WEBVTT 00:00:14.500 --> 00:00:18.652 In the last videos I talked about the derivatives of simple functions, 00:00:18.652 --> 00:00:22.455 and the goal was to have a clear picture or intuition to hold in 00:00:22.455 --> 00:00:26.199 your mind that actually explains where these formulas come from. 00:00:26.839 --> 00:00:31.387 But most of the functions you deal with in modeling the world involve mixing, 00:00:31.387 --> 00:00:35.176 combining, or tweaking these simple functions in some other way, 00:00:35.176 --> 00:00:39.782 so our next step is to understand how you take derivatives of more complicated 00:00:39.781 --> 00:00:40.539 combinations. 00:00:41.280 --> 00:00:43.995 Again, I don't want these to be something to memorize, 00:00:43.994 --> 00:00:47.599 I want you to have a clear picture in mind for where each one comes from. 00:00:49.520 --> 00:00:53.600 Now, this really boils down into three basic ways to combine functions. 00:00:54.100 --> 00:00:56.591 You can add them together, you can multiply them, 00:00:56.591 --> 00:00:59.780 and you can throw one inside the other, known as composing them. 00:01:00.600 --> 00:01:03.829 Sure, you could say subtracting them, but really that's just 00:01:03.829 --> 00:01:07.219 multiplying the second by negative one and adding them together. 00:01:08.239 --> 00:01:11.164 Likewise, dividing functions doesn't really add anything, 00:01:11.164 --> 00:01:14.844 because that's the same as plugging one inside the function, one over x, 00:01:14.843 --> 00:01:16.759 and then multiplying the two together. 00:01:17.659 --> 00:01:20.664 So really, most functions you come across just involve layering 00:01:20.664 --> 00:01:23.200 together these three different types of combinations, 00:01:23.200 --> 00:01:26.439 though there's not really a bound on how monstrous things can become. 00:01:27.099 --> 00:01:31.298 But as long as you know how derivatives play with just those three combination types, 00:01:31.299 --> 00:01:34.376 you'll always be able to take it step by step and peel through 00:01:34.376 --> 00:01:36.719 the layers for any kind of monstrous expression. 00:01:38.719 --> 00:01:42.540 So the question is, if you know the derivative of two functions, 00:01:42.540 --> 00:01:45.774 what is the derivative of their sum, of their product, 00:01:45.774 --> 00:01:48.420 and of the function composition between them? 00:01:50.319 --> 00:01:54.259 The sum rule is easiest, if somewhat tongue-twisting to say out loud. 00:01:54.840 --> 00:01:58.600 The derivative of a sum of two functions is the sum of their derivatives. 00:01:59.799 --> 00:02:03.700 But it's worth warming up with this example by really thinking through 00:02:03.700 --> 00:02:07.105 what it means to take a derivative of a sum of two functions, 00:02:07.105 --> 00:02:11.170 since the derivative patterns for products and function composition won't 00:02:11.169 --> 00:02:15.619 be so straightforward, and they're going to require this kind of deeper thinking. 00:02:16.699 --> 00:02:21.199 For example, let's think about this function f of x equals sine of x plus x squared. 00:02:22.199 --> 00:02:25.211 It's a function where, for every input, you add together 00:02:25.211 --> 00:02:27.959 the values of sine of x and x squared at that point. 00:02:29.759 --> 00:02:34.048 For example, let's say at x equals 0.5, the height of the sine 00:02:34.049 --> 00:02:38.201 graph is given by this vertical bar, and the height of the x 00:02:38.201 --> 00:02:42.560 squared parabola is given by this slightly smaller vertical bar. 00:02:44.379 --> 00:02:47.319 And their sum is the length you get by just stacking them together. 00:02:48.520 --> 00:02:53.939 For the derivative, you want to ask what happens as you nudge that input slightly, 00:02:53.938 --> 00:02:56.419 maybe increasing it up to 0.5 plus dx. 00:02:57.560 --> 00:03:02.920 The difference in the value of f between those two places is what we call df. 00:03:04.360 --> 00:03:08.978 And when you picture it like this, I think you'll agree that the total 00:03:08.978 --> 00:03:13.271 change in the height is whatever the change to the sine graph is, 00:03:13.270 --> 00:03:18.799 what we might call d sine of x, plus whatever the change to x squared is, dx squared. 00:03:22.240 --> 00:03:27.540 We know that the derivative of sine is cosine, and remember what that means. 00:03:27.919 --> 00:03:33.299 It means that this little change, d sine of x, is about cosine of x times dx. 00:03:33.780 --> 00:03:37.711 It's proportional to the size of our initial nudge dx, 00:03:37.711 --> 00:03:43.359 and the proportionality constant equals cosine of whatever input we started at. 00:03:43.979 --> 00:03:48.072 Likewise, because the derivative of x squared is 2x, 00:03:48.072 --> 00:03:53.939 the change in the height of the x squared graph is 2x times whatever dx was. 00:03:55.599 --> 00:04:00.475 So rearranging df divided by dx, the ratio of the tiny change to 00:04:00.475 --> 00:04:04.752 the sum function to the tiny change in x that caused it, 00:04:04.752 --> 00:04:10.079 is indeed cosine of x plus 2x, the sum of the derivatives of its parts. 00:04:11.520 --> 00:04:15.329 But like I said, things are a bit different for products, 00:04:15.329 --> 00:04:19.139 and let's think through why in terms of tiny nudges again. 00:04:20.060 --> 00:04:23.139 In this case, I don't think graphs are our best bet for visualizing things. 00:04:23.819 --> 00:04:27.040 Pretty commonly in math, at a lot of levels of math really, 00:04:27.040 --> 00:04:29.617 if you're dealing with a product of two things, 00:04:29.617 --> 00:04:32.140 it helps to understand it as some kind of area. 00:04:33.079 --> 00:04:36.014 In this case, maybe you try to configure some mental setup 00:04:36.014 --> 00:04:39.000 of a box where the side lengths are sine of x and x squared. 00:04:39.879 --> 00:04:41.040 But what would that mean? 00:04:42.319 --> 00:04:46.606 Well, since these are functions, you might think of those sides as adjustable, 00:04:46.607 --> 00:04:49.972 dependent on the value of x, which maybe you think of as this 00:04:49.971 --> 00:04:52.739 number that you can just freely adjust up and down. 00:04:53.740 --> 00:04:56.812 So getting a feel for what this means, focus on 00:04:56.812 --> 00:05:00.139 that top side who changes as the function sine of x. 00:05:01.060 --> 00:05:05.305 As you change this value of x up from 0, it increases up to 00:05:05.305 --> 00:05:09.127 a length of 1 as sine of x moves up towards its peak, 00:05:09.127 --> 00:05:13.939 and after that it starts to decrease as sine of x comes down from 1. 00:05:15.100 --> 00:05:18.580 And in the same way, that height there is always changing as x squared. 00:05:20.079 --> 00:05:25.800 So f of x, defined as the product of these two functions, is the area of this box. 00:05:27.060 --> 00:05:30.120 And for the derivative, let's think about how 00:05:30.120 --> 00:05:33.180 a tiny change to x by dx influences that area. 00:05:33.839 --> 00:05:36.279 What is that resulting change in area df? 00:05:39.000 --> 00:05:44.115 Well, the nudge dx caused that width to change by some small d sine of x, 00:05:44.115 --> 00:05:47.919 and it caused that height to change by some dx squared. 00:05:50.180 --> 00:05:53.649 And this gives us three little snippets of new area, 00:05:53.649 --> 00:05:58.033 a thin rectangle on the bottom whose area is its width, sine of x, 00:05:58.033 --> 00:06:00.259 times its thin height, dx squared. 00:06:01.779 --> 00:06:06.406 And there's this thin rectangle on the right, whose area is its height, 00:06:06.406 --> 00:06:09.299 x squared, times its thin width, d sine of x. 00:06:10.740 --> 00:06:14.139 And there's also this little bit in the corner, but we can ignore that. 00:06:14.439 --> 00:06:17.856 Its area is ultimately proportional to dx squared, 00:06:17.857 --> 00:06:22.480 and as we've seen before, that becomes negligible as dx goes to zero. 00:06:23.939 --> 00:06:27.375 I mean, this whole setup is very similar to what I showed last video, 00:06:27.375 --> 00:06:28.699 with the x squared diagram. 00:06:29.459 --> 00:06:32.704 And just like then, keep in mind that I'm using somewhat beefy 00:06:32.704 --> 00:06:35.899 changes here to draw things, just so we can actually see them. 00:06:36.360 --> 00:06:39.818 But in principle, dx is something very very small, 00:06:39.817 --> 00:06:44.699 and that means that dx squared and d sine of x are also very very small. 00:06:45.980 --> 00:06:51.175 So, applying what we know about the derivative of sine and of x squared, 00:06:51.175 --> 00:06:55.660 that tiny change, dx squared, is going to be about 2x times dx. 00:06:56.360 --> 00:07:01.580 And that tiny change, d sine of x, well that's going to be about cosine of x times dx. 00:07:02.920 --> 00:07:09.019 As usual, we divide out by that dx to see that the ratio we want, df divided by dx, 00:07:09.019 --> 00:07:12.504 is sine of x times the derivative of x squared, 00:07:12.504 --> 00:07:15.699 plus x squared times the derivative of sine. 00:07:17.959 --> 00:07:21.259 And nothing we've done here is specific to sine or to x squared. 00:07:21.579 --> 00:07:25.359 This same line of reasoning would work for any two functions, g and h. 00:07:27.000 --> 00:07:29.310 And sometimes people like to remember this pattern with 00:07:29.310 --> 00:07:31.539 a certain mnemonic that you kind of sing in your head. 00:07:32.220 --> 00:07:33.680 Left d right, right d left. 00:07:34.399 --> 00:07:38.812 In this example, where we have sine of x times x squared, left d right, 00:07:38.812 --> 00:07:43.778 means you take that left function, sine of x, times the derivative of the right, 00:07:43.778 --> 00:07:44.759 in this case 2x. 00:07:45.480 --> 00:07:48.876 Then you add on right d left, that right function, 00:07:48.875 --> 00:07:52.939 x squared, times the derivative of the left one, cosine of x. 00:07:54.360 --> 00:07:57.271 Now out of context, presented as a rule to remember, 00:07:57.271 --> 00:08:00.019 I think this would feel pretty strange, don't you? 00:08:00.740 --> 00:08:03.381 But when you actually think of this adjustable box, 00:08:03.380 --> 00:08:05.819 you can see what each of those terms represents. 00:08:06.579 --> 00:08:10.971 Left d right is the area of that little bottom rectangle, 00:08:10.971 --> 00:08:15.439 and right d left is the area of that rectangle on the side. 00:08:20.160 --> 00:08:23.847 By the way, I should mention that if you multiply by a constant, 00:08:23.846 --> 00:08:26.739 say 2 times sine of x, things end up a lot simpler. 00:08:27.399 --> 00:08:30.875 The derivative is just the same as the constant multiplied by 00:08:30.875 --> 00:08:34.519 the derivative of the function, in this case 2 times cosine of x. 00:08:35.558 --> 00:08:40.178 I'll leave it to you to pause and ponder and verify that makes sense. 00:08:41.918 --> 00:08:46.533 Aside from addition and multiplication, the other common way to combine functions, 00:08:46.533 --> 00:08:49.201 and believe me, this one comes up all the time, 00:08:49.201 --> 00:08:52.259 is to shove one inside the other, function composition. 00:08:53.220 --> 00:08:56.898 For example, maybe we take the function x squared and shove it 00:08:56.898 --> 00:09:00.460 inside sine of x to get this new function, sine of x squared. 00:09:01.399 --> 00:09:04.079 What do you think the derivative of that new function is? 00:09:05.299 --> 00:09:09.146 To think this one through, I'll choose yet another way to visualize things, 00:09:09.147 --> 00:09:12.540 just to emphasize that in creative math, we've got lots of options. 00:09:13.320 --> 00:09:18.591 I'll put up three different number lines, the top one is going to hold the value of x, 00:09:18.591 --> 00:09:21.440 the second one is going to hold the x squared, 00:09:21.440 --> 00:09:25.500 and the third line is going to hold the value of sine of x squared. 00:09:26.460 --> 00:09:30.310 That is, the function x squared gets you from line 1 to line 2, 00:09:30.309 --> 00:09:33.500 and the function sine gets you from line 2 to line 3. 00:09:34.840 --> 00:09:39.581 As I shift around this value of x, maybe moving it up to the value 3, 00:09:39.581 --> 00:09:45.340 that second value stays pegged to whatever x squared is, in this case moving up to 9. 00:09:46.200 --> 00:09:49.355 That bottom value, being sine of x squared, is 00:09:49.355 --> 00:09:52.580 going to go to whatever sine of 9 happens to be. 00:09:54.899 --> 00:10:00.399 So, for the derivative, let's again start by nudging that x value by some little dx. 00:10:01.539 --> 00:10:04.799 I always think that it's helpful to think of x as starting 00:10:04.799 --> 00:10:07.839 at some actual concrete number, maybe 1.5 in this case. 00:10:08.759 --> 00:10:14.737 The resulting nudge to that second value, the change in x squared caused by such a dx, 00:10:14.738 --> 00:10:15.700 is dx squared. 00:10:16.960 --> 00:10:21.062 We could expand this like we have before, as 2x times dx, 00:10:21.062 --> 00:10:25.307 which for our specific input would be 2 times 1.5 times dx, 00:10:25.307 --> 00:10:30.120 but it helps to keep things written as dx squared, at least for now. 00:10:31.019 --> 00:10:36.384 In fact, I'm going to go one step further, give a new name to this x squared, 00:10:36.384 --> 00:10:41.200 maybe h, so instead of writing dx squared for this nudge, we write dh. 00:10:42.620 --> 00:10:47.259 This makes it easier to think about that third value, which is now pegged at sine of h. 00:10:48.200 --> 00:10:53.680 Its change is d sine of h, the tiny change caused by the nudge dh. 00:10:55.000 --> 00:11:00.134 By the way, the fact that it's moving to the left while the dh bump is going to the right 00:11:00.134 --> 00:11:05.039 just means that this change, d sine of h, is going to be some kind of negative number. 00:11:06.139 --> 00:11:09.639 Once again, we can use our knowledge of the derivative of the sine. 00:11:10.500 --> 00:11:14.419 This d sine of h is going to be about cosine of h times dh. 00:11:15.240 --> 00:11:18.639 That's what it means for the derivative of sine to be cosine. 00:11:19.539 --> 00:11:23.771 Unfolding things, we can replace that h with x squared again, 00:11:23.772 --> 00:11:29.780 so we know that the bottom nudge will be a size of cosine of x squared times dx squared. 00:11:31.039 --> 00:11:32.480 Let's unfold things even further. 00:11:32.840 --> 00:11:38.100 That intermediate nudge dx squared is going to be about 2x times dx. 00:11:39.059 --> 00:11:41.445 It's always a good habit to remind yourself of 00:11:41.446 --> 00:11:43.680 what an expression like this actually means. 00:11:44.340 --> 00:11:48.578 In this case, where we started at x equals 1.5 up top, 00:11:48.577 --> 00:11:54.512 this whole expression is telling us that the size of the nudge on that third 00:11:54.513 --> 00:12:00.754 line is going to be about cosine of 1.5 squared times 2 times 1.5 times whatever 00:12:00.754 --> 00:12:02.220 the size of dx was. 00:12:02.720 --> 00:12:05.112 It's proportional to the size of dx, and this 00:12:05.111 --> 00:12:07.919 derivative is giving us that proportionality constant. 00:12:10.919 --> 00:12:12.559 Notice what we came out with here. 00:12:12.960 --> 00:12:15.855 We have the derivative of the outside function, 00:12:15.855 --> 00:12:19.235 and it's still taking in the unaltered inside function, 00:12:19.235 --> 00:12:23.220 and then multiplying it by the derivative of that inside function. 00:12:25.820 --> 00:12:29.220 Again, there's nothing special about sine of x or x squared. 00:12:29.740 --> 00:12:33.501 If you have any two functions, g of x and h of x, 00:12:33.501 --> 00:12:37.263 the derivative of their composition, g of h of x, 00:12:37.264 --> 00:12:43.659 is going to be the derivative of g evaluated on h, multiplied by the derivative of h. 00:12:47.139 --> 00:12:50.899 This pattern right here is what we usually call the chain rule. 00:12:52.039 --> 00:12:57.679 Notice for the derivative of g, I'm writing it as dg dh instead of dg dx. 00:12:58.679 --> 00:13:02.234 On the symbolic level, this is a reminder that the thing you plug 00:13:02.235 --> 00:13:06.060 into that derivative is still going to be that intermediary function h. 00:13:07.019 --> 00:13:09.745 But more than that, it's an important reflection of what 00:13:09.745 --> 00:13:12.519 this derivative of the outer function actually represents. 00:13:13.200 --> 00:13:18.449 Remember, in our three line setup, when we took the derivative of the sine on 00:13:18.448 --> 00:13:23.899 that bottom, we expanded the size of that nudge, d sine, as cosine of h times dh. 00:13:24.940 --> 00:13:27.496 This was because we didn't immediately know how 00:13:27.495 --> 00:13:29.840 the size of that bottom nudge depended on x. 00:13:30.419 --> 00:13:32.599 That's kind of the whole thing we were trying to figure out. 00:13:33.259 --> 00:13:37.360 But we could take the derivative with respect to that intermediate variable, h. 00:13:38.100 --> 00:13:41.916 That is, figure out how to express the size of that nudge on the third 00:13:41.916 --> 00:13:45.680 line as some multiple of dh, the size of the nudge on the second line. 00:13:46.580 --> 00:13:50.700 It was only after that that we unfolded further by figuring out what dh was. 00:13:53.320 --> 00:13:58.727 In this chain rule expression, we're saying, look at the ratio between a tiny change in 00:13:58.726 --> 00:14:02.351 g, the final output, to a tiny change in h that caused it, 00:14:02.351 --> 00:14:04.379 h being the value we plug into g. 00:14:05.320 --> 00:14:08.680 Then multiply that by the tiny change in h, divided 00:14:08.679 --> 00:14:11.199 by the tiny change in x that caused it. 00:14:12.299 --> 00:14:15.481 So notice, those dh's cancel out, and they give us a ratio 00:14:15.481 --> 00:14:19.473 between the change in that final output and the change to the input that, 00:14:19.474 --> 00:14:22.280 through a certain chain of events, brought it about. 00:14:23.860 --> 00:14:26.980 And that cancellation of dh is not just a notational trick. 00:14:26.980 --> 00:14:30.350 That is a genuine reflection of what's going on with the 00:14:30.350 --> 00:14:33.899 tiny nudges that underpin everything we do with derivatives. 00:14:36.299 --> 00:14:39.877 So those are the three basic tools to have in your belt to handle 00:14:39.878 --> 00:14:43.240 derivatives of functions that combine a lot of smaller things. 00:14:43.840 --> 00:14:47.379 You've got the sum rule, the product rule, and the chain rule. 00:14:48.399 --> 00:14:51.922 And I'll be honest with you, there is a big difference between knowing 00:14:51.922 --> 00:14:54.551 what the chain rule is and what the product rule is, 00:14:54.551 --> 00:14:58.620 and actually being fluent with applying them in even the most hairy of situations. 00:14:59.480 --> 00:15:03.100 Watching videos, any videos, about the mechanics of calculus is 00:15:03.100 --> 00:15:06.892 never going to substitute for practicing those mechanics yourself, 00:15:06.892 --> 00:15:10.400 and building up the muscles to do these computations yourself. 00:15:11.240 --> 00:15:13.600 I really wish I could offer to do that for you, 00:15:13.600 --> 00:15:17.440 but I'm afraid the ball is in your court, my friend, to seek out the practice. 00:15:18.039 --> 00:15:20.940 What I can offer, and what I hope I have offered, 00:15:20.941 --> 00:15:23.960 is to show you where these rules actually come from. 00:15:24.139 --> 00:15:27.774 To show that they're not just something to be memorized and hammered away, 00:15:27.774 --> 00:15:31.264 but they're natural patterns, things that you too could have discovered 00:15:31.264 --> 00:15:34.560 just by patiently thinking through what a derivative actually means.