WEBVTT 00:00:10.320 --> 00:00:13.160 Let me share with you something I found particularly 00:00:13.160 --> 00:00:16.000 weird when I was a student first learning calculus. 00:00:16.780 --> 00:00:21.539 Let's say you have a circle with radius 5 centered at the origin of the xy plane. 00:00:22.140 --> 00:00:26.333 This is something defined with the equation x2 plus y2 equals 5 squared, 00:00:26.333 --> 00:00:30.643 that is, all the points on the circle are a distance 5 from the origin as 00:00:30.643 --> 00:00:34.896 encapsulated by the Pythagorean theorem, where the sum of the squares of 00:00:34.896 --> 00:00:39.439 the two legs on this triangle equals the square of the hypotenuse, 5 squared. 00:00:40.460 --> 00:00:44.923 And suppose you want to find the slope of a tangent line to the circle, 00:00:44.923 --> 00:00:47.060 maybe at the point xy equals 3,4. 00:00:48.140 --> 00:00:51.790 Now if you're savvy with geometry, you might already know that this 00:00:51.790 --> 00:00:55.659 tangent line is perpendicular to the radius touching it at that point. 00:00:56.380 --> 00:00:59.530 But let's say you don't already know that, or maybe you want a 00:00:59.530 --> 00:01:02.679 technique that generalizes to curves other than just circles. 00:01:03.619 --> 00:01:07.332 As with other problems about the slopes of tangent lines to curves, 00:01:07.332 --> 00:01:11.379 the key thought here is to zoom in close enough that the curve basically 00:01:11.379 --> 00:01:16.200 looks just like its own tangent line, and then ask about a tiny step along that curve. 00:01:17.000 --> 00:01:21.430 The y component of that little step is what you might call dy, 00:01:21.430 --> 00:01:27.719 and the x component is dx, so the slope we want is the rise over run, dy divided by dx. 00:01:28.480 --> 00:01:31.642 But unlike other tangent slope problems in calculus, 00:01:31.641 --> 00:01:36.871 this curve is not the graph of a function, so we can't just take a simple derivative, 00:01:36.871 --> 00:01:41.980 asking about the size of some tiny nudge to the output of a function caused by some 00:01:41.980 --> 00:01:43.500 tiny nudge to the input. 00:01:44.019 --> 00:01:47.920 x is not an input, and y is not an output, they're both 00:01:47.921 --> 00:01:51.680 just interdependent values related by some equation. 00:01:52.819 --> 00:01:58.247 This is what's called an implicit curve, it's just the set of all points x, 00:01:58.248 --> 00:02:03.820 y that satisfy some property written in terms of the two variables, x and y. 00:02:04.900 --> 00:02:08.518 The procedure for how you actually find dy, dx for curves like 00:02:08.518 --> 00:02:12.020 this is the thing I found very weird as a calculus student. 00:02:12.659 --> 00:02:19.103 You take the derivative of both sides like this, for x squared you write 2x times dx, 00:02:19.103 --> 00:02:24.334 and similarly y squared becomes 2y times dy, and then the derivative 00:02:24.334 --> 00:02:28.200 of that constant 5 squared on the right is just 0. 00:02:29.520 --> 00:02:32.100 Now you can see why this feels a little strange, right? 00:02:32.560 --> 00:02:37.069 What does it mean to take the derivative of an expression that has multiple 00:02:37.069 --> 00:02:41.639 variables in it, and why is it that we're tacking on dy and dx in this way? 00:02:42.400 --> 00:02:46.078 But if you just blindly move forward with what you get, 00:02:46.078 --> 00:02:51.228 you can rearrange this equation and find an expression for dy divided by dx, 00:02:51.228 --> 00:02:55.240 which in this case comes out to be negative x divided by y. 00:02:56.039 --> 00:02:59.850 So at the point with coordinates x, y equals 3, 4, 00:02:59.850 --> 00:03:04.120 that slope would be negative 3 divided by 4, evidently. 00:03:05.060 --> 00:03:08.860 This strange process is called implicit differentiation. 00:03:09.620 --> 00:03:12.663 Don't worry, I have an explanation for how you can interpret 00:03:12.663 --> 00:03:16.060 taking a derivative of an expression with two variables like this. 00:03:16.580 --> 00:03:20.986 But first I want to set aside this particular problem and show how it's connected 00:03:20.985 --> 00:03:25.500 to a different type of calculus problem, something called a related rates problem. 00:03:26.319 --> 00:03:30.391 Imagine a 5 meter long ladder held up against a wall where the top 00:03:30.391 --> 00:03:33.353 of the ladder starts 4 meters above the ground, 00:03:33.353 --> 00:03:38.719 which by the Pythagorean theorem means that the bottom is 3 meters away from the wall. 00:03:39.620 --> 00:03:42.700 And let's say it's slipping down in such a way that the top 00:03:42.699 --> 00:03:45.780 of the ladder is dropping at a rate of 1 meter per second. 00:03:46.759 --> 00:03:50.221 The question is, in that initial moment, what's the rate at 00:03:50.221 --> 00:03:53.800 which the bottom of the ladder is moving away from the wall? 00:03:55.000 --> 00:03:56.199 It's interesting, right? 00:03:56.479 --> 00:04:00.355 That distance from the bottom of the ladder to the wall is 100% 00:04:00.355 --> 00:04:04.539 determined by the distance from the top of the ladder to the floor. 00:04:05.120 --> 00:04:08.843 So we should have enough information to figure out how the rates of 00:04:08.843 --> 00:04:12.344 change for each of those values actually depend on each other, 00:04:12.344 --> 00:04:16.180 but it might not be entirely clear how exactly you relate those two. 00:04:16.800 --> 00:04:21.566 First things first, it's always nice to give names to the quantities that we care about, 00:04:21.565 --> 00:04:25.789 so let's label that distance from the top of the ladder to the ground y of t, 00:04:25.790 --> 00:04:28.660 written as a function of time because it's changing. 00:04:29.680 --> 00:04:33.900 Likewise, label the distance between the bottom of the ladder and the wall x of t. 00:04:34.819 --> 00:04:39.480 The key equation that relates these terms is the Pythagorean theorem, 00:04:39.480 --> 00:04:43.060 x of t squared plus y of t squared equals 5 squared. 00:04:43.920 --> 00:04:48.540 What makes that a powerful equation to use is that it's true at all points of time. 00:04:50.300 --> 00:04:54.158 One way that you could solve this would be to isolate x of t, 00:04:54.158 --> 00:04:59.346 and then figure out what y of t has to be based on that 1 m per second drop rate, 00:04:59.346 --> 00:05:03.584 and you could take the derivative of the resulting function dx dt, 00:05:03.584 --> 00:05:07.000 the rate at which x is changing with respect to time. 00:05:07.860 --> 00:05:10.979 That's fine, it involves a couple layers of using the chain rule, 00:05:10.978 --> 00:05:15.248 and it'll definitely work for you, but I want to show a different way that you can think 00:05:15.249 --> 00:05:16.400 about the same problem. 00:05:17.319 --> 00:05:21.079 This left hand side of the equation is a function of time, right? 00:05:21.439 --> 00:05:26.100 It just so happens to equal a constant, meaning the value evidently doesn't change 00:05:26.100 --> 00:05:30.535 while time passes, but it's still written as an expression dependent on time, 00:05:30.535 --> 00:05:35.140 which means we can manipulate it like any other function that has t as an input. 00:05:36.060 --> 00:05:40.196 In particular, we can take a derivative of this left hand side, 00:05:40.196 --> 00:05:45.184 which is a way of saying if I let a little bit of time pass, some small dt, 00:05:45.184 --> 00:05:49.386 which causes y to slightly decrease and x to slightly increase, 00:05:49.386 --> 00:05:51.880 how much does this expression change? 00:05:53.000 --> 00:05:55.990 On the one hand, we know that the derivative should be 0, 00:05:55.990 --> 00:06:00.555 since the expression is a constant, and constants don't care about your tiny nudges in 00:06:00.555 --> 00:06:02.339 time, they just remain unchanged. 00:06:03.079 --> 00:06:06.519 But on the other hand, what do you get when you compute this derivative? 00:06:08.019 --> 00:06:14.120 Well, the derivative of x of t squared is 2 times x of t times the derivative of x. 00:06:14.439 --> 00:06:16.980 That's the chain rule that I talked about last video. 00:06:17.620 --> 00:06:23.667 2x dx represents the size of a change to x squared caused by some change to x, 00:06:23.666 --> 00:06:26.379 and then we're dividing out by dt. 00:06:27.500 --> 00:06:30.754 Likewise, the rate at which y of t squared is 00:06:30.754 --> 00:06:34.659 changing is 2 times y of t times the derivative of y. 00:06:35.740 --> 00:06:38.403 Now evidently, this whole expression must be 0, 00:06:38.403 --> 00:06:42.823 and that's an equivalent way of saying that x squared plus y squared must not 00:06:42.822 --> 00:06:44.579 change while the ladder moves. 00:06:45.879 --> 00:06:49.927 At the very start, time t equals 0, the height, y of t, 00:06:49.927 --> 00:06:53.680 is 4 meters, and that distance x of t is 3 meters. 00:06:54.480 --> 00:06:59.588 And since the top of the ladder is dropping at a rate of 1 meter per second, 00:06:59.588 --> 00:07:03.420 that derivative, dy dt, is negative 1 meters per second. 00:07:04.459 --> 00:07:08.847 Now, this gives us enough information to isolate the derivative, dx dt, 00:07:08.848 --> 00:07:13.360 and when you work it out, it comes out to be 4 thirds meters per second. 00:07:14.379 --> 00:07:17.925 The reason I bring up this ladder problem is that I want you to compare 00:07:17.925 --> 00:07:21.520 it to the problem of finding the slope of a tangent line to the circle. 00:07:22.360 --> 00:07:26.619 In both cases, we had the equation x squared plus y squared equals 5 squared, 00:07:26.619 --> 00:07:31.320 and in both cases we ended up taking the derivative of each side of this expression. 00:07:32.199 --> 00:07:36.173 But for the ladder question, these expressions were functions of time, 00:07:36.173 --> 00:07:38.783 so taking the derivative has a clear meaning, 00:07:38.783 --> 00:07:42.360 it's the rate at which the expression changes as time changes. 00:07:43.259 --> 00:07:48.437 But what makes the circle situation strange is that rather than saying that a small 00:07:48.437 --> 00:07:52.305 amount of time dt has passed, which causes x and y to change, 00:07:52.305 --> 00:07:56.798 the derivative just has these tiny nudges dx and dy just floating free, 00:07:56.798 --> 00:07:59.980 not tied to some other common variable, like time. 00:08:01.139 --> 00:08:02.979 Let me show you a nice way to think about this. 00:08:03.240 --> 00:08:07.439 Let's give this expression x squared plus y squared a name, maybe s. 00:08:08.240 --> 00:08:11.060 s is essentially a function of two variables. 00:08:11.879 --> 00:08:15.659 It takes every point xy on the plane and associates it with a number. 00:08:16.620 --> 00:08:19.660 For points on the circle, that number happens to be 25. 00:08:20.560 --> 00:08:24.399 If you stepped off the circle away from the center, that value would be bigger. 00:08:25.060 --> 00:08:29.689 For other points xy closer to the origin, that value would be smaller. 00:08:29.689 --> 00:08:34.846 Now what it means to take a derivative of this expression, a derivative of s, 00:08:34.846 --> 00:08:40.336 is to consider a tiny change to both of these variables, some tiny change dx to x, 00:08:40.336 --> 00:08:45.890 and some tiny change dy to y, and not necessarily one that keeps you on the circle, 00:08:45.890 --> 00:08:50.519 by the way, it's just any tiny step in any direction of the xy plane. 00:08:51.519 --> 00:08:55.019 From there you ask how much does the value of s change? 00:08:56.000 --> 00:09:01.631 That difference, the difference in the value of s before the nudge and after the nudge, 00:09:01.631 --> 00:09:03.379 is what I'm writing as ds. 00:09:04.480 --> 00:09:09.882 For example, in this picture we're starting off at a point where 00:09:09.881 --> 00:09:15.030 x equals 3 and where y equals 4, and let's just say that the 00:09:15.030 --> 00:09:20.179 step I drew has dx at negative 0.02 and dy at negative 0.01. 00:09:21.120 --> 00:09:28.233 Then the decrease in s, the amount that x squared plus y squared changes over that step, 00:09:28.232 --> 00:09:34.779 would be about 2 times 3 times negative 0.02 plus 2 times 4 times negative 0.01. 00:09:35.600 --> 00:09:40.800 That's what this derivative expression, 2x dx plus 2y dy, actually means. 00:09:41.379 --> 00:09:46.657 It's a recipe for telling you how much the value x squared plus y squared changes as 00:09:46.657 --> 00:09:52.060 determined by the point xy where you start and the tiny step dx dy that you take. And 00:09:53.080 --> 00:09:56.631 As with all things derivative, this is only an approximation, 00:09:56.630 --> 00:10:01.579 but it's one that gets truer and truer for smaller and smaller choices of dx and dy. 00:10:02.500 --> 00:10:07.052 The key point here is that when you restrict yourself to steps along the circle, 00:10:07.052 --> 00:10:11.720 you're essentially saying you want to ensure that this value of s doesn't change. 00:10:12.240 --> 00:10:16.519 It starts at a value of 25 and you want to keep it at a value of 25. 00:10:17.179 --> 00:10:19.099 That is, ds should be 0. 00:10:20.200 --> 00:10:25.090 So setting the expression 2x dx plus 2y dy equal to 0 is the condition 00:10:25.090 --> 00:10:29.700 under which one of these tiny steps actually stays on the circle. 00:10:30.620 --> 00:10:32.460 Again, this is only an approximation. 00:10:33.039 --> 00:10:36.429 Speaking more precisely, that condition is what keeps you 00:10:36.429 --> 00:10:39.879 on the tangent line of the circle, not the circle itself. 00:10:40.580 --> 00:10:43.900 But for tiny enough steps, those are essentially the same thing. 00:10:45.179 --> 00:10:47.761 Of course, there's nothing special about the expression 00:10:47.761 --> 00:10:49.779 x squared plus y squared equals 5 squared. 00:10:50.440 --> 00:10:53.521 It's always nice to think through more examples, 00:10:53.520 --> 00:10:57.500 so let's consider this expression sin of x times y2 equals x. 00:10:58.159 --> 00:11:01.639 This corresponds to a whole bunch of u-shaped curves on the plane. 00:11:02.419 --> 00:11:06.879 And those curves, remember, represent all of the points xy where the 00:11:06.879 --> 00:11:11.340 value of sine of x times y squared happens to equal the value of x. 00:11:16.000 --> 00:11:19.557 Now imagine taking some tiny step with components dx and dy, 00:11:19.557 --> 00:11:22.700 and not necessarily one that keeps you on the curve. 00:11:23.820 --> 00:11:27.661 Taking the derivative of each side of this equation will tell 00:11:27.660 --> 00:11:31.439 us how much the value of that side changes during the step. 00:11:32.460 --> 00:11:35.526 On the left side, the product rule that we talked through last 00:11:35.525 --> 00:11:38.839 video tells us that this should be left d right plus right d left. 00:11:39.480 --> 00:11:44.956 That is sine of x times the change to y squared, which is 2y times dy, 00:11:44.956 --> 00:11:50.980 plus y squared times the change to sine of x, which is cosine of x times dx. 00:11:52.019 --> 00:11:56.942 The right side is simply x, so the size of a change to that value is exactly dx, 00:11:56.942 --> 00:11:57.620 right? Now 00:11:59.159 --> 00:12:04.317 setting these two sides equal to each other is a way of saying whatever your tiny 00:12:04.317 --> 00:12:09.093 step with coordinates dx and dy is, if it's going to keep us on the curve, 00:12:09.094 --> 00:12:14.252 the values of both the left hand side and the right hand side must change by the 00:12:14.251 --> 00:12:15.079 same amount. 00:12:15.639 --> 00:12:18.860 That's the only way this top equation can remain true. 00:12:20.220 --> 00:12:23.735 From there, depending on what problem you're trying to solve, 00:12:23.735 --> 00:12:26.442 you have something to work with algebraically, 00:12:26.442 --> 00:12:31.110 and maybe the most common goal is to try to figure out what dy divided by dx is. 00:12:33.210 --> 00:12:37.671 As a final example here, I want to show you how you can use this technique 00:12:37.671 --> 00:12:41.710 of implicit differentiation to figure out new derivative formulas. 00:12:42.629 --> 00:12:46.099 I've mentioned that the derivative of e to the x is itself, 00:12:46.099 --> 00:12:50.626 but what about the derivative of its inverse function, the natural log of x? 00:12:50.625 --> 00:12:55.269 Well the graph of the natural log of x can be thought of as an implicit curve. 00:12:56.049 --> 00:13:00.829 It's all of the points xy on the plane where y happens to equal ln of x. 00:13:01.549 --> 00:13:04.606 It just happens to be the case that the x's and y's of this 00:13:04.606 --> 00:13:08.129 equation aren't as intermingled as they were in our other examples. 00:13:09.350 --> 00:13:15.409 The slope of this graph, dy divided by dx, should be the derivative of ln of x, right? 00:13:16.649 --> 00:13:20.500 Well to find that, first rearrange this equation 00:13:20.500 --> 00:13:24.029 y equals ln of x to be e to the y equals x. 00:13:24.649 --> 00:13:30.850 This is exactly what the natural log of x means, it's saying e to the what equals x. 00:13:31.870 --> 00:13:35.962 Since we know the derivative of e to the y, we can take the derivative 00:13:35.962 --> 00:13:39.471 of both sides here, effectively asking how a tiny step with 00:13:39.471 --> 00:13:43.389 components dx and dy changes the value of each one of these sides. 00:13:44.529 --> 00:13:50.073 To ensure that a step stays on the curve, the change to the left side of the equation, 00:13:50.073 --> 00:13:54.650 which is e to the y times dy, must equal the change to the right side, 00:13:54.650 --> 00:13:56.649 which in this case is just dx. 00:13:57.870 --> 00:14:03.445 Rearranging, that means dy divided by dx, the slope of our graph, 00:14:03.445 --> 00:14:06.190 equals 1 divided by e to the y. 00:14:06.909 --> 00:14:11.755 When we're on the curve, e to the y is by definition the same thing as x, 00:14:11.755 --> 00:14:14.610 so evidently this slope is 1 divided by x. 00:14:15.830 --> 00:14:19.926 And of course, an expression for the slope of a graph of a function 00:14:19.926 --> 00:14:24.084 written in terms of x like this is the derivative of that function, 00:14:24.083 --> 00:14:27.629 so evidently the derivative of ln of x is 1 divided by x. 00:14:32.610 --> 00:14:37.134 By the way, all of this is a little sneak peek into multivariable calculus, 00:14:37.134 --> 00:14:40.935 where you consider functions that have multiple inputs and how 00:14:40.934 --> 00:14:43.829 they change as you tweak those multiple inputs. 00:14:44.870 --> 00:14:48.878 The key, as always, is to have a clear image in your head of what 00:14:48.878 --> 00:14:53.070 tiny nudges are at play, and how exactly they depend on each other. 00:14:54.529 --> 00:14:56.881 Next up, I'm going to be talking about limits, 00:14:56.881 --> 00:14:59.949 and how they're used to formalize the idea of a derivative. 00:15:17.490 --> 00:15:22.730 Thank you.