00:29:50.790 --> 00:29:53.320 align:middle line:90% 00:29:53.320 --> 00:30:04.910 align:middle line:84% And so it will look like this that it will again fluctuate. 00:30:04.910 --> 00:30:10.390 align:middle line:84% It will fluctuate, but it will fluctuate much more rapidly. 00:30:10.390 --> 00:30:14.173 align:middle line:84% So if this here, say, was 1 over pi. 00:30:14.173 --> 00:30:17.370 align:middle line:90% 00:30:17.370 --> 00:30:24.480 align:middle line:84% And 1 over pi is the same as pi, there, it would be 0. 00:30:24.480 --> 00:30:26.746 align:middle line:90% And it would like fluctuate. 00:30:26.746 --> 00:30:30.490 align:middle line:90% 00:30:30.490 --> 00:30:32.350 align:middle line:84% So it will look like something like this, 00:30:32.350 --> 00:30:33.700 align:middle line:90% but it will fluctuate. 00:30:33.700 --> 00:30:36.330 align:middle line:90% 00:30:36.330 --> 00:30:43.350 align:middle line:84% And then, as you're going away, it will fluctuate less often. 00:30:43.350 --> 00:30:50.490 align:middle line:84% So as you're going down towards 0, it will fluctuate a lot. 00:30:50.490 --> 00:30:54.520 align:middle line:84% Because sine always is between minus 1 and 1, 00:30:54.520 --> 00:30:58.660 align:middle line:84% the value here will always be between minus 1 and 1. 00:30:58.660 --> 00:31:04.300 align:middle line:84% But as you're going towards 0 from the right or from the left, 00:31:04.300 --> 00:31:07.520 align:middle line:84% it's going to fluctuate more and more. 00:31:07.520 --> 00:31:10.915 align:middle line:84% As you're going away from 0, it's fluctuating less and less. 00:31:10.915 --> 00:31:14.600 align:middle line:90% 00:31:14.600 --> 00:31:17.640 align:middle line:84% So this is the function I'm interested in. 00:31:17.640 --> 00:31:27.950 align:middle line:84% But when you multiply through by x, then you see, 00:31:27.950 --> 00:31:31.040 align:middle line:84% when you multiply through by x, it's fluctuating, 00:31:31.040 --> 00:31:34.400 align:middle line:84% but it's always fluctuating between minus 1 and 1. 00:31:34.400 --> 00:31:41.180 align:middle line:84% If you multiply it by x, then as you're going to 0, 00:31:41.180 --> 00:31:48.230 align:middle line:84% this here will squeeze down, and the fluctuation will be 0 at 0. 00:31:48.230 --> 00:31:52.580 align:middle line:84% So it's easy to see that this function here-- so is indeed 00:31:52.580 --> 00:31:55.070 align:middle line:90% continuous. 00:31:55.070 --> 00:31:57.390 align:middle line:84% But let's look at the difference quotient. 00:31:57.390 --> 00:31:59.160 align:middle line:90% So the difference quotient-- 00:31:59.160 --> 00:32:08.300 align:middle line:90% 00:32:08.300 --> 00:32:12.080 align:middle line:84% so if we look at the difference quotient, so we're looking at-- 00:32:12.080 --> 00:32:15.170 align:middle line:84% and we just look at the difference quotient at 0. 00:32:15.170 --> 00:32:19.490 align:middle line:84% So this is the difference quotient. 00:32:19.490 --> 00:32:23.630 align:middle line:90% And so it's x sine 1 over x. 00:32:23.630 --> 00:32:26.605 align:middle line:84% The value at 0 by definition was 0. 00:32:26.605 --> 00:32:30.380 align:middle line:90% 00:32:30.380 --> 00:32:34.250 align:middle line:84% And so you see that now this here, 00:32:34.250 --> 00:32:39.980 align:middle line:84% this here becomes just x sine 1 over x divided by x. 00:32:39.980 --> 00:32:42.690 align:middle line:84% And so you see that this here cancel with that. 00:32:42.690 --> 00:32:45.950 align:middle line:90% So you just get sine 1 over x. 00:32:45.950 --> 00:32:49.020 align:middle line:84% And so you see that now, as x goes to-- 00:32:49.020 --> 00:32:53.870 align:middle line:84% so when x here goes to 0, this here fluctuates. 00:32:53.870 --> 00:33:07.620 align:middle line:84% This here fluctuates between minus 1 and 1. 00:33:07.620 --> 00:33:10.020 align:middle line:84% But it doesn't have a limit because it's 00:33:10.020 --> 00:33:11.580 align:middle line:90% fluctuating wider and wider. 00:33:11.580 --> 00:33:16.120 align:middle line:90% So this is an example again-- 00:33:16.120 --> 00:33:21.780 align:middle line:84% this is another example of a function that is continuous at 0 00:33:21.780 --> 00:33:23.320 align:middle line:90% but not differentiable at 0. 00:33:23.320 --> 00:33:25.145 align:middle line:84% It's actually much worse than the previous. 00:33:25.145 --> 00:33:30.880 align:middle line:90% 00:33:30.880 --> 00:33:37.290 align:middle line:84% The previous example that was-- this example is much more tame. 00:33:37.290 --> 00:33:39.790 align:middle line:90% It is not differentiable at 0. 00:33:39.790 --> 00:33:44.460 align:middle line:84% But if you were to restrict to just positive things, 00:33:44.460 --> 00:33:47.380 align:middle line:90% then it would be differentiable. 00:33:47.380 --> 00:33:50.160 align:middle line:84% You can also define derivative-- so, 00:33:50.160 --> 00:33:53.020 align:middle line:84% in general, when we define derivative, 00:33:53.020 --> 00:33:56.410 align:middle line:84% we let it be defined in a neighborhood of that point, 00:33:56.410 --> 00:33:57.970 align:middle line:90% so on either side. 00:33:57.970 --> 00:34:00.280 align:middle line:84% But if you were considering where you were, 00:34:00.280 --> 00:34:04.450 align:middle line:84% allowing the function just to be defined at this point 00:34:04.450 --> 00:34:07.630 align:middle line:84% and then to the right, say, or to the left, 00:34:07.630 --> 00:34:10.690 align:middle line:84% then this function would be differentiable. 00:34:10.690 --> 00:34:15.639 align:middle line:84% But in this example here, it doesn't 00:34:15.639 --> 00:34:19.060 align:middle line:84% matter if you restrict the interval to just to the right 00:34:19.060 --> 00:34:19.850 align:middle line:90% or to the left. 00:34:19.850 --> 00:34:21.280 align:middle line:90% It's still not differentiable. 00:34:21.280 --> 00:34:26.980 align:middle line:84% So this says something about this example, something 00:34:26.980 --> 00:34:32.230 align:middle line:84% about that it fluctuates wildly and that is causing it 00:34:32.230 --> 00:34:35.920 align:middle line:90% not to be differentiable. 00:34:35.920 --> 00:34:40.100 align:middle line:90% Again, if you defined it-- 00:34:40.100 --> 00:34:43.570 align:middle line:84% if you didn't have this kind of-- the x here 00:34:43.570 --> 00:34:47.080 align:middle line:84% is serving as some of-- it's dampening 00:34:47.080 --> 00:34:49.040 align:middle line:90% how much it's fluctuating. 00:34:49.040 --> 00:34:54.340 align:middle line:84% So if there wasn't an x here, then this function wouldn't-- it 00:34:54.340 --> 00:34:58.090 align:middle line:84% would just fluctuate, and it wouldn't even be continuous 00:34:58.090 --> 00:34:59.400 align:middle line:90% at x0. 00:34:59.400 --> 00:35:03.360 align:middle line:84% But once I put this x on, then it becomes continuous. 00:35:03.360 --> 00:35:07.890 align:middle line:84% If I put another x, so if I'm looking 00:35:07.890 --> 00:35:16.528 align:middle line:84% at f of x equal to x squared times sine 1 over x. 00:35:16.528 --> 00:35:18.070 align:middle line:84% Suppose I'm looking at this function. 00:35:18.070 --> 00:35:23.260 align:middle line:84% So now I have dampened the fluctuation even more. 00:35:23.260 --> 00:35:26.670 align:middle line:84% And so if I came looking at the difference quotient-- 00:35:26.670 --> 00:35:29.730 align:middle line:90% so I'm looking at this function. 00:35:29.730 --> 00:35:31.210 align:middle line:90% I should be more careful. 00:35:31.210 --> 00:35:36.610 align:middle line:84% So again, at 0, I need to define it separately. 00:35:36.610 --> 00:35:43.270 align:middle line:84% So I define a function that is 0 at x equal to 0. 00:35:43.270 --> 00:35:49.880 align:middle line:84% And then it is x squared times sine 1 over x otherwise. 00:35:49.880 --> 00:35:53.980 align:middle line:90% 00:35:53.980 --> 00:35:56.010 align:middle line:84% If I'm looking at this function, then I 00:35:56.010 --> 00:36:03.490 align:middle line:84% claim that this function is, in fact, differentiable at x at 0. 00:36:03.490 --> 00:36:06.850 align:middle line:90% And so let's look at that. 00:36:06.850 --> 00:36:11.350 align:middle line:84% So again, I'm looking at the difference quotient. 00:36:11.350 --> 00:36:14.090 align:middle line:90% And the value at 0 is 0. 00:36:14.090 --> 00:36:15.195 align:middle line:90% So I have this. 00:36:15.195 --> 00:36:18.130 align:middle line:90% 00:36:18.130 --> 00:36:25.060 align:middle line:84% And so this here is now just x squared sine 1 over x. 00:36:25.060 --> 00:36:27.020 align:middle line:90% This thing here is then x. 00:36:27.020 --> 00:36:30.700 align:middle line:84% And so you see that this difference quotient is just 00:36:30.700 --> 00:36:31.720 align:middle line:90% this. 00:36:31.720 --> 00:36:38.630 align:middle line:84% But this here does converge to 0 as x goes to 0. 00:36:38.630 --> 00:36:41.650 align:middle line:84% So this function is actually differentiable. 00:36:41.650 --> 00:36:48.100 align:middle line:84% It's fluctuates, but it's more mild the way it fluctuates. 00:36:48.100 --> 00:36:54.140 align:middle line:84% Now let me get back to the rules. 00:36:54.140 --> 00:36:59.410 align:middle line:84% So we proved one rule that was this sum rule. 00:36:59.410 --> 00:37:03.080 align:middle line:90% 00:37:03.080 --> 00:37:05.480 align:middle line:84% Then the second rule we're interested in 00:37:05.480 --> 00:37:08.070 align:middle line:84% is when we're looking at the Leibniz rule. 00:37:08.070 --> 00:37:11.180 align:middle line:84% So we're looking at the product of these two functions. 00:37:11.180 --> 00:37:17.510 align:middle line:84% And each function is assumed to be differentiable at x0. 00:37:17.510 --> 00:37:22.100 align:middle line:84% And then we claim that the product function is also 00:37:22.100 --> 00:37:24.240 align:middle line:90% differentiable at x0. 00:37:24.240 --> 00:37:28.700 align:middle line:84% And this is the formula for the derivative. 00:37:28.700 --> 00:37:32.750 align:middle line:84% To prove this one here, I need these facts 00:37:32.750 --> 00:37:36.480 align:middle line:84% that we have now proved that if it is differentiable, 00:37:36.480 --> 00:37:39.380 align:middle line:84% then it's also continuous at that point. 00:37:39.380 --> 00:37:41.695 align:middle line:90% So I'm going to use that fact. 00:37:41.695 --> 00:37:51.410 align:middle line:90% 00:37:51.410 --> 00:38:01.790 align:middle line:84% So this is proof of Leibniz rule. 00:38:01.790 --> 00:38:08.010 align:middle line:90% 00:38:08.010 --> 00:38:11.050 align:middle line:90% So we have these two functions. 00:38:11.050 --> 00:38:13.537 align:middle line:90% We're looking at f of x. 00:38:13.537 --> 00:38:14.870 align:middle line:90% We form the difference quotient. 00:38:14.870 --> 00:38:22.230 align:middle line:90% 00:38:22.230 --> 00:38:25.685 align:middle line:84% So this is the difference quotient, x minus x0. 00:38:25.685 --> 00:38:29.580 align:middle line:90% 00:38:29.580 --> 00:38:34.170 align:middle line:84% We do this trick that we've done a bunch of times before, 00:38:34.170 --> 00:38:39.090 align:middle line:84% when we also looked at products if we had two sequences, 00:38:39.090 --> 00:38:41.430 align:middle line:84% and we looked at the sequence that 00:38:41.430 --> 00:38:43.720 align:middle line:90% was the product of these two. 00:38:43.720 --> 00:38:48.510 align:middle line:90% So just a reminder that-- 00:38:48.510 --> 00:38:49.660 align:middle line:90% I mean, it's not like-- 00:38:49.660 --> 00:38:59.440 align:middle line:90% 00:38:59.440 --> 00:39:04.340 align:middle line:84% so remember that if you had a sequence an and a sequence bn, 00:39:04.340 --> 00:39:08.890 align:middle line:84% and you knew that an converges to a and bn converges 00:39:08.890 --> 00:39:18.520 align:middle line:84% to b, then the product here converges to a times b. 00:39:18.520 --> 00:39:23.320 align:middle line:84% And when we prove that, we use this little trick 00:39:23.320 --> 00:39:29.320 align:middle line:84% that we're splitting the difference between this 00:39:29.320 --> 00:39:30.830 align:middle line:90% and this. 00:39:30.830 --> 00:39:34.690 align:middle line:84% And so I'm going to do exactly the same trick here. 00:39:34.690 --> 00:39:37.070 align:middle line:84% So I'm looking at this thing here. 00:39:37.070 --> 00:39:43.940 align:middle line:84% And so I write this thing here as f of x times g of x minus-- 00:39:43.940 --> 00:39:48.410 align:middle line:84% and it doesn't really matter where you're inserting it. 00:39:48.410 --> 00:39:51.450 align:middle line:84% So this here is splitting the difference. 00:39:51.450 --> 00:39:53.020 align:middle line:90% And then I have to add it again. 00:39:53.020 --> 00:40:02.180 align:middle line:90% 00:40:02.180 --> 00:40:06.590 align:middle line:84% So this is certainly the same, just subtracting it 00:40:06.590 --> 00:40:10.100 align:middle line:90% and adding it right back. 00:40:10.100 --> 00:40:13.170 align:middle line:90% And now I split this into two. 00:40:13.170 --> 00:40:16.110 align:middle line:90% So I write this thing here. 00:40:16.110 --> 00:40:20.150 align:middle line:84% So this fraction, I split into two. 00:40:20.150 --> 00:40:23.880 align:middle line:84% And the first one, I take this one here together. 00:40:23.880 --> 00:40:27.060 align:middle line:90% And you can factor out a g of x. 00:40:27.060 --> 00:40:36.080 align:middle line:84% You can write it like this, f of x minus f of x0, x minus x0. 00:40:36.080 --> 00:40:39.290 align:middle line:90% So I'm just here. 00:40:39.290 --> 00:40:44.540 align:middle line:84% I'm factoring out the g of x, and then I'm left with this. 00:40:44.540 --> 00:40:47.600 align:middle line:84% And this is keeping the denominator, like that. 00:40:47.600 --> 00:40:49.930 align:middle line:90% This is this part here. 00:40:49.930 --> 00:40:52.480 align:middle line:84% And then I have the other part here. 00:40:52.480 --> 00:40:56.235 align:middle line:84% But there, you see that the f of x0 is factoring out. 00:40:56.235 --> 00:40:58.780 align:middle line:90% 00:40:58.780 --> 00:41:10.180 align:middle line:84% And then I have g of x minus g of x0, x minus x0, like that. 00:41:10.180 --> 00:41:17.170 align:middle line:84% But now you see that as x goes to x0, well, 00:41:17.170 --> 00:41:20.830 align:middle line:90% g was continuous at x0. 00:41:20.830 --> 00:41:25.400 align:middle line:84% So this thing here, as x goes to x0, 00:41:25.400 --> 00:41:29.930 align:middle line:84% this thing here goes to g of x0-- 00:41:29.930 --> 00:41:30.620 align:middle line:90% this one here. 00:41:30.620 --> 00:41:32.810 align:middle line:84% That was because g was continuous. 00:41:32.810 --> 00:41:35.930 align:middle line:84% Because it was differentiable, so it was also continuous. 00:41:35.930 --> 00:41:38.510 align:middle line:84% This here goes to the derivative. 00:41:38.510 --> 00:41:41.350 align:middle line:90% 00:41:41.350 --> 00:41:47.780 align:middle line:84% And the second thing here this is just constant. 00:41:47.780 --> 00:41:55.260 align:middle line:84% And this goes to the derivative, like that. 00:41:55.260 --> 00:42:01.040 align:middle line:84% So what we have proven this here is as x goes to x0. 00:42:01.040 --> 00:42:04.400 align:middle line:84% So you see that we have proven that this difference 00:42:04.400 --> 00:42:09.350 align:middle line:84% quotient here for the product has a limit. 00:42:09.350 --> 00:42:11.970 align:middle line:90% And this is what the limit is. 00:42:11.970 --> 00:42:15.200 align:middle line:84% And so this means that the product is differentiable. 00:42:15.200 --> 00:42:17.820 align:middle line:90% And this here is the derivative. 00:42:17.820 --> 00:42:21.180 align:middle line:84% And that's exactly the Leibniz formula, 00:42:21.180 --> 00:42:24.510 align:middle line:84% which was this over here, this formula here. 00:42:24.510 --> 00:42:27.320 align:middle line:90% 00:42:27.320 --> 00:42:34.490 align:middle line:84% The next thing is that we want to prove the next rule is 00:42:34.490 --> 00:42:37.370 align:middle line:90% like this quotient rule. 00:42:37.370 --> 00:42:40.165 align:middle line:84% So let me prove the quotient rule. 00:42:40.165 --> 00:42:58.320 align:middle line:90% 00:42:58.320 --> 00:43:04.680 align:middle line:84% So Leibniz and Newton was the one that 00:43:04.680 --> 00:43:09.993 align:middle line:90% was responsible for calculus. 00:43:09.993 --> 00:43:13.770 align:middle line:90% 00:43:13.770 --> 00:43:16.110 align:middle line:84% And so then the next thing I want to prove 00:43:16.110 --> 00:43:17.950 align:middle line:90% is this quotient rule. 00:43:17.950 --> 00:43:27.870 align:middle line:90% 00:43:27.870 --> 00:43:31.535 align:middle line:84% And so, again, you have these two functions, f and g. 00:43:31.535 --> 00:43:36.360 align:middle line:90% 00:43:36.360 --> 00:43:49.190 align:middle line:84% And they are differentiable at x0. 00:43:49.190 --> 00:43:58.430 align:middle line:84% And I'm also assuming that g here at x0 is not equal to x. 00:43:58.430 --> 00:44:02.800 align:middle line:84% Let's just assume that g is not equal to 0 everywhere. 00:44:02.800 --> 00:44:08.420 align:middle line:84% It's different from 0 everywhere. 00:44:08.420 --> 00:44:13.870 align:middle line:90% 00:44:13.870 --> 00:44:18.280 align:middle line:84% And then the quotient rule says that if you're 00:44:18.280 --> 00:44:25.210 align:middle line:84% looking at the quotient here and you take the derivative of the-- 00:44:25.210 --> 00:44:29.320 align:middle line:84% so it say that if you're looking at the quotient here-- 00:44:29.320 --> 00:44:33.910 align:middle line:84% so the quotient will say that if you're looking at this thing 00:44:33.910 --> 00:44:37.700 align:middle line:84% here, this now makes sense because g does not vanish. 00:44:37.700 --> 00:44:48.470 align:middle line:84% This here is differentiable at x0. 00:44:48.470 --> 00:44:56.480 align:middle line:84% And the derivative of this function at x0 00:44:56.480 --> 00:44:59.270 align:middle line:84% is given by the following formula, 00:44:59.270 --> 00:45:07.700 align:middle line:84% that it is the derivative of f at x0 times g of x0 00:45:07.700 --> 00:45:20.100 align:middle line:84% minus f at x0 times g prime at x0 divided by g squared. 00:45:20.100 --> 00:45:23.460 align:middle line:90% So g of x minus 0 times itself-- 00:45:23.460 --> 00:45:24.965 align:middle line:90% I'll write that as a square. 00:45:24.965 --> 00:45:28.370 align:middle line:90% 00:45:28.370 --> 00:45:30.530 align:middle line:84% Write this here again-- this here, 00:45:30.530 --> 00:45:35.750 align:middle line:84% this is just g of x0 times g of x0. 00:45:35.750 --> 00:45:40.360 align:middle line:90% 00:45:40.360 --> 00:45:44.475 align:middle line:90% So now let's try to prove that. 00:45:44.475 --> 00:46:09.550 align:middle line:90% 00:46:09.550 --> 00:46:14.380 align:middle line:84% So we're looking at f of x like this. 00:46:14.380 --> 00:46:19.540 align:middle line:84% At x minus-- so this is the difference version-- 00:46:19.540 --> 00:46:24.920 align:middle line:90% at x0 divided by x minus x0. 00:46:24.920 --> 00:46:26.530 align:middle line:90% That's the difference equation. 00:46:26.530 --> 00:46:40.490 align:middle line:84% And so written out, this is f of x g of x minus f of x0 00:46:40.490 --> 00:46:46.580 align:middle line:90% divided by g of x0 x minus x0. 00:46:46.580 --> 00:46:52.110 align:middle line:84% So let me put this on a fraction with a common denominator. 00:46:52.110 --> 00:46:56.730 align:middle line:84% So if I do that, then this thing here is-- 00:46:56.730 --> 00:47:01.940 align:middle line:84% so then I'm multiplying by g of x0 00:47:01.940 --> 00:47:05.690 align:middle line:84% both in the numerator and the denominator here. 00:47:05.690 --> 00:47:12.350 align:middle line:84% So I'm getting f of x times g of x0. 00:47:12.350 --> 00:47:14.690 align:middle line:84% And then here, I'm multiplying by g 00:47:14.690 --> 00:47:18.470 align:middle line:84% of x in both numerator and denominator, so getting 00:47:18.470 --> 00:47:22.670 align:middle line:90% this thing here, g of x. 00:47:22.670 --> 00:47:33.230 align:middle line:84% And then it becomes x minus x0 and then g of x, g of x0, 00:47:33.230 --> 00:47:33.775 align:middle line:90% like that. 00:47:33.775 --> 00:47:37.260 align:middle line:90% 00:47:37.260 --> 00:47:43.900 align:middle line:84% And so now you can write this thing here as-- 00:47:43.900 --> 00:47:49.770 align:middle line:84% and so now you do the same trick as before. 00:47:49.770 --> 00:47:54.540 align:middle line:84% I mean, I could have gotten a way slightly easier 00:47:54.540 --> 00:47:59.730 align:middle line:84% with looking at a special case of this quotient rule, 00:47:59.730 --> 00:48:02.260 align:middle line:84% where f was a constant function 1. 00:48:02.260 --> 00:48:04.480 align:middle line:90% So I could do that first. 00:48:04.480 --> 00:48:07.890 align:middle line:84% And then the general case, I could then from-- 00:48:07.890 --> 00:48:13.750 align:middle line:84% get from this special case together with Leibniz rule. 00:48:13.750 --> 00:48:16.420 align:middle line:84% It's most often I want to do it together anyway, 00:48:16.420 --> 00:48:18.400 align:middle line:84% but if you have to do it together, 00:48:18.400 --> 00:48:22.050 align:middle line:84% you have to use the same trick that we just did. 00:48:22.050 --> 00:48:39.280 align:middle line:84% So we're using that over here, so f of g of x0 minus 00:48:39.280 --> 00:48:57.520 align:middle line:84% and then f of x0, g of x0, Plus f of x0 g of x0 00:48:57.520 --> 00:49:03.170 align:middle line:84% minus f of x0 g of x-- so this is, again, 00:49:03.170 --> 00:49:05.900 align:middle line:84% where I'm subtracting in the denominator or something, 00:49:05.900 --> 00:49:09.220 align:middle line:84% and then I'm adding it back up immediately. 00:49:09.220 --> 00:49:19.260 align:middle line:90% So I have this, like that. 00:49:19.260 --> 00:49:21.940 align:middle line:90% 00:49:21.940 --> 00:49:27.800 align:middle line:84% And so now if you do that, then you see that-- 00:49:27.800 --> 00:49:33.080 align:middle line:84% and now you split it up, again, into two fractions. 00:49:33.080 --> 00:49:35.630 align:middle line:90% So we split it up. 00:49:35.630 --> 00:49:40.490 align:middle line:84% And so this thing here, this thing here, but there, 00:49:40.490 --> 00:49:43.590 align:middle line:84% you see that there's a g of x here that factor out. 00:49:43.590 --> 00:49:47.420 align:middle line:90% So this is equal to g of x0. 00:49:47.420 --> 00:49:54.080 align:middle line:84% And then you have f of x minus f of x0 and then 00:49:54.080 --> 00:50:03.680 align:middle line:84% x minus x0, g of x, g of x0, like that. 00:50:03.680 --> 00:50:09.020 align:middle line:84% And for the second one, you see that there, you also 00:50:09.020 --> 00:50:11.120 align:middle line:90% have something that factor out. 00:50:11.120 --> 00:50:17.420 align:middle line:90% And this is f of x0, like that. 00:50:17.420 --> 00:50:23.840 align:middle line:84% And you're getting g of x0 minus g 00:50:23.840 --> 00:50:37.840 align:middle line:84% of x over x minus x0, g of x, g of x0, like that. 00:50:37.840 --> 00:50:46.030 align:middle line:84% And so now I can make this looks nicer. 00:50:46.030 --> 00:50:50.440 align:middle line:84% I can, this one here, cancel with that. 00:50:50.440 --> 00:50:53.050 align:middle line:84% And so I can write this thing here 00:50:53.050 --> 00:51:03.820 align:middle line:84% as 1 over g of x f of x minus f of x0 divided by x minus x0. 00:51:03.820 --> 00:51:06.150 align:middle line:90% So let's take care of this here. 00:51:06.150 --> 00:51:08.710 align:middle line:90% 00:51:08.710 --> 00:51:16.100 align:middle line:84% Here, I will write it as I will take this one here out. 00:51:16.100 --> 00:51:26.200 align:middle line:84% So I'll write it as f of x0 of x times g of x0. 00:51:26.200 --> 00:51:29.090 align:middle line:84% And then I'm left with this thing here. 00:51:29.090 --> 00:51:31.430 align:middle line:84% And what I don't love about this thing here 00:51:31.430 --> 00:51:34.340 align:middle line:84% is it's not quite the difference quotient 00:51:34.340 --> 00:51:37.070 align:middle line:84% because the order of these two is wrong. 00:51:37.070 --> 00:51:39.930 align:middle line:90% So I switched the order. 00:51:39.930 --> 00:51:46.170 align:middle line:84% But if I switch the order, then I have to pick up a minus sign. 00:51:46.170 --> 00:51:47.880 align:middle line:90% So I get minus here. 00:51:47.880 --> 00:51:58.650 align:middle line:84% And now it is g of x, g of x0, x minus x0, like that. 00:51:58.650 --> 00:52:05.150 align:middle line:84% And now you see that as x goes to x0, 00:52:05.150 --> 00:52:12.620 align:middle line:84% so now as x goes to x0, well, this one here, the function 00:52:12.620 --> 00:52:13.530 align:middle line:90% was differentiable. 00:52:13.530 --> 00:52:14.730 align:middle line:90% So it's also continuous. 00:52:14.730 --> 00:52:18.740 align:middle line:90% So this goes to 1 over g of x0. 00:52:18.740 --> 00:52:25.640 align:middle line:84% This one here just goes to the derivative at x0. 00:52:25.640 --> 00:52:29.010 align:middle line:84% This one here goes to-- again, because the function 00:52:29.010 --> 00:52:33.540 align:middle line:84% is continuous, this goes to minus f of x0. 00:52:33.540 --> 00:52:36.495 align:middle line:84% And then it's what I wrote before as g squared. 00:52:36.495 --> 00:52:40.920 align:middle line:90% It's g of x0 times itself. 00:52:40.920 --> 00:52:47.910 align:middle line:84% And then this thing here goes to the derivative of x0 at 0. 00:52:47.910 --> 00:52:53.410 align:middle line:84% And now if I want to, I can rewrite this thing here. 00:52:53.410 --> 00:52:56.470 align:middle line:84% If I want to put it back on the common denominator, 00:52:56.470 --> 00:53:00.570 align:middle line:84% then I have to multiply in the numerator and denominator 00:53:00.570 --> 00:53:02.320 align:middle line:90% by g of x0. 00:53:02.320 --> 00:53:10.170 align:middle line:84% If I do this, I get f prime of x0, g of x0, 00:53:10.170 --> 00:53:15.640 align:middle line:84% and then I get g squared of x0 in the denominator. 00:53:15.640 --> 00:53:25.170 align:middle line:84% And then here it's just those two minus f of x0 g prime of x0. 00:53:25.170 --> 00:53:29.440 align:middle line:84% And so you see that this here is exactly this formula 00:53:29.440 --> 00:53:31.540 align:middle line:90% here, this quotient rule. 00:53:31.540 --> 00:53:38.290 align:middle line:84% So this proof, that if you take two functions, the one 00:53:38.290 --> 00:53:41.270 align:middle line:84% you have in the denominator, if that is nonvanishing, 00:53:41.270 --> 00:53:43.600 align:middle line:90% So You can form the quotient. 00:53:43.600 --> 00:53:47.300 align:middle line:84% And if each of these two functions is differentiable, 00:53:47.300 --> 00:53:50.150 align:middle line:84% then the quotient is also differentiable. 00:53:50.150 --> 00:53:53.600 align:middle line:84% And this is the formula for the derivative. 00:53:53.600 --> 00:53:57.250 align:middle line:84% So now why is this already really, really useful? 00:53:57.250 --> 00:54:11.440 align:middle line:84% So we calculated-- so we started with the example of where 00:54:11.440 --> 00:54:13.880 align:middle line:90% f is a constant. 00:54:13.880 --> 00:54:17.050 align:middle line:84% And we saw that it's differentiable, 00:54:17.050 --> 00:54:19.060 align:middle line:90% and the derivative is 0. 00:54:19.060 --> 00:54:24.740 align:middle line:84% Then the second example we had was where the function was x, 00:54:24.740 --> 00:54:26.810 align:middle line:84% and we saw that it was differentiable 00:54:26.810 --> 00:54:29.580 align:middle line:90% and the derivative is 1. 00:54:29.580 --> 00:54:32.160 align:middle line:90% If you now take-- 00:54:32.160 --> 00:54:36.570 align:middle line:90% if you now take ax times-- 00:54:36.570 --> 00:54:39.860 align:middle line:84% so let's just take ax as the next example. 00:54:39.860 --> 00:54:45.300 align:middle line:84% This is like a constant function times this function. 00:54:45.300 --> 00:54:48.140 align:middle line:84% So Leibniz rule-- so if f of x is 00:54:48.140 --> 00:54:51.750 align:middle line:84% equal to a times x, where A is a constant, 00:54:51.750 --> 00:54:55.040 align:middle line:84% Leibniz will say that the product of two differentiable 00:54:55.040 --> 00:54:59.480 align:middle line:84% function is differentiable, and the derivative of this 00:54:59.480 --> 00:55:01.398 align:middle line:90% is the derivative of that. 00:55:01.398 --> 00:55:03.690 align:middle line:84% But the derivative of that, that's a constant function. 00:55:03.690 --> 00:55:04.380 align:middle line:90% So that's 0. 00:55:04.380 --> 00:55:06.300 align:middle line:84% So I don't even have to think about that. 00:55:06.300 --> 00:55:09.710 align:middle line:84% So it's just the a times the derivative 00:55:09.710 --> 00:55:10.710 align:middle line:90% of the second function. 00:55:10.710 --> 00:55:11.460 align:middle line:90% But that's one. 00:55:11.460 --> 00:55:15.660 align:middle line:84% So this here is the derivative of that. 00:55:15.660 --> 00:55:21.440 align:middle line:84% And the next example would be that you're looking at ax plus, 00:55:21.440 --> 00:55:22.765 align:middle line:90% say, b. 00:55:22.765 --> 00:55:25.050 align:middle line:90% f of x is equal to this. 00:55:25.050 --> 00:55:28.450 align:middle line:90% Then you can use the sum plus-- 00:55:28.450 --> 00:55:31.030 align:middle line:90% we know what each of these is. 00:55:31.030 --> 00:55:34.300 align:middle line:84% So you get that this one here is also differentiable. 00:55:34.300 --> 00:55:38.790 align:middle line:84% And the derivative of this function here is also just a. 00:55:38.790 --> 00:55:43.150 align:middle line:84% And you can continue like this, that if you take-- 00:55:43.150 --> 00:55:47.500 align:middle line:84% and if you now have a polynomial, 00:55:47.500 --> 00:55:50.640 align:middle line:84% suppose you take a to the power n. 00:55:50.640 --> 00:55:55.950 align:middle line:84% So if f of x is x to the power n, 00:55:55.950 --> 00:55:58.140 align:middle line:84% then you can think about this thing 00:55:58.140 --> 00:56:03.330 align:middle line:84% here as x times x times x times x. 00:56:03.330 --> 00:56:05.800 align:middle line:84% And you can use the Leibniz rule. 00:56:05.800 --> 00:56:08.770 align:middle line:84% And so if you use the Leibniz rule, 00:56:08.770 --> 00:56:12.720 align:middle line:84% so you just do it first in the case where f of x 00:56:12.720 --> 00:56:16.660 align:middle line:84% is equal to x squared, you can think about this as x times x. 00:56:16.660 --> 00:56:18.970 align:middle line:84% These are two differentiable functions. 00:56:18.970 --> 00:56:23.190 align:middle line:84% So the derivative here is now the derivative of this. 00:56:23.190 --> 00:56:26.130 align:middle line:84% So that's 1 times the other function. 00:56:26.130 --> 00:56:30.480 align:middle line:84% Plus you keep this one here and take the derivative of that. 00:56:30.480 --> 00:56:31.590 align:middle line:90% So this is this. 00:56:31.590 --> 00:56:35.420 align:middle line:90% So this is 2 times x. 00:56:35.420 --> 00:56:40.520 align:middle line:84% And so if you take this function here using Leibniz rule. 00:56:40.520 --> 00:56:43.700 align:middle line:84% In this case, the derivative is differentiable, 00:56:43.700 --> 00:56:50.065 align:middle line:84% and the derivative is n times x to the power n minus 1. 00:56:50.065 --> 00:56:55.340 align:middle line:90% 00:56:55.340 --> 00:56:58.280 align:middle line:84% And in fact, even more generally, 00:56:58.280 --> 00:57:05.300 align:middle line:84% just using the sum and the Leibniz rule, 00:57:05.300 --> 00:57:13.220 align:middle line:84% if you take any polynomial an xn plus an minus 1 xn 00:57:13.220 --> 00:57:20.130 align:middle line:84% minus 1 down to the last thing is being a constant-- if you 00:57:20.130 --> 00:57:24.000 align:middle line:84% take a polynomial like this just using the Leibniz 00:57:24.000 --> 00:57:29.190 align:middle line:84% rule and the sum rule, using those repeatedly, 00:57:29.190 --> 00:57:32.190 align:middle line:84% we know that this is differentiable 00:57:32.190 --> 00:57:34.890 align:middle line:84% and the derivative is just like the sum 00:57:34.890 --> 00:57:36.190 align:middle line:90% of the derivative of these. 00:57:36.190 --> 00:57:42.900 align:middle line:84% So this is going to be an times n times x to the power n minus 1 00:57:42.900 --> 00:57:51.420 align:middle line:84% plus n minus 1 times n minus 1 xn minus 2. 00:57:51.420 --> 00:57:54.550 align:middle line:84% And then this one here, the derivative is 0. 00:57:54.550 --> 00:57:58.590 align:middle line:84% So the last one you're getting something in nontrivial form 00:57:58.590 --> 00:58:01.840 align:middle line:90% is a1. 00:58:01.840 --> 00:58:04.590 align:middle line:84% So this is just using sum and Leibniz rule. 00:58:04.590 --> 00:58:21.190 align:middle line:84% And so this is just using the sum plus Leibniz rule 00:58:21.190 --> 00:58:21.910 align:middle line:90% repeatedly. 00:58:21.910 --> 00:58:28.400 align:middle line:90% 00:58:28.400 --> 00:58:33.430 align:middle line:84% Now likewise, if you take a rational function, 00:58:33.430 --> 00:58:36.560 align:middle line:84% so meaning that if you take something-- 00:58:36.560 --> 00:58:41.800 align:middle line:84% if f of x is equal to some polynomial-- 00:58:41.800 --> 00:58:45.760 align:middle line:90% 00:58:45.760 --> 00:58:54.640 align:middle line:84% so let's go p of x divided by q of x, so where this here 00:58:54.640 --> 00:59:03.790 align:middle line:90% is a polynomial and so is that. 00:59:03.790 --> 00:59:07.540 align:middle line:84% So now you can use the quotient rule. 00:59:07.540 --> 00:59:15.680 align:middle line:90% 00:59:15.680 --> 00:59:17.180 align:middle line:84% So the quotient rule will tell you 00:59:17.180 --> 00:59:21.170 align:middle line:84% that as long as you are looking at point where this here 00:59:21.170 --> 00:59:25.040 align:middle line:84% does not vanish, so then the quotient here is also 00:59:25.040 --> 00:59:28.250 align:middle line:84% differentiable, and it gives you a formula 00:59:28.250 --> 00:59:31.010 align:middle line:84% for the derivative in terms of the derivative of each 00:59:31.010 --> 00:59:31.950 align:middle line:90% of these two. 00:59:31.950 --> 00:59:35.480 align:middle line:84% And each of those two, they can be calculated like that. 00:59:35.480 --> 00:59:42.200 align:middle line:84% So you see, already now, we have quite the derivative 00:59:42.200 --> 00:59:43.620 align:middle line:90% of a lot of functions. 00:59:43.620 --> 00:59:47.040 align:middle line:84% We have of all function and of all rational functions. 00:59:47.040 --> 00:59:51.020 align:middle line:84% Rational functions, again, are quotients of a polynomial 00:59:51.020 --> 00:59:54.770 align:middle line:90% by other polynomial. 00:59:54.770 --> 01:00:00.680 align:middle line:84% So now the next rule that we want to talk about 01:00:00.680 --> 01:00:03.340 align:middle line:90% is the chain rule. 01:00:03.340 --> 01:00:07.220 align:middle line:90% 01:00:07.220 --> 01:00:09.950 align:middle line:90% So what is the chain rule? 01:00:09.950 --> 01:00:14.550 align:middle line:84% So the chain rule is a little bit different from those three 01:00:14.550 --> 01:00:16.410 align:middle line:90% rules here. 01:00:16.410 --> 01:00:22.200 align:middle line:84% And it is where you take the composition of functions. 01:00:22.200 --> 01:00:24.400 align:middle line:90% So the next is the chain rule. 01:00:24.400 --> 01:00:30.060 align:middle line:90% 01:00:30.060 --> 01:00:40.900 align:middle line:84% And again, this is about composition of functions. 01:00:40.900 --> 01:00:44.340 align:middle line:90% 01:00:44.340 --> 01:00:51.990 align:middle line:84% So I have a function f that goes, say, from-- 01:00:51.990 --> 01:00:54.180 align:middle line:84% to say that it goes from some interval a 01:00:54.180 --> 01:01:00.840 align:middle line:84% to b into another interval from c to d. 01:01:00.840 --> 01:01:06.300 align:middle line:84% And then I have a function g that goes from this interval 01:01:06.300 --> 01:01:20.450 align:middle line:84% here into R because g is defined whatever the image of f 01:01:20.450 --> 01:01:25.950 align:middle line:84% always, if it's in the set, this is where g is defined. 01:01:25.950 --> 01:01:33.350 align:middle line:84% Then you have that you can form if you take some x in here-- 01:01:33.350 --> 01:01:39.450 align:middle line:84% so if x here is in a to b, then this here makes sense. 01:01:39.450 --> 01:01:43.020 align:middle line:90% And this is now an element here. 01:01:43.020 --> 01:01:45.120 align:middle line:90% So this is where g is defined. 01:01:45.120 --> 01:01:47.360 align:middle line:90% So this here makes sense. 01:01:47.360 --> 01:01:51.060 align:middle line:84% And this is now what we call the composition. 01:01:51.060 --> 01:01:52.920 align:middle line:90% And we write it like this. 01:01:52.920 --> 01:01:58.220 align:middle line:84% It's important to write it in the correct order. 01:01:58.220 --> 01:02:00.430 align:middle line:84% And so this is the composition function. 01:02:00.430 --> 01:02:08.330 align:middle line:90% 01:02:08.330 --> 01:02:10.870 align:middle line:84% So now the chain rule is the following. 01:02:10.870 --> 01:02:23.730 align:middle line:90% 01:02:23.730 --> 01:02:25.790 align:middle line:84% So the chain rule is a foreign theorem. 01:02:25.790 --> 01:02:36.480 align:middle line:90% 01:02:36.480 --> 01:02:53.310 align:middle line:84% Then if f is differentiable at x0 01:02:53.310 --> 01:03:09.010 align:middle line:84% and g is differentiable at y0, where 01:03:09.010 --> 01:03:17.660 align:middle line:84% y0 is equal to the value of f at x0, then the composition-- 01:03:17.660 --> 01:03:20.650 align:middle line:84% then the composition of these two functions 01:03:20.650 --> 01:03:25.720 align:middle line:84% is differentiable at x0, and the derivative 01:03:25.720 --> 01:03:31.270 align:middle line:84% is equal to the derivative of g at y0 times 01:03:31.270 --> 01:03:34.940 align:middle line:90% the derivative of f at x0. 01:03:34.940 --> 01:03:36.790 align:middle line:90% So this is the chain rule. 01:03:36.790 --> 01:03:39.760 align:middle line:84% The chain rule says that the composition is also 01:03:39.760 --> 01:03:43.720 align:middle line:84% differentiable at x0, and this is 01:03:43.720 --> 01:03:47.840 align:middle line:90% the formula for the derivative. 01:03:47.840 --> 01:03:50.230 align:middle line:90% Let's try to prove that. 01:03:50.230 --> 01:03:54.940 align:middle line:84% And so, in this case, it's useful to divide up 01:03:54.940 --> 01:03:56.883 align:middle line:90% to two cases. 01:03:56.883 --> 01:04:03.580 align:middle line:90% 01:04:03.580 --> 01:04:07.520 align:middle line:84% And the two cases to divide up to-- so let 01:04:07.520 --> 01:04:12.030 align:middle line:84% me just look at a special case first, 01:04:12.030 --> 01:04:16.940 align:middle line:84% and then once we understand that, the general case will 01:04:16.940 --> 01:04:17.960 align:middle line:90% be clear. 01:04:17.960 --> 01:04:20.370 align:middle line:90% So let's make one assumption. 01:04:20.370 --> 01:04:21.415 align:middle line:90% So let's try to prove it. 01:04:21.415 --> 01:04:25.310 align:middle line:90% 01:04:25.310 --> 01:04:34.736 align:middle line:84% But we assume first that the derivative-- 01:04:34.736 --> 01:04:37.250 align:middle line:90% 01:04:37.250 --> 01:04:41.630 align:middle line:84% I'm going to assume that the derivative at x0 01:04:41.630 --> 01:04:43.410 align:middle line:90% is not equal to 0. 01:04:43.410 --> 01:04:48.350 align:middle line:84% So I'm assuming that, first, what does this guarantee me? 01:04:48.350 --> 01:04:53.150 align:middle line:84% Well, remember that the derivative is the limit 01:04:53.150 --> 01:04:54.355 align:middle line:90% of this difference quotient. 01:04:54.355 --> 01:04:58.220 align:middle line:90% 01:04:58.220 --> 01:05:01.880 align:middle line:90% So this is the limit. 01:05:01.880 --> 01:05:06.060 align:middle line:84% If this here is not equal to 0, this 01:05:06.060 --> 01:05:09.790 align:middle line:84% means that when x is very close to x0, 01:05:09.790 --> 01:05:15.900 align:middle line:84% this thing here is very close to this thing that 01:05:15.900 --> 01:05:17.530 align:middle line:90% is not equal to 0. 01:05:17.530 --> 01:05:22.140 align:middle line:84% But then it means that if you are very close to x0, 01:05:22.140 --> 01:05:25.660 align:middle line:90% f of x cannot be equal to. 01:05:25.660 --> 01:05:31.800 align:middle line:84% So if you have this here, this implies that f of x 01:05:31.800 --> 01:05:36.360 align:middle line:90% is not equal to x0 for-- 01:05:36.360 --> 01:05:52.320 align:middle line:84% sorry, it's not equal to f of x0 for x sufficiently close to x0. 01:05:52.320 --> 01:05:55.540 align:middle line:84% I'm always assuming, of course, that x is not 01:05:55.540 --> 01:05:59.820 align:middle line:90% but x is not equal to 0. 01:05:59.820 --> 01:06:04.760 align:middle line:84% So I'm going to first make this assumption. 01:06:04.760 --> 01:06:07.630 align:middle line:84% And if I make this assumption, what I get out of it 01:06:07.630 --> 01:06:11.030 align:middle line:84% is that this thing here is not equal to that. 01:06:11.030 --> 01:06:12.740 align:middle line:90% Why is this useful? 01:06:12.740 --> 01:06:17.590 align:middle line:84% Because now I can actually look at this difference and I can 01:06:17.590 --> 01:06:18.800 align:middle line:90% divide by it. 01:06:18.800 --> 01:06:33.220 align:middle line:84% So I'm going to just to make things a little bit clearer. 01:06:33.220 --> 01:06:35.153 align:middle line:90% So I have my function. 01:06:35.153 --> 01:06:39.400 align:middle line:90% 01:06:39.400 --> 01:06:42.220 align:middle line:90% So this here is f. 01:06:42.220 --> 01:06:45.010 align:middle line:90% And here I have g. 01:06:45.010 --> 01:06:47.500 align:middle line:90% Here, I have x0. 01:06:47.500 --> 01:06:52.780 align:middle line:84% But under f, it's taken to this, which is f of x0. 01:06:52.780 --> 01:06:56.900 align:middle line:84% But this I also will denote by y0. 01:06:56.900 --> 01:06:59.930 align:middle line:84% So I'm denoting just to make things clearer, 01:06:59.930 --> 01:07:03.870 align:middle line:90% y0 is just the value here. 01:07:03.870 --> 01:07:09.180 align:middle line:90% And then I will use for-- 01:07:09.180 --> 01:07:13.260 align:middle line:90% f of x I will denote by y. 01:07:13.260 --> 01:07:17.560 align:middle line:84% So now I'm looking at the difference quotient. 01:07:17.560 --> 01:07:20.550 align:middle line:84% So I'm looking at f of-- sorry, I'm 01:07:20.550 --> 01:07:34.650 align:middle line:84% looking at g of f of x minus g of f of x0 x minus x0. 01:07:34.650 --> 01:07:37.620 align:middle line:84% That's what I'm supposed to look at. 01:07:37.620 --> 01:07:45.160 align:middle line:84% And I can write this thing here as g of y 01:07:45.160 --> 01:07:47.680 align:middle line:90% because this here I also call y. 01:07:47.680 --> 01:07:53.190 align:middle line:84% So you can write it like this g of y0. 01:07:53.190 --> 01:07:58.930 align:middle line:84% And then I will write it as y minus y0, 01:07:58.930 --> 01:08:01.330 align:middle line:84% but I didn't have this in the denominator 01:08:01.330 --> 01:08:05.860 align:middle line:84% up here, so I have to multiply it back on. 01:08:05.860 --> 01:08:10.630 align:middle line:84% And then I still have to divide by that. 01:08:10.630 --> 01:08:13.060 align:middle line:90% So I can write it like this. 01:08:13.060 --> 01:08:15.357 align:middle line:90% This here I'm happy with. 01:08:15.357 --> 01:08:16.899 align:middle line:84% Get to this in just a second, but I'm 01:08:16.899 --> 01:08:18.609 align:middle line:90% happy with this expression. 01:08:18.609 --> 01:08:23.359 align:middle line:84% The other one I really want to put it back in, in terms of f. 01:08:23.359 --> 01:08:25.005 align:middle line:90% So I'm writing this in here. 01:08:25.005 --> 01:08:33.399 align:middle line:90% 01:08:33.399 --> 01:08:35.319 align:middle line:84% And this thing here, I really want 01:08:35.319 --> 01:08:37.700 align:middle line:90% to put it back in terms of f. 01:08:37.700 --> 01:08:44.649 align:middle line:84% So if I do that, I write it as f of dx minus f of x0 x minus x0. 01:08:44.649 --> 01:08:47.560 align:middle line:90% 01:08:47.560 --> 01:08:56.859 align:middle line:84% But now you see that as x goes to x0, well, 01:08:56.859 --> 01:09:03.689 align:middle line:84% as x goes to x0, because the function f was continuous at x0, 01:09:03.689 --> 01:09:09.229 align:middle line:84% it means that if x goes to x0, then the value at x, which is y, 01:09:09.229 --> 01:09:11.069 align:middle line:90% will go to the value at x0. 01:09:11.069 --> 01:09:20.363 align:middle line:84% So when x goes to x0, we have that y goes to y0. 01:09:20.363 --> 01:09:23.779 align:middle line:90% 01:09:23.779 --> 01:09:31.290 align:middle line:84% So this means that this thing here, this here converges. 01:09:31.290 --> 01:09:34.050 align:middle line:84% That's just because the function f is differentiable. 01:09:34.050 --> 01:09:37.620 align:middle line:84% So this here will converge to the derivative. 01:09:37.620 --> 01:09:44.660 align:middle line:84% So the second factor here is converging to f prime at x0. 01:09:44.660 --> 01:09:47.660 align:middle line:84% The first factor is-- but now you see that we just 01:09:47.660 --> 01:09:50.729 align:middle line:90% argued that y goes to y0. 01:09:50.729 --> 01:09:53.340 align:middle line:84% So this is also a difference quotient. 01:09:53.340 --> 01:09:55.070 align:middle line:84% But it's a difference quotient for g. 01:09:55.070 --> 01:09:58.710 align:middle line:84% And since y goes to y0, this difference quotient 01:09:58.710 --> 01:10:04.365 align:middle line:84% here goes to the derivative of g at y0. 01:10:04.365 --> 01:10:07.740 align:middle line:90% 01:10:07.740 --> 01:10:11.110 align:middle line:84% But that's exactly the formula that we wanted to prove, 01:10:11.110 --> 01:10:13.530 align:middle line:84% except that we made one assumption 01:10:13.530 --> 01:10:21.180 align:middle line:84% that the derivative of f at x0 was not equal to 0. 01:10:21.180 --> 01:10:26.400 align:middle line:84% We made that assumption only so that this number here 01:10:26.400 --> 01:10:28.540 align:middle line:90% was not equal to 0. 01:10:28.540 --> 01:10:32.250 align:middle line:90% And so we could divide by it. 01:10:32.250 --> 01:10:40.890 align:middle line:84% But actually, otherwise, if you, for some reason, 01:10:40.890 --> 01:10:48.300 align:middle line:84% if y was not equal to y0, but then 01:10:48.300 --> 01:10:51.000 align:middle line:84% this formula here would still hold. 01:10:51.000 --> 01:11:00.190 align:middle line:84% So it's only the possibility that y could be equal to y0. 01:11:00.190 --> 01:11:02.590 align:middle line:84% But now you see that if you were just 01:11:02.590 --> 01:11:07.970 align:middle line:84% looking at the difference quotient g of f-- 01:11:07.970 --> 01:11:13.900 align:middle line:90% g of y minus-- 01:11:13.900 --> 01:11:19.690 align:middle line:90% sorry, g of f of x minus-- 01:11:19.690 --> 01:11:22.070 align:middle line:84% so this is the difference quotient. 01:11:22.070 --> 01:11:31.910 align:middle line:84% That is the one we really have to look at for the composition. 01:11:31.910 --> 01:11:36.670 align:middle line:90% And this is what we also call-- 01:11:36.670 --> 01:11:39.730 align:middle line:84% this thing here is what we called y0. 01:11:39.730 --> 01:11:41.775 align:middle line:84% This thing here is what we call y. 01:11:41.775 --> 01:11:45.250 align:middle line:90% 01:11:45.250 --> 01:11:54.690 align:middle line:84% And the only case we haven't dealt with is where f prime of 0 01:11:54.690 --> 01:12:03.540 align:middle line:84% is equal to 0, and y is equal to y0. 01:12:03.540 --> 01:12:06.420 align:middle line:90% This here, again, was f of x. 01:12:06.420 --> 01:12:09.390 align:middle line:84% And this here was equal to f of x0. 01:12:09.390 --> 01:12:15.210 align:middle line:84% So this is only this case here we haven't dealt with. 01:12:15.210 --> 01:12:21.840 align:middle line:84% But you see that if it so happened that f of x 01:12:21.840 --> 01:12:25.680 align:middle line:84% was equal to f of 0, then this difference quotient here, 01:12:25.680 --> 01:12:28.800 align:middle line:84% the denominator would trivially be 0. 01:12:28.800 --> 01:12:33.720 align:middle line:84% And so you want to prove we have that if this here was not 01:12:33.720 --> 01:12:36.180 align:middle line:84% equal to 0, then we have that this is 01:12:36.180 --> 01:12:40.230 align:middle line:90% converging to the derivative. 01:12:40.230 --> 01:12:44.220 align:middle line:84% And the only case we have to look at where this derivative is 01:12:44.220 --> 01:12:48.010 align:middle line:90% 0-- so we are claiming that-- 01:12:48.010 --> 01:12:50.580 align:middle line:84% so the only case we are left with that we 01:12:50.580 --> 01:12:57.640 align:middle line:84% haven't dealt with is, first of all, where the derivative-- 01:12:57.640 --> 01:12:58.790 align:middle line:90% so let's see. 01:12:58.790 --> 01:13:03.910 align:middle line:84% So let me just write down here clearly. 01:13:03.910 --> 01:13:14.295 align:middle line:84% The remaining case for the chain rule-- 01:13:14.295 --> 01:13:20.620 align:middle line:90% 01:13:20.620 --> 01:13:25.525 align:middle line:84% so the many cases that the derivative at x0 is 0. 01:13:25.525 --> 01:13:29.680 align:middle line:90% 01:13:29.680 --> 01:13:33.490 align:middle line:84% And we have already observed that if the derivative is 01:13:33.490 --> 01:13:42.140 align:middle line:90% 0 but y is not equal to y0-- 01:13:42.140 --> 01:13:46.870 align:middle line:84% if this is the case, then actually g 01:13:46.870 --> 01:13:55.370 align:middle line:90% of f of x minus g of f of x0-- 01:13:55.370 --> 01:13:59.660 align:middle line:90% this thing here still converged. 01:13:59.660 --> 01:14:03.860 align:middle line:84% In this case here, where this is the case, 01:14:03.860 --> 01:14:06.990 align:middle line:90% this has still converged to 0. 01:14:06.990 --> 01:14:12.480 align:middle line:84% So we still have the chain rule in that case 01:14:12.480 --> 01:14:14.460 align:middle line:90% if this here was not the case. 01:14:14.460 --> 01:14:16.520 align:middle line:84% The only thing we have to worry about 01:14:16.520 --> 01:14:19.980 align:middle line:90% is what if y was equal to y0? 01:14:19.980 --> 01:14:24.500 align:middle line:84% But if y was equal to y0, when we're looking at the difference 01:14:24.500 --> 01:14:25.500 align:middle line:90% quotient. 01:14:25.500 --> 01:14:27.930 align:middle line:84% This is the difference quotient, after all. 01:14:27.930 --> 01:14:31.770 align:middle line:84% And so in that case, this thing here is y0. 01:14:31.770 --> 01:14:33.030 align:middle line:90% This is y0. 01:14:33.030 --> 01:14:35.870 align:middle line:84% But in that case, the difference quotient 01:14:35.870 --> 01:14:37.860 align:middle line:90% is actually zero already. 01:14:37.860 --> 01:14:42.930 align:middle line:84% So of course, it's going to be very close to 0. 01:14:42.930 --> 01:14:44.310 align:middle line:90% It actually is 0. 01:14:44.310 --> 01:14:49.170 align:middle line:84% So you see that the remaining case, that's trivial. 01:14:49.170 --> 01:14:52.020 align:middle line:84% But we wanted to first exclude it 01:14:52.020 --> 01:14:55.360 align:middle line:84% because we wanted to be able to divide by this difference. 01:14:55.360 --> 01:15:01.410 align:middle line:84% And then once we have seen that under that assumption, 01:15:01.410 --> 01:15:04.980 align:middle line:84% we got the formula we wanted, then we go back 01:15:04.980 --> 01:15:13.050 align:middle line:84% and then we say that not only the remaining issue, first 01:15:13.050 --> 01:15:17.440 align:middle line:84% of all, was the case where the derivative was 0, 01:15:17.440 --> 01:15:24.060 align:middle line:84% but in fact, the only case that was potentially a problem 01:15:24.060 --> 01:15:26.410 align:middle line:90% was when y was equal to y0. 01:15:26.410 --> 01:15:29.280 align:middle line:84% But for that, the difference quotient 01:15:29.280 --> 01:15:32.260 align:middle line:90% was actually trivially 0. 01:15:32.260 --> 01:15:34.505 align:middle line:90% So that's the chain rule. 01:15:34.505 --> 01:15:38.250 align:middle line:90% 01:15:38.250 --> 01:15:41.130 align:middle line:84% And so we only have a few minutes left. 01:15:41.130 --> 01:15:45.360 align:middle line:84% And I won't quite be able to get through all of the stuff. 01:15:45.360 --> 01:15:47.490 align:middle line:84% But I can get through one more lemma. 01:15:47.490 --> 01:15:50.480 align:middle line:90% 01:15:50.480 --> 01:15:54.525 align:middle line:84% And then I'll pick up from here next time. 01:15:54.525 --> 01:15:58.960 align:middle line:90% 01:15:58.960 --> 01:16:00.620 align:middle line:90% The notes are already posted. 01:16:00.620 --> 01:16:06.220 align:middle line:84% But so the last bit of the notes we'll do next time. 01:16:06.220 --> 01:16:10.810 align:middle line:90% So the following lemma here-- 01:16:10.810 --> 01:16:18.100 align:middle line:84% so if you have a function, say, from R to R, 01:16:18.100 --> 01:16:31.900 align:middle line:84% and you have that x0 is a local max or min-- 01:16:31.900 --> 01:16:37.850 align:middle line:84% so f is a function from R to R that is differentiable. 01:16:37.850 --> 01:16:44.540 align:middle line:90% 01:16:44.540 --> 01:16:57.950 align:middle line:84% And x0 is a local max or local min for f, 01:16:57.950 --> 01:17:08.300 align:middle line:84% then the derivative at x0 is equal to 0. 01:17:08.300 --> 01:17:12.080 align:middle line:84% Let me remind you what a local max and local min is. 01:17:12.080 --> 01:17:16.140 align:middle line:90% So if you take a function-- 01:17:16.140 --> 01:17:20.270 align:middle line:84% so local max, similarly for local min, 01:17:20.270 --> 01:17:24.140 align:middle line:84% if you have a function here, you have here x0, 01:17:24.140 --> 01:17:30.120 align:middle line:84% a local max means that in a neighborhood of this point, 01:17:30.120 --> 01:17:34.860 align:middle line:84% then the value at x0 is the maximum. 01:17:34.860 --> 01:17:54.340 align:middle line:84% So local max at x0 means that there exists delta greater 01:17:54.340 --> 01:18:04.360 align:middle line:84% than 0 such that if you're looking at the interval 01:18:04.360 --> 01:18:11.650 align:middle line:84% to either side of x0, this interval, 01:18:11.650 --> 01:18:19.900 align:middle line:84% and you're looking at f here restricted to this subset, 01:18:19.900 --> 01:18:23.470 align:middle line:90% then x0 is a local max-- 01:18:23.470 --> 01:18:24.320 align:middle line:90% is a max. 01:18:24.320 --> 01:18:33.040 align:middle line:84% So this means that f of x0 is equal to the max of f, 01:18:33.040 --> 01:18:43.795 align:middle line:84% where x here is in this little interval here, like that. 01:18:43.795 --> 01:18:49.770 align:middle line:90% 01:18:49.770 --> 01:18:54.920 align:middle line:84% So the idea is that the function may look like this. 01:18:54.920 --> 01:18:57.510 align:middle line:90% And this here is x0. 01:18:57.510 --> 01:19:01.050 align:middle line:84% So you see that in a neighborhood like this. 01:19:01.050 --> 01:19:05.130 align:middle line:84% It looks like that you have that this x0 is a maximum. 01:19:05.130 --> 01:19:07.190 align:middle line:84% But that might even be somewhere else, 01:19:07.190 --> 01:19:09.890 align:middle line:90% where the value is much higher. 01:19:09.890 --> 01:19:15.360 align:middle line:84% So that's a local maximum, and similarly for a local minimum. 01:19:15.360 --> 01:19:18.630 align:middle line:84% And so I won't be able to actually prove this lemma. 01:19:18.630 --> 01:19:22.130 align:middle line:90% We will prove that next time. 01:19:22.130 --> 01:19:24.260 align:middle line:84% And then once we have that lemma, 01:19:24.260 --> 01:19:26.660 align:middle line:84% we'll prove what's called Rolle's theorem. 01:19:26.660 --> 01:19:29.970 align:middle line:84% And then we will use it to prove the mean value theorem. 01:19:29.970 --> 01:19:32.700 align:middle line:90% But I'll do that next time. 01:19:32.700 --> 01:19:37.225 align:middle line:84% But it's actually in the notes that I posted already. 01:19:37.225 --> 01:19:40.000 align:middle line:90%