WEBVTT 00:00:07.740 --> 00:00:11.853 The months ahead of you hold within them a lot of hard work, some neat examples, 00:00:11.852 --> 00:00:14.949 some not-so-neat examples, beautiful connections to physics, 00:00:14.949 --> 00:00:17.387 not-so-beautiful piles of formulas to memorize, 00:00:17.388 --> 00:00:20.943 plenty of moments of getting stuck and banging your head into a wall, 00:00:20.943 --> 00:00:24.548 a few nice aha moments sprinkled in as well, and some genuinely lovely 00:00:24.547 --> 00:00:27.239 graphical intuition to help guide you through it all. 00:00:27.629 --> 00:00:31.699 But if the course ahead of you is anything like my first introduction to calculus, 00:00:31.699 --> 00:00:34.543 or any of the first courses I've seen in the years since, 00:00:34.543 --> 00:00:38.712 there's one topic you will not see, but which I believe stands to greatly accelerate 00:00:38.713 --> 00:00:39.399 your learning. 00:00:40.280 --> 00:00:44.620 You see, almost all of the visual intuitions from that first year are based on graphs. 00:00:45.079 --> 00:00:49.659 The derivative is the slope of a graph, the integral is a certain area under that graph. 00:00:50.200 --> 00:00:54.035 But as you generalize calculus beyond functions whose inputs and outputs are 00:00:54.034 --> 00:00:58.019 simply numbers, it's not always possible to graph the function you're analyzing. 00:01:00.679 --> 00:01:04.653 So if all your intuitions for the fundamental ideas, like derivatives, 00:01:04.653 --> 00:01:08.794 are rooted too rigidly in graphs, it can make for a very tall and largely 00:01:08.793 --> 00:01:13.158 unnecessary conceptual hurdle between you and the more quote-unquote advanced 00:01:13.159 --> 00:01:17.580 topics like multivariable calculus and complex analysis, differential geometry. 00:01:18.840 --> 00:01:22.204 What I want to share with you is a way to think about derivatives, 00:01:22.204 --> 00:01:24.716 which I'll refer to as the transformational view, 00:01:24.716 --> 00:01:28.433 that generalizes more seamlessly into some of those more general contexts 00:01:28.433 --> 00:01:29.640 where calculus comes up. 00:01:29.879 --> 00:01:34.859 And then we'll use this alternate view to analyze a fun puzzle about repeated fractions. 00:01:35.459 --> 00:01:37.538 But first off, I just want to make sure we're all 00:01:37.539 --> 00:01:39.659 on the same page about what the standard visual is. 00:01:40.060 --> 00:01:44.936 If you were to graph a function, which simply takes real numbers as inputs and outputs, 00:01:44.936 --> 00:01:49.593 one of the first things you learn in a calculus course is that the derivative gives 00:01:49.593 --> 00:01:54.138 you the slope of this graph, where what we mean by that is that the derivative of 00:01:54.138 --> 00:01:58.240 the function is a new function which for every input x returns that slope. 00:01:59.519 --> 00:02:01.978 Now I'd encourage you not to think of this derivative 00:02:01.978 --> 00:02:04.439 as slope idea as being the definition of a derivative. 00:02:05.000 --> 00:02:07.555 Instead think of it as being more fundamentally about how 00:02:07.555 --> 00:02:10.419 sensitive the function is to tiny little nudges around the input. 00:02:11.020 --> 00:02:14.058 And the slope is just one way to think about that sensitivity 00:02:14.057 --> 00:02:16.900 relevant only to this particular way of viewing functions. 00:02:17.340 --> 00:02:19.816 I have not just another video, but a full series on this 00:02:19.816 --> 00:02:22.120 topic if it's something you want to learn more about. 00:02:22.599 --> 00:02:26.022 The basic idea behind the alternate visual for the derivative is to 00:02:26.022 --> 00:02:29.294 think of this function as mapping all of the input points on the 00:02:29.294 --> 00:02:32.819 number line to their corresponding outputs on a different number line. 00:02:33.400 --> 00:02:36.810 In this context, what the derivative gives you is a measure of how 00:02:36.810 --> 00:02:40.219 much the input space gets stretched or squished in various regions. 00:02:41.860 --> 00:02:46.673 That is, if you were to zoom in around a specific input and take a look at some 00:02:46.673 --> 00:02:51.485 evenly spaced points around it, the derivative of the function of that input is 00:02:51.485 --> 00:02:56.599 going to tell you how spread out or contracted those points become after the mapping. 00:02:57.939 --> 00:02:59.400 Here, a specific example helps. 00:02:59.740 --> 00:03:05.920 Take the function x2, it maps 1 to 1, 2 to 4, 3 to 9, and so on. 00:03:06.479 --> 00:03:09.219 You can also see how it acts on all of the points in between. 00:03:12.719 --> 00:03:16.760 If you were to zoom in on a little cluster of points around the input 1, 00:03:16.760 --> 00:03:19.639 and see where they land around the relevant output, 00:03:19.639 --> 00:03:23.736 which for this function also happens to be 1, you'd notice that they tend 00:03:23.736 --> 00:03:24.900 to get stretched out. 00:03:25.759 --> 00:03:29.019 In fact, it roughly looks like stretching out by a factor of 2. 00:03:29.659 --> 00:03:35.534 The closer you zoom in, the more this local behavior looks just like multiplying by a 00:03:35.534 --> 00:03:41.682 factor of 2. This is what it means for the derivative of x2 at the input x equals 1 to be 00:03:41.682 --> 00:03:41.819 2. 00:03:42.340 --> 00:03:45.400 It's what that fact looks like in the context of transformations. 00:03:46.460 --> 00:03:49.731 If you looked at a neighborhood of points around the input 3, 00:03:49.731 --> 00:03:52.159 they would get stretched out by a factor of 6. 00:03:52.740 --> 00:03:57.439 This is what it means for the derivative of this function at the input 3 to equal 6. 00:03:58.979 --> 00:04:03.613 Around the input 1 fourth, a small region tends to get contracted specifically by a 00:04:03.614 --> 00:04:08.360 factor of 1 half, and that's what it looks like for a derivative to be smaller than 1. 00:04:10.719 --> 00:04:12.599 The input 0 is interesting. 00:04:13.120 --> 00:04:15.617 Zooming in by a factor of 10, it doesn't really 00:04:15.617 --> 00:04:17.959 look like a constant stretching or squishing. 00:04:18.379 --> 00:04:21.680 For one thing, all of the outputs end up on the right positive side of things. 00:04:23.319 --> 00:04:27.694 As you zoom in closer and closer, by 100x, or by 1000x, 00:04:27.694 --> 00:04:33.396 it looks more and more like a small neighborhood of points around 0 just 00:04:33.396 --> 00:04:39.959 gets collapsed into 0 itself. This is what it looks like for the derivative to be 0. 00:04:40.500 --> 00:04:45.019 The local behavior looks more and more like multiplying the whole number line by 0. 00:04:45.680 --> 00:04:49.783 It doesn't have to completely collapse everything to a point at a particular zoom level, 00:04:49.783 --> 00:04:53.840 instead it's a matter of what the limiting behavior is as you zoom in closer and closer. 00:04:55.279 --> 00:04:58.959 It's also instructive to take a look at the negative inputs here. 00:05:00.699 --> 00:05:04.536 Things start to feel a little cramped since they collide with where all the positive 00:05:04.536 --> 00:05:08.057 input values go, and this is one of the downsides of thinking of functions as 00:05:08.057 --> 00:05:08.780 transformations. 00:05:09.399 --> 00:05:13.093 But for derivatives, we only really care about the local behavior anyway, 00:05:13.093 --> 00:05:15.639 what happens in a small range around a given input. 00:05:16.500 --> 00:05:20.189 Here, notice that the inputs in a little neighborhood around, say, 00:05:20.189 --> 00:05:24.100 negative 2, don't just get stretched out, they also get flipped around. 00:05:24.680 --> 00:05:28.132 Specifically, the action on such a neighborhood looks more 00:05:28.132 --> 00:05:31.819 and more like multiplying by negative 4 the closer you zoom in. 00:05:32.319 --> 00:05:35.599 This is what it looks like for the derivative of a function to be negative. 00:05:38.459 --> 00:05:40.951 And I think you get the point, this is all well and good, 00:05:40.951 --> 00:05:43.660 but let's see how this is actually useful in solving a problem. 00:05:44.259 --> 00:05:48.305 A friend of mine recently asked me a pretty fun question about the infinite 00:05:48.305 --> 00:05:52.137 fraction 1 plus 1 divided by 1 plus 1 divided by 1 plus 1 divided by 1, 00:05:52.137 --> 00:05:56.182 and clearly you watch math videos online, so maybe you've seen this before, 00:05:56.182 --> 00:05:59.961 but my friend's question actually cuts to something you might not have 00:05:59.961 --> 00:06:04.220 thought about before, relevant to the view of derivatives we're looking at here. 00:06:05.019 --> 00:06:09.661 The typical way you might evaluate an expression like this is to set it equal to x, 00:06:09.661 --> 00:06:13.639 and then notice that there is a copy of the full fraction inside itself. 00:06:14.699 --> 00:06:18.779 So you can replace that copy with another x, and then just solve for x. 00:06:19.439 --> 00:06:24.579 That is, what you want is to find a fixed point of the function 1 plus 1 divided by x. 00:06:27.160 --> 00:06:30.970 But here's the thing, there are actually two solutions for x, 00:06:30.970 --> 00:06:36.380 two special numbers where 1 plus 1 divided by that number gives you back the same thing. 00:06:36.939 --> 00:06:42.949 One is the golden ratio, phi, around 1.618, and the other is negative 0.618, 00:06:42.949 --> 00:06:46.540 which happens to be negative 1 divided by phi. 00:06:46.959 --> 00:06:49.681 I like to call this other number phi's little brother, 00:06:49.682 --> 00:06:52.900 since just about any property that phi has, this number also has. 00:06:53.560 --> 00:06:58.413 And this raises the question, would it be valid to say that the infinite 00:06:58.413 --> 00:07:03.600 fraction we saw is somehow also equal to phi's little brother, negative 0.618? 00:07:04.519 --> 00:07:08.812 Maybe you initially say, obviously not, everything on the left hand side is positive, 00:07:08.812 --> 00:07:11.259 so how could it possibly equal a negative number? 00:07:12.500 --> 00:07:17.100 Well, first we should be clear about what we actually mean by an expression like this. 00:07:17.779 --> 00:07:21.315 One way you could think about it, and it's not the only way, 00:07:21.315 --> 00:07:26.185 there's freedom for choice here, is to imagine starting with some constant, like 1, 00:07:26.185 --> 00:07:30.939 and then repeatedly applying the function 1 plus 1 divided by x, and then asking, 00:07:30.939 --> 00:07:33.259 what is this approach as you keep going? 00:07:36.040 --> 00:07:38.552 I mean, certainly symbolically what you get looks more and more 00:07:38.552 --> 00:07:41.300 like our infinite fraction, so maybe if you wanted to equal a number, 00:07:41.300 --> 00:07:43.420 you should ask what this series of numbers approaches. 00:07:45.120 --> 00:07:48.509 And if that's your view of things, maybe you start off with a negative number, 00:07:48.509 --> 00:07:51.300 so it's not so crazy for the whole expression to end up negative. 00:07:52.740 --> 00:07:55.836 After all, if you start with negative 1 divided by phi, 00:07:55.836 --> 00:07:59.985 then applying this function 1 plus 1 over x, you get back the same number, 00:07:59.985 --> 00:08:03.802 negative 1 divided by phi, so no matter how many times you apply it, 00:08:03.802 --> 00:08:05.740 you're staying fixed at this value. 00:08:07.819 --> 00:08:10.620 But even then, there is one reason you should 00:08:10.620 --> 00:08:13.420 view phi as the favorite brother in this pair. 00:08:14.019 --> 00:08:19.330 Here, try this, pull up a calculator of some kind, then start with any random number, 00:08:19.331 --> 00:08:22.728 and plug it into this function, 1 plus 1 divided by x, 00:08:22.728 --> 00:08:28.040 and plug that number into 1 plus 1 over x, and again, and again, and again, and again. 00:08:28.480 --> 00:08:33.158 No matter what constant you start with, you eventually end up at 1.618. 00:08:33.798 --> 00:08:38.481 Even if you start with a negative number, even one that's really close to phi's 00:08:38.481 --> 00:08:43.399 little brother, eventually it shies away from that value and jumps back over to phi. 00:08:50.820 --> 00:08:52.460 So, what's going on here? 00:08:52.799 --> 00:08:55.919 Why is one of these fixed points favored above the other one? 00:08:56.720 --> 00:09:00.158 Maybe you can already see how the transformational understanding of derivatives 00:09:00.158 --> 00:09:03.984 is helpful for understanding this setup, but for the sake of having a point of contrast, 00:09:03.984 --> 00:09:07.080 I want to show you how a problem like this is often taught using graphs. 00:09:07.919 --> 00:09:11.115 If you were to plug in some random input to this function, 00:09:11.115 --> 00:09:14.039 the y value tells you the corresponding output, right? 00:09:14.039 --> 00:09:17.862 So to think about plugging that output back into the function, 00:09:17.863 --> 00:09:22.050 you might first move horizontally until you hit the line y equals x, 00:09:22.049 --> 00:09:26.782 and that's going to give you a position where the x value corresponds to your 00:09:26.783 --> 00:09:28.240 previous y value, right? 00:09:28.919 --> 00:09:34.553 So then from there, you can move vertically to see what output this new x value has, 00:09:34.553 --> 00:09:35.879 and then you repeat. 00:09:36.340 --> 00:09:40.598 You move horizontally to the line y equals x to find a point whose x value is the same 00:09:40.597 --> 00:09:44.759 as the output you just got, and then you move vertically to apply the function again. 00:09:45.879 --> 00:09:48.285 Now personally, I think this is kind of an awkward way 00:09:48.285 --> 00:09:50.779 to think about repeatedly applying a function, don't you? 00:09:51.299 --> 00:09:53.803 I mean, it makes sense, but you kind of have to pause 00:09:53.803 --> 00:09:56.539 and think about it to remember which way to draw the lines. 00:09:57.120 --> 00:10:01.426 And you can, if you want, think through what conditions make this spiderweb 00:10:01.426 --> 00:10:05.280 process narrow in on a fixed point, versus propagating away from it. 00:10:05.860 --> 00:10:08.899 In fact, go ahead, pause right now, and try to think it through as an exercise. 00:10:09.240 --> 00:10:10.460 It has to do with slopes. 00:10:12.019 --> 00:10:15.818 Or if you want to skip the exercise for something that I think gives a much more 00:10:15.818 --> 00:10:19.620 satisfying understanding, think about how this function acts as a transformation. 00:10:22.279 --> 00:10:24.923 So I'm going to go ahead and start here by drawing a bunch of 00:10:24.923 --> 00:10:27.740 arrows to indicate where the various sampled input points will go. 00:10:28.320 --> 00:10:31.440 And side note, don't you think this gives a neat emergent pattern? 00:10:31.820 --> 00:10:35.020 I wasn't expecting this, but it was cool to see it pop up when animating. 00:10:35.019 --> 00:10:38.797 I guess the action of 1 divided by x gives this nice emergent circle, 00:10:38.797 --> 00:10:41.279 and then we're just shifting things over by 1. 00:10:42.039 --> 00:10:46.621 Anyway, I want you to think about what it means to repeatedly apply some function, 00:10:46.621 --> 00:10:48.719 like 1 plus 1 over x, in this context. 00:10:50.240 --> 00:10:53.590 Well after letting it map all of the inputs to the outputs, 00:10:53.590 --> 00:10:58.504 you could consider those as the new inputs, and then just apply the same process again, 00:10:58.504 --> 00:11:01.519 and then again, and do it however many times you want. 00:11:02.580 --> 00:11:06.523 Notice, in animating this with a few dots representing the sample points, 00:11:06.523 --> 00:11:11.320 it doesn't take many iterations at all before all of those dots kind of clump in around 1. 00:11:11.320 --> 00:11:12.000 618. 00:11:14.620 --> 00:11:18.355 Now remember, we know that 1.618 and its little brother, 00:11:18.355 --> 00:11:23.860 negative 0.618 on and on, stay fixed in place during each iteration of this process. 00:11:24.860 --> 00:11:27.480 But zoom in on a neighborhood around phi. 00:11:27.480 --> 00:11:32.788 During the map, points in that region get contracted around phi, 00:11:32.788 --> 00:11:39.403 meaning that the function 1 plus 1 over x has a derivative with a magnitude less 00:11:39.403 --> 00:11:41.120 than 1 at this input. 00:11:41.879 --> 00:11:45.200 In fact, this derivative works out to be around negative 0.38. 00:11:46.120 --> 00:11:50.312 So what that means is that each repeated application scrunches the neighborhood 00:11:50.312 --> 00:11:54.399 around this number smaller and smaller, like a gravitational pull towards phi. 00:11:54.960 --> 00:11:58.620 So now tell me what you think happens in the neighborhood of phi's little brother. 00:12:01.320 --> 00:12:05.426 Over there, the derivative actually has a magnitude larger than 1, 00:12:05.426 --> 00:12:08.920 so points near the fixed point are repelled away from it. 00:12:09.519 --> 00:12:11.598 And when you work it out, you can see that they get 00:12:11.599 --> 00:12:13.800 stretched by more than a factor of 2 in each iteration. 00:12:14.419 --> 00:12:17.674 They also get flipped around, because the derivative is negative here, 00:12:17.674 --> 00:12:20.839 but the salient fact for the sake of stability is just the magnitude. 00:12:23.440 --> 00:12:26.970 Mathematicians would call this right value a stable fixed point, 00:12:26.970 --> 00:12:29.360 and the left one is an unstable fixed point. 00:12:30.000 --> 00:12:33.408 Something is considered stable if when you perturb it just a little bit, 00:12:33.408 --> 00:12:37.100 it tends to come back towards where it started, rather than going away from it. 00:12:38.179 --> 00:12:40.777 So what we're seeing is a very useful little fact, 00:12:40.778 --> 00:12:45.312 that the stability of a fixed point is determined by whether or not the magnitude of its 00:12:45.312 --> 00:12:47.300 derivative is bigger or smaller than 1. 00:12:47.299 --> 00:12:50.479 This explains why phi always shows up in the numerical play, 00:12:50.480 --> 00:12:53.922 where you're just hitting enter on your calculator over and over, 00:12:53.922 --> 00:12:55.800 but phi's little brother never does. 00:12:56.460 --> 00:12:59.620 As to whether or not you want to consider phi's little brother a 00:12:59.620 --> 00:13:02.879 valid value of the infinite fraction, well that's really up to you. 00:13:03.259 --> 00:13:06.970 Everything we just showed suggests that if you think of this expression 00:13:06.970 --> 00:13:10.524 as representing a limiting process, then because every possible seed 00:13:10.524 --> 00:13:14.442 value other than phi's little brother gives you a series converging to phi, 00:13:14.442 --> 00:13:17.740 it does feel silly to put them on equal footing with each other. 00:13:18.259 --> 00:13:21.773 But maybe you don't think of it as a limit, maybe the kind of math 00:13:21.773 --> 00:13:25.600 you're doing lends itself to treating this as a purely algebraic object, 00:13:25.600 --> 00:13:29.220 like the solutions of a polynomial, which simply has multiple values. 00:13:30.340 --> 00:13:34.485 Anyway, that's beside the point, and my point here is not that viewing derivatives 00:13:34.485 --> 00:13:38.779 as this change in density is somehow better than the graphical intuition on the whole. 00:13:39.600 --> 00:13:42.204 In fact, picturing an entire function this way can be 00:13:42.203 --> 00:13:44.759 kind of clunky and impractical as compared to graphs. 00:13:45.340 --> 00:13:48.221 My point is that it deserves more of a mention in most of the 00:13:48.221 --> 00:13:50.918 introductory calculus courses, because it can help make a 00:13:50.918 --> 00:13:53.940 student's understanding of the derivative a little more flexible. 00:13:54.899 --> 00:13:58.357 Like I mentioned, the real reason I'd recommend you carry this perspective 00:13:58.357 --> 00:14:01.816 with you as you learn new topics is not so much for what it does with your 00:14:01.817 --> 00:14:05.000 understanding of single variable calculus, it's for what comes after.