WEBVTT 00:00:12.140 --> 00:00:16.414 Now that we've seen what a derivative means and what it has to do with rates of change, 00:00:16.414 --> 00:00:19.379 our next step is to learn how to actually compute these guys. 00:00:19.839 --> 00:00:22.939 As in, if I give you some kind of function with an explicit formula, 00:00:22.939 --> 00:00:26.039 you'd want to be able to find what the formula for its derivative is. 00:00:26.699 --> 00:00:30.341 Maybe it's obvious, but I think it's worth stating explicitly why this 00:00:30.341 --> 00:00:34.085 is an important thing to be able to do, why much of a calculus student's 00:00:34.085 --> 00:00:37.469 time ends up going towards grappling with derivatives of abstract 00:00:37.469 --> 00:00:41.060 functions rather than thinking about concrete rate of change problems. 00:00:42.219 --> 00:00:45.908 It's because a lot of real-world phenomena, the sort of things that 00:00:45.908 --> 00:00:49.543 we want to use calculus to analyze, are modeled using polynomials, 00:00:49.543 --> 00:00:53.559 trigonometric functions, exponentials, and other pure functions like that. 00:00:53.979 --> 00:00:58.284 So if you build up some fluency with the ideas of rates of change for those kinds of 00:00:58.284 --> 00:01:02.641 pure abstract functions, it gives you a language to more readily talk about the rates 00:01:02.642 --> 00:01:07.100 at which things change in concrete situations that you might be using calculus to model. 00:01:07.920 --> 00:01:12.141 But it is way too easy for this process to feel like just memorizing a list of rules, 00:01:12.141 --> 00:01:16.166 and if that happens, if you get that feeling, it's also easy to lose sight of the 00:01:16.165 --> 00:01:20.239 fact that derivatives are fundamentally about just looking at tiny changes to some 00:01:20.239 --> 00:01:24.019 quantity and how that relates to a resulting tiny change in another quantity. 00:01:24.780 --> 00:01:28.731 So in this video and in the next one, my aim is to show you how you can think 00:01:28.731 --> 00:01:31.671 about a few of these rules intuitively and geometrically, 00:01:31.671 --> 00:01:35.674 and I really want to encourage you to never forget that tiny nudges are at the 00:01:35.674 --> 00:01:36.739 heart of derivatives. 00:01:37.920 --> 00:01:41.280 Let's start with a simple function like f of x equals x squared. 00:01:41.620 --> 00:01:42.740 What if I asked you its derivative? 00:01:43.519 --> 00:01:47.037 That is, if you were to look at some value x, like x equals 2, 00:01:47.037 --> 00:01:50.332 and compare it to a value slightly bigger, just dx bigger, 00:01:50.332 --> 00:01:53.739 what's the corresponding change in the value of the function? 00:01:54.260 --> 00:01:54.700 dF. 00:01:55.620 --> 00:01:58.692 And in particular, what's dF divided by dx, the rate 00:01:58.692 --> 00:02:01.939 at which this function is changing per unit change in x. 00:02:03.159 --> 00:02:07.248 As a first step for intuition, we know that you can think of this ratio 00:02:07.248 --> 00:02:10.882 dF dx as the slope of a tangent line to the graph of x squared, 00:02:10.883 --> 00:02:15.200 and from that you can see that the slope generally increases as x increases. 00:02:15.840 --> 00:02:18.400 At zero, the tangent line is flat, and the slope is zero. 00:02:19.000 --> 00:02:21.259 At x equals 1, it's something a bit steeper. 00:02:22.599 --> 00:02:24.400 At x equals 2, it's steeper still. 00:02:25.120 --> 00:02:27.579 But looking at graphs isn't generally the best way 00:02:27.579 --> 00:02:30.040 to understand the precise formula for a derivative. 00:02:30.719 --> 00:02:34.933 For that, it's best to take a more literal look at what x squared actually means, 00:02:34.933 --> 00:02:38.840 and in this case let's go ahead and picture a square whose side length is x. 00:02:39.919 --> 00:02:43.119 If you increase x by some tiny nudge, some little dx, 00:02:43.120 --> 00:02:46.379 what's the resulting change in the area of that square? 00:02:47.719 --> 00:02:51.479 That slight change in area is what dF means in this context. 00:02:52.020 --> 00:02:55.777 It's the tiny increase to the value of f of x equals x squared, 00:02:55.776 --> 00:02:58.419 caused by increasing x by that tiny nudge dx. 00:02:59.360 --> 00:03:03.031 Now you can see that there's three new bits of area in this diagram, 00:03:03.031 --> 00:03:05.319 two thin rectangles and a minuscule square. 00:03:06.240 --> 00:03:10.106 The two thin rectangles each have side lengths of x and dx, 00:03:10.105 --> 00:03:13.780 so they account for 2 times x times dx units of new area. 00:03:18.240 --> 00:03:21.031 For example, let's say x was 3 and dx was 0.01, 00:03:21.031 --> 00:03:25.741 then that new area from these two thin rectangles would be 2 times 3 times 0.01, 00:03:25.741 --> 00:03:28.300 which is 0.06, about 6 times the size of dx. 00:03:29.699 --> 00:03:32.871 That little square there has an area of dx squared, 00:03:32.872 --> 00:03:36.960 but you should think of that as being really tiny, negligibly tiny. 00:03:37.699 --> 00:03:41.339 For example, if dx was 0.01, that would be only 0.0001, 00:03:41.340 --> 00:03:46.086 and keep in mind I'm drawing dx with a fair bit of width here just so we 00:03:46.086 --> 00:03:49.662 can actually see it, but always remember in principle, 00:03:49.662 --> 00:03:54.992 dx should be thought of as a truly tiny amount, and for those truly tiny amounts, 00:03:54.992 --> 00:03:59.674 a good rule of thumb is that you can ignore anything that includes a dx 00:03:59.674 --> 00:04:01.819 raised to a power greater than 1. 00:04:02.400 --> 00:04:05.879 That is, a tiny change squared is a negligible change. 00:04:07.500 --> 00:04:13.055 What this leaves us with is that dF is just some multiple of dx, and that multiple 2x, 00:04:13.055 --> 00:04:18.100 which you could also write as dF divided by dx, is the derivative of x squared. 00:04:19.040 --> 00:04:24.297 For example, if you were starting at x equals 3, then as you slightly increase x, 00:04:24.297 --> 00:04:29.683 the rate of change in the area per unit change in length added, dx squared over dx, 00:04:29.682 --> 00:04:34.427 would be 2 times 3, or 6, and if instead you were starting at x equals 5, 00:04:34.427 --> 00:04:38.980 then the rate of change would be 10 units of area per unit change in x. 00:04:41.220 --> 00:04:45.420 Let's go ahead and try a different simple function, f of x equals x cubed. 00:04:45.939 --> 00:04:48.038 This is going to be the geometric view of the stuff 00:04:48.038 --> 00:04:50.139 that I went through algebraically in the last video. 00:04:51.019 --> 00:04:55.640 What's nice here is that we can think of x cubed as the volume of an actual 00:04:55.641 --> 00:05:00.020 cube whose side lengths are x, and when you increase x by a tiny nudge, 00:05:00.019 --> 00:05:04.519 a tiny dx, the resulting increase in volume is what I have here in yellow. 00:05:04.860 --> 00:05:08.747 That represents all the volume in a cube with side lengths x plus dx 00:05:08.747 --> 00:05:12.579 that's not already in the original cube, the one with side length x. 00:05:13.579 --> 00:05:18.601 It's nice to think of this new volume as broken up into multiple components, 00:05:18.601 --> 00:05:22.385 but almost all of it comes from these three square faces, 00:05:22.386 --> 00:05:25.843 or said a little more precisely, as dx approaches 0, 00:05:25.843 --> 00:05:30.278 those three squares comprise a portion closer and closer to 100% of 00:05:30.278 --> 00:05:31.779 that new yellow volume. 00:05:33.839 --> 00:05:38.057 Each of those thin squares has a volume of x squared times dx, 00:05:38.057 --> 00:05:41.540 the area of the face times that little thickness dx. 00:05:42.220 --> 00:05:46.260 So in total this gives us 3x squared dx of volume change. 00:05:47.300 --> 00:05:51.062 And to be sure there are other slivers of volume here along the edges 00:05:51.062 --> 00:05:54.877 and that tiny one in the corner, but all of that volume is going to be 00:05:54.877 --> 00:05:58.639 proportional to dx squared, or dx cubed, so we can safely ignore them. 00:05:59.459 --> 00:06:03.466 Again this is ultimately because they're going to be divided by dx, 00:06:03.466 --> 00:06:07.117 and if there's still any dx remaining then those terms aren't 00:06:07.117 --> 00:06:10.300 going to survive the process of letting dx approach 0. 00:06:11.279 --> 00:06:14.434 What this means is that the derivative of x cubed, 00:06:14.435 --> 00:06:19.199 the rate at which x cubed changes per unit change of x, is 3 times x squared. 00:06:20.639 --> 00:06:25.185 What that means in terms of graphical intuition is that the slope of 00:06:25.185 --> 00:06:29.600 the graph of x cubed at every single point x is exactly 3x squared. 00:06:34.079 --> 00:06:38.786 And reasoning about that slope, it should make sense that this derivative is high on the 00:06:38.786 --> 00:06:42.805 left and then 0 at the origin and then high again as you move to the right, 00:06:42.805 --> 00:06:47.141 but just thinking in terms of the graph would never have landed us on the precise 00:06:47.141 --> 00:06:48.199 quantity 3x squared. 00:06:48.879 --> 00:06:53.060 For that we had to take a much more direct look at what x cubed actually means. 00:06:54.259 --> 00:06:57.560 Now in practice you wouldn't necessarily think of the square every 00:06:57.560 --> 00:06:59.926 time you're taking the derivative of x squared, 00:06:59.927 --> 00:07:03.278 nor would you necessarily think of this cube whenever you're taking 00:07:03.278 --> 00:07:04.560 the derivative of x cubed. 00:07:04.879 --> 00:07:08.399 Both of them fall under a pretty recognizable pattern for polynomial terms. 00:07:09.199 --> 00:07:13.341 The derivative of x to the fourth turns out to be 4x cubed, 00:07:13.341 --> 00:07:17.759 the derivative of x to the fifth is 5x to the fourth, and so on. 00:07:18.879 --> 00:07:22.791 Abstractly you'd write this as the derivative of x to 00:07:22.791 --> 00:07:26.559 the n for any power n is n times x to the n minus 1. 00:07:27.300 --> 00:07:30.560 This right here is what's known in the business as the power rule. 00:07:31.740 --> 00:07:35.913 In practice we all quickly just get jaded and think about this symbolically as 00:07:35.913 --> 00:07:39.769 the exponent hopping down in front, leaving behind one less than itself, 00:07:39.769 --> 00:07:44.259 rarely pausing to think about the geometric delights that underlie these derivatives. 00:07:45.240 --> 00:07:47.295 That's the kind of thing that happens when these tend 00:07:47.295 --> 00:07:49.199 to fall in the middle of much longer computations. 00:07:50.639 --> 00:07:53.327 But rather than tracking it all off to symbolic patterns, 00:07:53.327 --> 00:07:57.359 let's just take a moment and think about why this works for powers beyond just 2 and 3. 00:07:58.439 --> 00:08:02.773 When you nudge that input x, increasing it slightly to x plus dx, 00:08:02.773 --> 00:08:06.974 working out the exact value of that nudged output would involve 00:08:06.973 --> 00:08:10.519 multiplying together these n separate x plus dx terms. 00:08:11.339 --> 00:08:13.889 The full expansion would be really complicated, 00:08:13.889 --> 00:08:18.459 but part of the point of derivatives is that most of that complication can be ignored. 00:08:19.279 --> 00:08:22.019 The first term in your expansion is x to the n. 00:08:22.680 --> 00:08:25.584 This is analogous to the area of the original square, 00:08:25.584 --> 00:08:28.920 or the volume of the original cube from our previous examples. 00:08:30.819 --> 00:08:36.038 For the next terms in the expansion you can choose mostly x's with a single dx. 00:08:41.720 --> 00:08:46.803 Since there are n different parentheticals from which you could have chosen 00:08:46.803 --> 00:08:50.015 that single dx, this gives us n separate terms, 00:08:50.015 --> 00:08:53.159 all of which include n minus 1 x's times a dx, 00:08:53.159 --> 00:08:56.639 giving a value of x to the power n minus 1 times dx. 00:08:57.580 --> 00:09:02.820 This is analogous to how the majority of the new area in the square came from those 00:09:02.820 --> 00:09:07.997 two bars, each with area x times dx, or how the bulk of the new volume in the cube 00:09:07.996 --> 00:09:13.299 came from those three thin squares, each of which had a volume of x squared times dx. 00:09:14.539 --> 00:09:17.432 There will be many other terms of this expansion, 00:09:17.432 --> 00:09:21.250 but all of them are just going to be some multiple of dx squared, 00:09:21.250 --> 00:09:25.184 so we can safely ignore them, and what that means is that all but a 00:09:25.184 --> 00:09:29.349 negligible portion of the increase in the output comes from n copies of 00:09:29.350 --> 00:09:31.259 this x to the n minus 1 times dx. 00:09:31.940 --> 00:09:37.520 That's what it means for the derivative of x to the n to be n times x to the n minus 1. 00:09:38.960 --> 00:09:43.220 And even though, like I said in practice, you'll find yourself performing this 00:09:43.220 --> 00:09:47.911 derivative quickly and symbolically, imagining the exponent hopping down to the front, 00:09:47.910 --> 00:09:52.279 every now and then it's nice to just step back and remember why these rules work. 00:09:52.820 --> 00:09:56.879 Not just because it's pretty, and not just because it helps remind us that math 00:09:56.879 --> 00:10:00.331 actually makes sense and isn't just a pile of formulas to memorize, 00:10:00.331 --> 00:10:04.494 but because it flexes that very important muscle of thinking about derivatives in 00:10:04.494 --> 00:10:05.560 terms of tiny nudges. 00:10:07.500 --> 00:10:11.639 As another example, think of the function f of x equals 1 divided by x. 00:10:12.700 --> 00:10:16.738 Now on the hand you could just blindly try applying the power rule, 00:10:16.738 --> 00:10:20.540 since 1 divided by x is the same as writing x to the negative 1. 00:10:21.100 --> 00:10:24.432 That would involve letting the negative 1 hop down in front, 00:10:24.432 --> 00:10:27.439 leaving behind 1 less than itself, which is negative 2. 00:10:28.240 --> 00:10:31.443 But let's have some fun and see if we can reason about this geometrically, 00:10:31.443 --> 00:10:33.579 rather than just plugging it through some formula. 00:10:34.860 --> 00:10:40.180 The value 1 over x is asking what number multiplied by x equals 1. 00:10:40.960 --> 00:10:42.820 So here's how I'd like to visualize it. 00:10:42.820 --> 00:10:48.120 Imagine a little rectangular puddle of water sitting in two dimensions whose area is 1. 00:10:48.960 --> 00:10:53.766 And let's say that its width is x, which means that the height has to be 1 over x, 00:10:53.765 --> 00:10:55.620 since the total area of it is 1. 00:10:56.360 --> 00:11:01.039 So if x was stretched out to 2, then that height is forced down to 1 half. 00:11:01.779 --> 00:11:05.919 And if you increased x up to 3, then the other side has to be squished down to 1 third. 00:11:07.039 --> 00:11:10.679 This is a nice way to think about the graph of 1 over x, by the way. 00:11:11.279 --> 00:11:15.360 If you think of this width x of the puddle as being in the xy-plane, 00:11:15.360 --> 00:11:20.623 then that corresponding output 1 divided by x, the height of the graph above that point, 00:11:20.623 --> 00:11:24.940 is whatever the height of your puddle has to be to maintain an area of 1. 00:11:26.360 --> 00:11:29.358 So with this visual in mind, for the derivative, 00:11:29.357 --> 00:11:33.579 imagine nudging up that value of x by some tiny amount, some tiny dx. 00:11:34.580 --> 00:11:37.401 How must the height of this rectangle change so 00:11:37.400 --> 00:11:40.339 that the area of the puddle remains constant at 1? 00:11:41.340 --> 00:11:46.019 That is, increasing the width by dx adds some new area to the right here. 00:11:46.259 --> 00:11:50.355 So the puddle has to decrease in height by some d 1 over x, 00:11:50.355 --> 00:11:54.860 so that the area lost off of that top cancels out the area gained. 00:11:56.100 --> 00:11:59.259 You should think of that d 1 over x as being a negative amount, 00:11:59.259 --> 00:12:02.320 by the way, since it's decreasing the height of the rectangle. 00:12:03.539 --> 00:12:04.399 And you know what? 00:12:04.840 --> 00:12:07.027 I'm going to leave the last few steps here for you, 00:12:07.027 --> 00:12:09.720 for you to pause and ponder and work out an ultimate expression. 00:12:10.559 --> 00:12:14.120 And once you reason out what d of 1 over x divided by dx should be, 00:12:14.120 --> 00:12:17.838 I want you to compare it to what you would have gotten if you had just 00:12:17.839 --> 00:12:21.820 blindly applied the power rule, purely symbolically, to x to the negative 1. 00:12:23.980 --> 00:12:26.143 And while I'm encouraging you to pause and ponder, 00:12:26.143 --> 00:12:28.519 here's another fun challenge if you're feeling up to it. 00:12:29.059 --> 00:12:33.419 See if you can reason through what the derivative of the square root of x should be. 00:12:36.399 --> 00:12:40.052 To finish things off, I want to tackle one more type of function, 00:12:40.052 --> 00:12:44.259 trigonometric functions, and in particular let's focus on the sine function. 00:12:45.320 --> 00:12:48.230 So for this section I'm going to assume that you're already 00:12:48.230 --> 00:12:51.673 familiar with how to think about trig functions using the unit circle, 00:12:51.673 --> 00:12:54.100 the circle with a radius 1 centered at the origin. 00:12:55.240 --> 00:12:59.152 For a given value of theta, like say 0.8, you imagine yourself 00:12:59.152 --> 00:13:02.878 walking around the circle starting from the rightmost point 00:13:02.878 --> 00:13:06.480 until you've traversed that distance of 0.8 in arc length. 00:13:06.759 --> 00:13:11.717 This is the same thing as saying that the angle right here is exactly theta radians, 00:13:11.717 --> 00:13:13.759 since the circle has a radius of 1. 00:13:14.759 --> 00:13:20.013 Then what sine of theta means is the height of that point above the x-axis, 00:13:20.013 --> 00:13:24.507 and as your theta value increases and you walk around the circle 00:13:24.506 --> 00:13:28.239 your height bobs up and down between negative 1 and 1. 00:13:29.019 --> 00:13:33.615 So when you graph sine of theta versus theta you get this wave pattern, 00:13:33.615 --> 00:13:35.659 the quintessential wave pattern. 00:13:37.600 --> 00:13:40.311 And just from looking at this graph we can start to 00:13:40.311 --> 00:13:43.180 get a feel for the shape of the derivative of the sine. 00:13:44.019 --> 00:13:48.827 The slope at 0 is something positive since sine of theta is increasing there, 00:13:48.827 --> 00:13:54.191 and as we move to the right and sine of theta approaches its peak that slope goes down 00:13:54.191 --> 00:13:54.500 to 0. 00:13:55.720 --> 00:13:58.340 Then the slope is negative for a little while, 00:13:58.340 --> 00:14:03.080 while the sine is decreasing before coming back up to 0 as the sine graph levels out. 00:14:04.460 --> 00:14:07.432 And as you continue thinking this through and drawing it out, 00:14:07.432 --> 00:14:11.173 if you're familiar with the graph of trig functions you might guess that this 00:14:11.173 --> 00:14:13.668 derivative graph should be exactly cosine of theta, 00:14:13.668 --> 00:14:17.264 since all the peaks and valleys line up perfectly with where the peaks and 00:14:17.264 --> 00:14:19.279 valleys for the cosine function should be. 00:14:20.340 --> 00:14:23.910 And spoiler alert, the derivative is in fact the cosine of theta, 00:14:23.909 --> 00:14:27.860 but aren't you a little curious about why it's precisely cosine of theta? 00:14:28.240 --> 00:14:32.158 I mean you could have all sorts of functions with peaks and valleys at the same points 00:14:32.158 --> 00:14:34.365 that have roughly the same shape, but who knows, 00:14:34.365 --> 00:14:38.103 maybe the derivative of sine could have turned out to be some entirely new type of 00:14:38.102 --> 00:14:40.399 function that just happens to have a similar shape. 00:14:41.600 --> 00:14:44.831 Well just like the previous examples, a more exact understanding 00:14:44.831 --> 00:14:48.662 of the derivative requires looking at what the function actually represents, 00:14:48.662 --> 00:14:51.100 rather than looking at the graph of the function. 00:14:52.399 --> 00:14:54.995 So think back to that walk around the unit circle, 00:14:54.995 --> 00:14:58.966 having traversed an arc with length theta and thinking about sine of theta as 00:14:58.966 --> 00:15:00.240 the height of that point. 00:15:01.700 --> 00:15:06.247 Now zoom into that point on the circle and consider a slight nudge of d theta 00:15:06.246 --> 00:15:10.620 along their circumference, a tiny step in your walk around the unit circle. 00:15:11.480 --> 00:15:14.639 How much does that tiny step change the sine of theta? 00:15:15.440 --> 00:15:20.420 How much does this increase d theta of arc length increase the height above the x-axis? 00:15:21.639 --> 00:15:26.181 Well zoomed in close enough, the circle basically looks like a straight line in this 00:15:26.181 --> 00:15:30.777 neighborhood, so let's go ahead and think of this right triangle where the hypotenuse 00:15:30.777 --> 00:15:34.891 of that right triangle represents the nudge d theta along the circumference, 00:15:34.890 --> 00:15:39.539 and that left side here represents the change in height, the resulting d sine of theta. 00:15:40.139 --> 00:15:44.184 Now this tiny triangle is actually similar to this larger triangle here, 00:15:44.184 --> 00:15:48.840 with the defining angle theta and whose hypotenuse is the radius of the circle with 00:15:48.841 --> 00:15:49.340 length 1. 00:15:50.960 --> 00:15:55.940 Specifically this little angle right here is precisely equal to theta radians. 00:15:57.419 --> 00:16:00.519 Now think about what the derivative of sine is supposed to mean. 00:16:01.220 --> 00:16:05.585 It's the ratio between that d sine of theta, the tiny change to the height, 00:16:05.585 --> 00:16:09.320 divided by d theta, the tiny change to the input of the function. 00:16:10.519 --> 00:16:14.052 And from the picture we can see that that's the ratio between the 00:16:14.052 --> 00:16:17.960 length of the side adjacent to the angle theta divided by the hypotenuse. 00:16:18.799 --> 00:16:21.517 Well let's see, adjacent divided by hypotenuse, 00:16:21.518 --> 00:16:26.220 that's exactly what the cosine of theta means, that's the definition of the cosine. 00:16:27.539 --> 00:16:30.222 So this gives us two different really nice ways of 00:16:30.222 --> 00:16:32.959 thinking about how the derivative of sine is cosine. 00:16:33.139 --> 00:16:36.641 One of them is looking at the graph and getting a loose feel for the shape of 00:16:36.642 --> 00:16:40.280 things based on thinking about the slope of the sine graph at every single point. 00:16:41.100 --> 00:16:45.399 And the other is a more precise line of reasoning looking at the unit circle itself. 00:16:47.080 --> 00:16:49.276 For those of you that like to pause and ponder, 00:16:49.275 --> 00:16:52.846 see if you can try a similar line of reasoning to find what the derivative of 00:16:52.846 --> 00:16:54.220 the cosine of theta should be. 00:16:56.320 --> 00:16:59.496 In the next video I'll talk about how you can take derivatives 00:16:59.495 --> 00:17:02.470 of functions who combine simple functions like these ones, 00:17:02.470 --> 00:17:06.000 either as sums or products or function compositions, things like that. 00:17:06.559 --> 00:17:09.643 And similar to this video the goal is going to be to understand each one 00:17:09.643 --> 00:17:13.358 geometrically in a way that makes it intuitively reasonable and somewhat more memorable.