WEBVTT 00:00:15.259 --> 00:00:18.960 The goal here is simple, explain what a derivative is. 00:00:19.160 --> 00:00:21.658 The thing is though, there's some subtlety to this topic, 00:00:21.658 --> 00:00:24.200 and a lot of potential for paradoxes if you're not careful. 00:00:24.780 --> 00:00:27.551 So a secondary goal is that you have an appreciation 00:00:27.551 --> 00:00:30.219 for what those paradoxes are and how to avoid them. 00:00:31.219 --> 00:00:35.591 You see, it's common for people to say that the derivative measures an instantaneous 00:00:35.591 --> 00:00:39.759 rate of change, but when you think about it, that phrase is actually an oxymoron. 00:00:40.240 --> 00:00:43.542 Change is something that happens between separate points in time, 00:00:43.542 --> 00:00:46.646 and when you blind yourself to all but just a single instant, 00:00:46.646 --> 00:00:48.600 there's not really any room for change. 00:00:49.500 --> 00:00:51.840 You'll see what I mean more as we get into it, 00:00:51.840 --> 00:00:56.022 but when you appreciate that a phrase like instantaneous rate of change is actually 00:00:56.021 --> 00:01:00.103 nonsense, I think it makes you appreciate just how clever the fathers of calculus 00:01:00.103 --> 00:01:02.991 were in capturing the idea that phrase is meant to evoke, 00:01:02.991 --> 00:01:05.980 but with a perfectly sensible piece of math, the derivative. 00:01:07.540 --> 00:01:11.877 As our central example, I want you to imagine a car that starts at some point A, 00:01:11.876 --> 00:01:15.839 speeds up, and then slows down to a stop at some point B 100 meters away, 00:01:15.840 --> 00:01:19.000 and let's say it all happens over the course of 10 seconds. 00:01:20.519 --> 00:01:23.899 That's the setup to have in mind as we lay out what the derivative is. 00:01:23.900 --> 00:01:29.099 Well, we could graph this motion, letting the vertical axis represent the 00:01:29.099 --> 00:01:34.579 distance traveled, and the horizontal axis represent time, so at each time t, 00:01:34.578 --> 00:01:38.723 represented with a point somewhere on the horizontal axis, 00:01:38.724 --> 00:01:44.134 the height of the graph tells us how far the car has traveled in total after 00:01:44.134 --> 00:01:45.540 that amount of time. 00:01:46.760 --> 00:01:50.160 It's pretty common to name a distance function like this s of t. 00:01:50.159 --> 00:01:52.706 I would use the letter d for distance, but that 00:01:52.706 --> 00:01:55.359 guy already has another full time job in calculus. 00:01:56.500 --> 00:01:59.760 Initially, the curve is quite shallow, since the car is slow to start. 00:02:00.280 --> 00:02:04.340 During that first second, the distance it travels doesn't change that much. 00:02:04.980 --> 00:02:07.582 For the next few seconds, as the car speeds up, 00:02:07.581 --> 00:02:10.454 the distance traveled in a given second gets larger, 00:02:10.455 --> 00:02:13.219 which corresponds to a steeper slope in this graph. 00:02:13.800 --> 00:02:17.520 Then towards the end, when it slows down, that curve shallows out again. 00:02:20.759 --> 00:02:25.516 If we were to plot the car's velocity in meters per second as a function of time, 00:02:25.516 --> 00:02:27.199 it might look like this bump. 00:02:27.860 --> 00:02:30.000 At early times, the velocity is very small. 00:02:30.460 --> 00:02:34.224 Up to the middle of the journey, the car builds up to some maximum velocity, 00:02:34.223 --> 00:02:36.619 covering a relatively large distance each second. 00:02:37.659 --> 00:02:39.919 Then it slows back down towards a speed of zero. 00:02:41.379 --> 00:02:44.180 These two curves are definitely related to each other. 00:02:44.840 --> 00:02:47.159 If you change the specific distance vs. 00:02:47.259 --> 00:02:50.299 time function, you'll have some different velocity vs. 00:02:50.419 --> 00:02:51.079 time function. 00:02:51.759 --> 00:02:55.039 What we want to understand is the specifics of that relationship. 00:02:55.680 --> 00:02:59.099 Exactly how does velocity depend on a distance vs. 00:02:59.400 --> 00:02:59.819 time function? 00:03:01.939 --> 00:03:04.681 To do that, it's worth taking a moment to think 00:03:04.681 --> 00:03:07.539 critically about what exactly velocity means here. 00:03:08.379 --> 00:03:11.879 Intuitively, we all might know what velocity at a given moment means, 00:03:11.879 --> 00:03:14.979 it's just whatever the car's speedometer shows in that moment. 00:03:17.180 --> 00:03:21.457 Intuitively, it might make sense that the car's velocity should be higher at times when 00:03:21.457 --> 00:03:25.639 this distance function is steeper, when the car traverses more distance per unit time. 00:03:26.699 --> 00:03:30.719 But the funny thing is, velocity at a single moment makes no sense. 00:03:31.360 --> 00:03:34.895 If I show you a picture of a car, just a snapshot in an instant, 00:03:34.895 --> 00:03:38.540 and I ask you how fast it's going, you'd have no way of telling me. 00:03:39.620 --> 00:03:42.379 What you'd need are two separate points in time to compare. 00:03:43.180 --> 00:03:47.310 That way you can compute whatever the change in distance across those times is, 00:03:47.310 --> 00:03:48.860 divided by the change in time. 00:03:49.560 --> 00:03:49.740 Right? 00:03:49.819 --> 00:03:54.159 I mean, that's what velocity is, it's the distance traveled per unit time. 00:03:55.620 --> 00:03:59.069 So how is it that we're looking at a function for velocity that 00:03:59.069 --> 00:04:02.359 only takes in a single value of t, a single snapshot in time? 00:04:02.900 --> 00:04:04.280 It's weird, isn't it? 00:04:04.280 --> 00:04:07.868 We want to associate individual points in time with a velocity, 00:04:07.868 --> 00:04:12.300 but actually computing velocity requires comparing two separate points in time. 00:04:14.639 --> 00:04:17.399 If that feels strange and paradoxical, good! 00:04:17.920 --> 00:04:20.959 You're grappling with the same conflicts that the fathers of calculus did. 00:04:21.379 --> 00:04:25.339 And if you want a deep understanding for rates of change, not just for a moving car, 00:04:25.339 --> 00:04:29.346 but for all sorts of things in science, you're going to need to resolve this apparent 00:04:29.346 --> 00:04:29.719 paradox. 00:04:32.199 --> 00:04:34.705 First, I think it's best to talk about the real world, 00:04:34.706 --> 00:04:36.939 and then we'll go into a purely mathematical one. 00:04:37.540 --> 00:04:40.460 Let's think about what the car's speedometer is probably doing. 00:04:41.199 --> 00:04:43.932 At some point, say 3 seconds into the journey, 00:04:43.932 --> 00:04:48.757 the speedometer might measure how far the car goes in a very small amount of time, 00:04:48.757 --> 00:04:52.420 maybe the distance traveled between 3 seconds and 3.01 seconds. 00:04:53.360 --> 00:04:57.514 Then it could compute the speed in meters per second as that tiny 00:04:57.514 --> 00:05:01.860 distance traversed in meters divided by that tiny time, 0.01 seconds. 00:05:02.899 --> 00:05:05.555 That is, a physical car just side-steps the paradox and 00:05:05.555 --> 00:05:08.259 doesn't actually compute speed at a single point in time. 00:05:08.779 --> 00:05:11.679 It computes speed during a very small amount of time. 00:05:13.180 --> 00:05:18.687 So let's call that difference in time dt, which you might think of as 0.01 seconds, 00:05:18.687 --> 00:05:22.360 and let's call that resulting difference in distance ds. 00:05:22.959 --> 00:05:26.743 So the velocity at some point in time is ds divided by dt, 00:05:26.744 --> 00:05:30.400 the tiny change in distance over the tiny change in time. 00:05:31.579 --> 00:05:35.339 Graphically, you can imagine zooming in on some point of this distance vs. 00:05:35.500 --> 00:05:37.680 time graph above t equals 3. 00:05:38.560 --> 00:05:43.143 That dt is a small step to the right, since time is on the horizontal axis, 00:05:43.142 --> 00:05:47.001 and that ds is the resulting change in the height of the graph, 00:05:47.002 --> 00:05:50.439 since the vertical axis represents the distance traveled. 00:05:51.220 --> 00:05:55.472 So ds divided by dt is something you can think of as the rise 00:05:55.471 --> 00:05:59.519 over run slope between two very close points on this graph. 00:06:00.699 --> 00:06:03.439 Of course, there's nothing special about the value t equals 3. 00:06:03.939 --> 00:06:06.797 We could apply this to any other point in time, 00:06:06.797 --> 00:06:10.665 so we consider this expression ds over dt to be a function of t, 00:06:10.665 --> 00:06:15.605 something where I can give you a time t and you can give me back the value of this 00:06:15.605 --> 00:06:18.879 ratio at that time, the velocity as a function of time. 00:06:19.600 --> 00:06:22.838 For example, when I had the computer draw this bump curve here, 00:06:22.838 --> 00:06:27.240 the one representing the velocity function, here's what I had the computer actually do. 00:06:27.939 --> 00:06:32.620 First, I chose a small value for dt, I think in this case it was 0.01. 00:06:33.439 --> 00:06:38.207 Then I had the computer look at a whole bunch of times t between 0 and 10, 00:06:38.208 --> 00:06:41.386 and compute the distance function s at t plus dt, 00:06:41.386 --> 00:06:44.820 and then subtract off the value of that function at t. 00:06:45.420 --> 00:06:51.064 In other words, that's the difference in the distance traveled between the given time, 00:06:51.064 --> 00:06:53.660 t, and the time 0.01 seconds after that. 00:06:54.519 --> 00:06:58.305 Then you can just divide that difference by the change in time, dt, 00:06:58.305 --> 00:07:02.480 and that gives you velocity in meters per second around each point in time. 00:07:04.420 --> 00:07:08.721 So with a formula like this, you could give the computer any curve representing any 00:07:08.721 --> 00:07:12.920 distance function s of t, and it could figure out the curve representing velocity. 00:07:13.540 --> 00:07:17.437 Now would be a good time to pause, reflect, and make sure this idea 00:07:17.437 --> 00:07:21.622 of relating distance to velocity by looking at tiny changes makes sense, 00:07:21.622 --> 00:07:25.520 because we're going to tackle the paradox of the derivative head on. 00:07:27.480 --> 00:07:32.771 This idea of ds over dt, a tiny change in the value of the function s divided by 00:07:32.771 --> 00:07:38.000 the tiny change in the input that caused it, that's almost what a derivative is. 00:07:38.699 --> 00:07:43.908 And even though a car's speedometer will actually look at a concrete change in time, 00:07:43.908 --> 00:07:49.055 like 0.01 seconds, and even though the drawing program here is looking at an actual 00:07:49.055 --> 00:07:54.447 concrete change in time, in pure math the derivative is not this ratio ds over dt for a 00:07:54.447 --> 00:07:59.962 specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt 00:07:59.963 --> 00:08:00.760 approaches 0. 00:08:02.540 --> 00:08:07.333 Luckily there is a really nice visual understanding for what it means to ask what 00:08:07.333 --> 00:08:11.075 this ratio approaches, Remember, for any specific choice of dt, 00:08:11.074 --> 00:08:15.810 this ratio ds over dt is the slope of a line passing through two separate points 00:08:15.810 --> 00:08:16.980 on the graph, right? 00:08:17.740 --> 00:08:22.357 Well as dt approaches 0, and as those two points approach each other, 00:08:22.357 --> 00:08:26.314 the slope of the line approaches the slope of a line that's 00:08:26.314 --> 00:08:30.140 tangent to the graph at whatever point t we're looking at. 00:08:30.579 --> 00:08:33.928 So the true honest-to-goodness pure math derivative is not the 00:08:33.928 --> 00:08:37.119 rise over run slope between two nearby points on the graph, 00:08:37.119 --> 00:08:41.000 it's equal to the slope of a line tangent to the graph at a single point. 00:08:42.360 --> 00:08:45.864 Now notice what I'm not saying, I'm not saying that the derivative is 00:08:45.864 --> 00:08:49.420 whatever happens when dt is infinitely small, whatever that would mean. 00:08:50.000 --> 00:08:52.340 Nor am I saying that you plug in 0 for dt. 00:08:53.039 --> 00:08:58.899 This dt is always a finitely small non-zero value, it's just that it approaches 0 is all. 00:09:03.620 --> 00:09:04.960 I think that's really clever. 00:09:05.379 --> 00:09:08.277 Even though change in an instant makes no sense, 00:09:08.277 --> 00:09:12.003 this idea of letting dt approach 0 is a really sneaky backdoor 00:09:12.003 --> 00:09:16.379 way to talk reasonably about the rate of change at a single point in time. 00:09:17.019 --> 00:09:17.519 Isn't that neat? 00:09:18.059 --> 00:09:20.495 It's kind of flirting with the paradox of change in 00:09:20.495 --> 00:09:22.980 an instant without ever needing to actually touch it. 00:09:23.299 --> 00:09:25.743 And it comes with such a nice visual intuition too, 00:09:25.744 --> 00:09:28.660 as the slope of a tangent line to a single point on the graph. 00:09:30.159 --> 00:09:33.098 And because change in an instant still makes no sense, 00:09:33.099 --> 00:09:37.427 I think it's healthiest for you to think of this slope not as some instantaneous 00:09:37.427 --> 00:09:41.543 rate of change, but instead as the best constant approximation for a rate of 00:09:41.543 --> 00:09:42.720 change around a point. 00:09:44.340 --> 00:09:46.940 By the way, it's worth saying a couple words on notation here. 00:09:47.340 --> 00:09:51.743 Throughout this video I've been using dt to refer to a tiny change in t with 00:09:51.743 --> 00:09:55.403 some actual size, and ds to refer to the resulting change in s, 00:09:55.403 --> 00:09:59.807 which again has an actual size, and this is because that's how I want you to 00:09:59.807 --> 00:10:00.779 think about them. 00:10:01.659 --> 00:10:05.795 But the convention in calculus is that whenever you're using the letter d like this, 00:10:05.796 --> 00:10:08.910 you're kind of announcing your intention that eventually you're 00:10:08.909 --> 00:10:11.100 going to see what happens as dt approaches 0. 00:10:11.919 --> 00:10:16.828 For example, the honest-to-goodness pure math derivative is written as ds divided by dt, 00:10:16.828 --> 00:10:19.697 even though it's technically not a fraction per se, 00:10:19.697 --> 00:10:23.779 but whatever that fraction approaches for smaller and smaller nudges in t. 00:10:25.779 --> 00:10:27.679 I think a specific example should help here. 00:10:28.259 --> 00:10:31.374 You might think that asking about what this ratio approaches 00:10:31.374 --> 00:10:35.303 for smaller and smaller values would make it much more difficult to compute, 00:10:35.303 --> 00:10:37.500 but weirdly it kind of makes things easier. 00:10:38.200 --> 00:10:43.160 Let's say you have a given distance vs time function that happens to be exactly t cubed. 00:10:43.159 --> 00:10:47.771 So after 1 second the car has traveled 1 cubed equals 1 meters, 00:10:47.772 --> 00:10:52.240 after 2 seconds it's traveled 2 cubed, or 8 meters, and so on. 00:10:53.019 --> 00:10:55.635 Now what I'm about to do might seem somewhat complicated, 00:10:55.635 --> 00:10:57.800 but once the dust settles it really is simpler, 00:10:57.801 --> 00:11:01.680 and more importantly it's the kind of thing you only ever have to do once in calculus. 00:11:03.100 --> 00:11:06.951 Let's say you wanted to compute the velocity, ds divided by dt, 00:11:06.951 --> 00:11:09.299 at some specific time, like t equals 2. 00:11:09.940 --> 00:11:13.257 For right now let's think of dt as having an actual size, 00:11:13.256 --> 00:11:16.459 some concrete nudge, we'll let it go to 0 in just a bit. 00:11:17.139 --> 00:11:22.323 The tiny change in distance between 2 seconds and 2 plus dt 00:11:22.323 --> 00:11:27.939 seconds is s of 2 plus dt minus s of 2, and we divide that by dt. 00:11:28.620 --> 00:11:34.659 Since our function is t cubed, that numerator looks like 2 plus dt cubed minus 2 cubed. 00:11:35.259 --> 00:11:38.100 And this is something we can work out algebraically. 00:11:38.100 --> 00:11:42.320 Again, bear with me, there's a reason I'm showing you the details here. 00:11:42.799 --> 00:11:49.843 When you expand that top, what you get is 2 cubed plus 3 times 2 squared dt 00:11:49.844 --> 00:11:57.260 plus 3 times 2 times dt squared plus dt cubed, and all of that is minus 2 cubed. 00:11:58.379 --> 00:12:01.961 Now there's a lot of terms, and I want you to remember that it looks like a mess, 00:12:01.961 --> 00:12:02.879 but it does simplify. 00:12:03.779 --> 00:12:05.899 Those 2 cubed terms cancel out. 00:12:06.519 --> 00:12:11.606 Everything remaining here has a dt in it, and since there's a dt on the bottom there, 00:12:11.606 --> 00:12:13.559 many of those cancel out as well. 00:12:14.279 --> 00:12:19.682 What this means is that the ratio ds divided by dt has boiled down into 00:12:19.682 --> 00:12:24.860 3 times 2 squared plus 2 different terms that each have a dt in them. 00:12:25.580 --> 00:12:30.129 So if we ask what happens as dt approaches 0, representing the idea of looking at a 00:12:30.129 --> 00:12:34.679 smaller and smaller change in time, we can just completely ignore those other terms. 00:12:36.100 --> 00:12:39.250 By eliminating the need to think about a specific dt, 00:12:39.250 --> 00:12:43.100 we've eliminated a lot of the complication in the full expression. 00:12:43.879 --> 00:12:47.360 So what we're left with is this nice clean 3 times 2 squared. 00:12:48.360 --> 00:12:52.490 You can think of that as meaning that the slope of a line tangent to 00:12:52.490 --> 00:12:56.919 the point at t equals 2 of this graph is exactly 3 times 2 squared, or 12. 00:12:57.820 --> 00:13:01.060 And of course, there's nothing special about the time t equals 2. 00:13:01.559 --> 00:13:04.720 We could more generally say that the derivative 00:13:04.721 --> 00:13:08.080 of t cubed as a function of t is 3 times t squared. 00:13:10.740 --> 00:13:13.220 Now take a step back, because that's beautiful. 00:13:13.820 --> 00:13:16.280 The derivative is this crazy complicated idea. 00:13:16.600 --> 00:13:19.623 We've got tiny changes in distance over tiny changes in time, 00:13:19.623 --> 00:13:22.208 but instead of looking at any specific one of those, 00:13:22.207 --> 00:13:24.500 we're talking about what that thing approaches. 00:13:24.500 --> 00:13:26.980 I mean, that's a lot to think about. 00:13:27.639 --> 00:13:31.559 And yet what we've come out with is such a simple expression, 3 times t squared. 00:13:32.960 --> 00:13:36.060 And in practice, you wouldn't go through all this algebra each time. 00:13:36.419 --> 00:13:40.413 Knowing that the derivative of t cubed is 3t squared is one of those things that all 00:13:40.413 --> 00:13:44.500 calculus students learn how to do immediately without having to re-derive it each time. 00:13:45.059 --> 00:13:48.339 And in the next video, I'm going to show you a nice way to think about 00:13:48.340 --> 00:13:51.759 this and a couple other derivative formulas in really nice geometric ways. 00:13:52.500 --> 00:13:56.384 But the point I want to make by showing you all of the algebraic guts 00:13:56.384 --> 00:14:00.326 here is that when you consider the tiny change in distance caused by a 00:14:00.326 --> 00:14:04.600 tiny change in time for some specific value of dt, you'd have kind of a mess. 00:14:05.259 --> 00:14:08.902 But when you consider what that ratio approaches as dt approaches 0, 00:14:08.902 --> 00:14:13.020 it lets you ignore much of that mess, and it really does simplify the problem. 00:14:13.779 --> 00:14:16.720 That right there is kind of the heart of why calculus becomes useful. 00:14:18.019 --> 00:14:21.528 Another reason to show you a concrete derivative like this is that it 00:14:21.528 --> 00:14:25.088 sets the stage for an example of the kind of paradoxes that come about 00:14:25.089 --> 00:14:28.700 if you believe too much in the illusion of instantaneous rate of change. 00:14:30.000 --> 00:14:34.812 So think about the actual car traveling according to this t cubed distance function, 00:14:34.812 --> 00:14:38.720 and consider its motion at the moment t equals 0, right at the start. 00:14:39.700 --> 00:14:43.379 Now ask yourself whether or not the car is moving at that time. 00:14:45.559 --> 00:14:50.298 On the one hand, we can compute its speed at that point using the derivative, 00:14:50.298 --> 00:14:53.700 3t squared, which for time t equals 0 works out to be 0. 00:14:54.779 --> 00:14:59.990 Visually, this means that the tangent line to the graph at that point is perfectly flat, 00:14:59.990 --> 00:15:03.269 so the car's quote-unquote instantaneous velocity is 0, 00:15:03.269 --> 00:15:06.139 and that suggests that obviously it's not moving. 00:15:07.159 --> 00:15:11.860 But on the other hand, if it doesn't start moving at time 0, when does it start moving? 00:15:12.580 --> 00:15:14.540 Really, pause and ponder that for a moment. 00:15:15.100 --> 00:15:17.779 Is the car moving at time t equals 0? 00:15:22.600 --> 00:15:23.379 Do you see the paradox? 00:15:24.259 --> 00:15:26.000 The issue is that the question makes no sense. 00:15:26.539 --> 00:15:30.439 It references the idea of change in a moment, but that doesn't actually exist. 00:15:30.860 --> 00:15:32.600 That's just not what the derivative measures. 00:15:33.480 --> 00:15:38.369 What it means for the derivative of a distance function to be 0 is that the best 00:15:38.369 --> 00:15:43.320 constant approximation for the car's velocity around that point is 0 m per second. 00:15:44.080 --> 00:15:47.580 For example, if you look at an actual change in time, 00:15:47.580 --> 00:15:51.080 say between time 0 and 0.1 seconds, the car does move. 00:15:51.500 --> 00:15:53.700 It moves 0.001 m. 00:15:54.600 --> 00:15:59.853 That's very small, and importantly, it's very small compared to the change in time, 00:15:59.852 --> 00:16:02.979 giving an average speed of only 0.01 m per second. 00:16:03.679 --> 00:16:08.704 And remember, what it means for the derivative of this motion to be 0 is that 00:16:08.705 --> 00:16:13.860 for smaller and smaller nudges in time, this ratio of m per second approaches 0. 00:16:14.840 --> 00:16:16.720 But that's not to say that the car is static. 00:16:17.539 --> 00:16:20.966 Approximating its movement with a constant velocity of 0 is, 00:16:20.966 --> 00:16:22.820 after all, just an approximation. 00:16:24.340 --> 00:16:29.211 So whenever you hear people refer to the derivative as an instantaneous rate of change, 00:16:29.211 --> 00:16:33.472 a phrase which is intrinsically oxymoronic, I want you to think of that as a 00:16:33.472 --> 00:16:37.679 conceptual shorthand for the best constant approximation for rate of change. 00:16:39.179 --> 00:16:42.126 In the next couple videos, I'll be talking more about the derivative, 00:16:42.126 --> 00:16:45.241 what it looks like in different contexts, how do you actually compute it, 00:16:45.241 --> 00:16:48.399 why is it useful, things like that, focusing on visual intuition as always.