WEBVTT 00:00:05.040 --> 00:00:07.120 Somebody tell me where are these two things? 00:00:09.240 --> 00:00:15.693 Point-slope form. Okay, point-slope form. Yes, close. They are definitely point- 00:00:15.693 --> 00:00:16.580 slope form. 00:00:26.140 --> 00:00:37.578 But usually we don't put dy dx here. What do we normally put? M. M is the slope. 00:00:37.578 --> 00:00:47.300 Since I have dy dx or F prime, that's the slope of the tangent line. 00:00:47.300 --> 00:01:11.793 Okay. These are simply equations of the tangent line. Equations of the tangent line 00:01:11.793 --> 00:01:16.220 to y at x1, y1. 00:01:20.440 --> 00:01:27.600 So some function y. And why do I notice the function y for the top one? Because of 00:01:27.600 --> 00:01:34.760 this, is the derivative of this some function y, which is not the same function y. 00:01:35.800 --> 00:01:39.665 This notation is a little bit better on bottom because this is definitely the 00:01:39.665 --> 00:01:41.120 equation of the tangent line. 00:01:54.740 --> 00:02:07.820 To the function F at a comma F of a. 00:02:09.960 --> 00:02:15.549 It's just using two different notations. Both slopes are derivatives. So that's the 00:02:15.549 --> 00:02:20.600 slope of the tangent line. So these are the equations of the tangent lines. 00:02:25.220 --> 00:02:33.260 At either x1, y1 of this function y or a comma F of a. Same thing. 00:02:33.500 --> 00:02:35.660 What the function is this? 00:02:36.600 --> 00:02:37.600 Polynomial. 00:02:38.440 --> 00:02:39.440 Polynomial. 00:02:39.440 --> 00:02:41.200 Okay. Find the equation. 00:02:48.220 --> 00:02:49.260 The tangent line. 00:02:49.260 --> 00:02:49.320 Sub urban. 00:02:53.400 --> 00:02:54.040 One. 00:02:54.760 --> 00:02:58.280 X equals zero. 00:03:01.140 --> 00:03:01.320 Oh. 00:03:02.380 --> 00:03:09.580 It's all right. 00:03:16.040 --> 00:03:17.460 So what do we got to do first? 00:03:22.000 --> 00:03:29.319 So to find the equation of the tangent line, are we going to use this one or this 00:03:29.319 --> 00:03:29.680 one? 00:03:37.620 --> 00:03:40.120 The bottom, why the bottom? 00:03:44.300 --> 00:03:49.120 More importantly, this is not y equals, this is f of x equals. 00:03:51.340 --> 00:03:59.400 Okay, so we need to find f prime of a, the slope of the tangent line. 00:03:59.820 --> 00:04:02.040 To find f prime of a, we need to find what? 00:04:05.880 --> 00:04:08.380 f prime of x for any x value. 00:04:08.640 --> 00:04:17.936 Six x to the fifth minus thirty-five x to the fourth plus sixteen x cubed minus 00:04:17.936 --> 00:04:18.760 three x 00:04:18.760 --> 00:04:23.160 squared, plus six x plus two. 00:04:24.080 --> 00:04:24.420 There we go. 00:04:27.920 --> 00:04:33.760 Very simple, using a bunch of power rules and sum-and-difference rules. 00:04:34.760 --> 00:04:39.860 Okay, and then we need f prime of a, right? 00:04:40.580 --> 00:04:49.280 f prime of 0, 0 minus 0 plus 0 minus 0 plus 0 plus 2. 00:04:50.740 --> 00:04:52.420 f prime of 0 is 2. 00:04:56.560 --> 00:05:02.100 And the only other thing we need is f of 0. 00:05:02.980 --> 00:05:04.400 The y value. 00:05:06.960 --> 00:05:09.500 0, 0, 0, 0, 0 minus 8. 00:05:15.740 --> 00:05:24.420 So y minus f of 0 equals f prime of 0 times x minus 0. 00:05:29.980 --> 00:05:31.420 What was the slope? Two. 00:05:49.300 --> 00:05:55.000 It's kind of ugly. It looks like it only has two zeros, two real zeros. 00:05:56.040 --> 00:05:59.220 How many complex zeros does this have? 00:06:04.580 --> 00:06:07.220 What's the degree of this polynomial we wrote down? 00:06:07.220 --> 00:06:08.120 Six. 00:06:08.360 --> 00:06:08.500 Six. 00:06:11.980 --> 00:06:13.060 Guaranteed to have six. 00:06:14.320 --> 00:06:20.780 There's a maximum of five turning points or minimums and maximums. 00:06:23.820 --> 00:06:29.011 Anyways, we found the slope of the, or sorry, we found the equation of the tangent 00:06:29.011 --> 00:06:30.340 line, which was what? 00:06:30.340 --> 00:06:33.700 Y equals, say again? 00:06:35.440 --> 00:06:37.500 Two x minus six? 00:06:38.680 --> 00:06:39.740 Minus eight. 00:06:42.000 --> 00:06:43.840 Okay, we see it's tangent right there. 00:06:47.280 --> 00:06:48.920 If I zoom in, 00:06:53.440 --> 00:06:56.640 you can't even tell the difference between those right there, right? 00:06:56.640 --> 00:07:05.120 They only touch at one single part. Right there at zero negative 8. 00:07:08.960 --> 00:07:13.280 Nearby x equals zero, when I zoom in, we're only looking at very small x values. 00:07:15.660 --> 00:07:18.280 Nearby, the two functions look exactly the same. 00:07:19.240 --> 00:07:20.580 They're just slightly off. 00:07:20.580 --> 00:07:22.800 We agreed? Okay. 00:07:23.110 --> 00:07:25.910 So this is a crazy function. 00:07:42.770 --> 00:07:45.710 Near x equals zero. 00:07:48.110 --> 00:07:53.890 This crazy function is approximately this function. 00:07:56.830 --> 00:08:05.910 f of x is approximately 2x minus 8, which is a very simple linear function. 00:08:09.190 --> 00:08:10.210 Okay. 00:08:15.130 --> 00:08:15.410 So, 00:08:17.110 --> 00:08:19.530 if we want to approximate 00:08:24.970 --> 00:08:26.770 f of, let's say, 00:08:27.050 --> 00:08:28.050 point one. 00:08:31.790 --> 00:08:35.330 Actually, let's say we want to find f of point one. 00:08:37.970 --> 00:08:39.690 It would take a lot of work, right? 00:08:40.670 --> 00:08:47.081 Point one to the sixth power minus seven times point one to the fifth power plus 00:08:47.081 --> 00:08:52.050 four times point one to the fourth power minus point one cubed 00:08:52.050 --> 00:08:59.330 plus three times point one squared plus two times point one minus eight. 00:09:00.890 --> 00:09:04.430 We're going to take a lot of work and probably need a calculator. 00:09:05.510 --> 00:09:06.210 Yeah. Okay. 00:09:08.550 --> 00:09:13.730 We'll plug that in in a second, but the approximation is very simple. 00:09:15.950 --> 00:09:18.210 It's we plug it into this function. 00:09:18.610 --> 00:09:20.970 Two times point one minus eight, 00:09:22.710 --> 00:09:27.870 which is what point two minus eight, which is seven point. 00:09:31.390 --> 00:09:33.430 Negative seven point eight. 00:09:33.430 --> 00:09:35.370 Seven point eight. Is that right? Yeah. Okay. 00:09:38.470 --> 00:09:42.330 So if we want to find values of f 00:09:44.390 --> 00:09:48.510 nearby the x equals zero, like point one is pretty close, point zero. 00:09:50.250 --> 00:09:55.550 It's very simple. We find the equation of the tangent line. 00:09:57.170 --> 00:10:01.350 It's simply plug that value into the equation of the tangent line. 00:10:03.590 --> 00:10:07.850 So this is called the is called linear approximation. 00:10:09.330 --> 00:10:11.450 Actually, it's called three different things. 00:10:13.090 --> 00:10:14.090 Linear approximation. 00:10:19.130 --> 00:10:24.730 Okay. We approximated some very complicated function with a simple linear function. 00:10:25.190 --> 00:10:27.110 That's why it's called linear approximation. 00:10:27.890 --> 00:10:33.610 Or it's called linearization. 00:10:38.010 --> 00:10:40.170 Why is it called that? 00:10:42.690 --> 00:10:47.290 We turn a nonlinear function into a linear function. 00:10:48.890 --> 00:10:52.770 So it's the linearization of the function. We turn it to a linear function. 00:10:54.070 --> 00:11:02.330 Okay. We linearize it. It's also called the tangent line approximation. 00:11:06.350 --> 00:11:08.450 Why is it called that? 00:11:09.110 --> 00:11:10.410 It's the tangent line. 00:11:11.370 --> 00:11:15.135 We're approximated with the tangent slope, or sorry, the equation of the tangent 00:11:15.135 --> 00:11:15.370 line. 00:11:16.510 --> 00:11:18.530 It all means the same thing. 00:11:25.050 --> 00:11:30.550 So this is it. Any function. 00:11:36.130 --> 00:11:42.530 Well, actually before I write that down, if we take this thing right here 00:11:46.110 --> 00:11:49.490 and solve for this guy, which is our linear equation, 00:11:51.610 --> 00:11:57.810 we get f prime of a times x minus a plus f of a. 00:11:59.470 --> 00:12:02.770 This is the equation of the tangent line. 00:12:03.870 --> 00:12:09.890 And this linear function here, y, is our linear function. 00:12:11.150 --> 00:12:23.710 Okay. So it is f of x is approximately f prime of a times x minus a plus f of a. 00:12:30.530 --> 00:12:34.750 This is known as the linearization of a function or linear approximation. 00:12:34.750 --> 00:12:44.590 Let's just call it linear approximation of f. 00:12:45.070 --> 00:12:52.750 But it's only the linear approximation of f at x equals whatever number a is. 00:12:53.010 --> 00:13:02.550 Let's find the linear approximation for the same function at x equals 1. 00:13:05.710 --> 00:13:19.185 Okay. At x equals 1, f of x is approximately f prime of 1 times x minus 1 plus f of 00:13:19.185 --> 00:13:19.510 1. 00:13:20.610 --> 00:13:27.717 And we just have to find two things, the derivative at 1 and the function value at 00:13:27.717 --> 00:13:27.890 1. 00:13:28.550 --> 00:13:31.030 It's only different than the previous problem. 00:13:35.010 --> 00:13:40.710 So where we're at? Let's do that up here real quick. 00:13:44.630 --> 00:13:46.710 This is going to take a little bit more work. 00:13:50.850 --> 00:14:00.110 f prime of 1 is 6 minus 35 plus 16 minus 3 plus 6 plus 2. 00:14:00.450 --> 00:14:01.710 Could somebody do that work? 00:14:02.310 --> 00:14:07.450 So f of 1, what did we say it was? 00:14:07.670 --> 00:14:08.090 Negative 6. 00:14:08.370 --> 00:14:10.790 f prime of 1 is? 00:14:12.210 --> 00:14:12.530 Negative 8. 00:14:12.750 --> 00:14:13.290 Negative 8. 00:14:13.830 --> 00:14:25.430 So our function is approximately negative 8 times x minus 1 plus minus 8 minus 6. 00:14:30.670 --> 00:14:33.630 You just want to keep it real simple. 00:14:33.970 --> 00:14:37.130 Negative eight x plus two. 00:14:56.790 --> 00:14:59.830 And let's approximate at.9. 00:14:59.970 --> 00:15:01.130 Is that close to 1? 00:15:06.330 --> 00:15:07.290 Oh. 00:15:08.410 --> 00:15:19.770 What is that negative.72 plus 2, which is 1 point? 00:15:26.430 --> 00:15:27.770 1.28, is that right? 00:15:39.270 --> 00:15:42.890 Can somebody confirm my basics arithmetic? 00:15:46.110 --> 00:15:49.090 Is that negative 7.2? 00:15:49.630 --> 00:15:49.850 Yes. 00:15:51.130 --> 00:16:00.890 Is that negative 5.2? 00:16:02.070 --> 00:16:03.170 There we go. 00:16:04.760 --> 00:16:10.160 Sorry, it was y minus, or sorry, y equals, what did we get? 00:16:11.540 --> 00:16:12.440 Negative 8. 00:16:22.440 --> 00:16:22.960 Okay. 00:16:33.840 --> 00:16:36.220 So this is the approximation right here. 00:16:36.780 --> 00:16:37.700 Let me turn off the old one. 00:16:39.300 --> 00:16:39.820 Okay. 00:16:39.970 --> 00:16:43.170 Again, it's only the approximation near x equals 1. 00:16:44.450 --> 00:16:44.850 Okay. 00:16:45.490 --> 00:16:49.568 Every different x value, we have a different linear equation, different change in 00:16:49.568 --> 00:16:51.230 line, different linear equations. 00:16:54.890 --> 00:16:55.270 Oops. 00:17:00.510 --> 00:17:00.530 Okay. 00:17:00.730 --> 00:17:01.670 Let's do some more examples. 00:17:01.830 --> 00:17:03.770 Let's keep it simple. 00:17:04.990 --> 00:17:08.290 Again, our function. 00:17:10.510 --> 00:17:18.493 Just the point slope formula, solve for the y, it's f prime of a times x minus a 00:17:18.493 --> 00:17:19.690 plus f of a. 00:17:22.410 --> 00:17:23.970 This should say approximation. 00:17:34.270 --> 00:17:37.070 What is f prime? 00:18:04.250 --> 00:18:07.050 What is f prime? 00:18:12.790 --> 00:18:29.410 f of x is approximately one times x minus zero plus zero. 00:18:29.410 --> 00:18:36.670 Sorry, let's just say f of x is approximately x. 00:18:38.870 --> 00:18:41.750 This is our linear function. 00:18:42.950 --> 00:18:53.430 So sine of x is approximately just our function x near x equals 0. 00:18:54.450 --> 00:18:56.490 In fact, let's look at that real quick. 00:18:56.550 --> 00:18:57.830 This is kind of handy. 00:18:59.750 --> 00:19:04.190 For example, we're going to find sine of 0.2. 00:19:05.110 --> 00:19:06.670 We definitely need a calculator for that, right? 00:19:08.310 --> 00:19:10.850 Can somebody take out the calculator and tell me what that's equal to? 00:19:13.330 --> 00:19:15.850 It's approximately 0.2. 00:19:16.030 --> 00:19:18.690 Oh, got to make sure we're in what mode? 00:19:19.430 --> 00:19:22.470 It has to be in radians. Let's see what this is. 00:19:22.850 --> 00:19:23.110 Oh, it is. 00:19:26.670 --> 00:19:29.750 So sine of 0.2 is about 0.198. 00:19:32.150 --> 00:19:32.850 Slightly less. 00:19:37.550 --> 00:19:39.730 But 0.2 is pretty good approximation. 00:19:40.730 --> 00:19:44.070 It's kind of cool. 00:19:44.230 --> 00:19:45.990 It's a complicated function, right? 00:19:45.990 --> 00:19:47.370 Well, it's not that complicated. 00:19:51.830 --> 00:19:52.910 What's that like that? 00:19:54.290 --> 00:19:55.330 See if you can do this. 00:19:55.990 --> 00:19:58.490 Find the equation. 00:20:00.390 --> 00:20:12.110 Find the linear approximation at x equals 1. 00:20:27.750 --> 00:20:29.150 Okay. 00:20:51.590 --> 00:20:53.790 What's the derivative of the natural? 00:20:58.310 --> 00:20:58.830 Okay. 00:20:58.970 --> 00:21:01.750 Which is just one. That's right. 00:21:06.390 --> 00:21:08.010 And so that's the derivative. 00:21:10.110 --> 00:21:12.870 So f prime of one is simply one over one. 00:21:15.310 --> 00:21:18.270 And what is the natural log of one, everybody? 00:21:18.270 --> 00:21:19.410 Zero. 00:21:23.270 --> 00:21:33.290 So our function, ln x, is approximately one times 00:21:33.890 --> 00:21:35.990 X minus one plus zero. 00:21:41.190 --> 00:21:43.490 There is x minus one. 00:21:43.850 --> 00:21:47.210 Which is that line right there. 00:21:54.350 --> 00:21:59.850 So let's approximate ln. What's the number near one? 00:22:01.290 --> 00:22:02.830 One point two. 00:22:08.210 --> 00:22:12.390 That is 1.2 minus one. 00:22:13.590 --> 00:22:18.030 Let's see what ln of 1.2 is. 00:22:28.950 --> 00:22:31.250 Sorry, ln of one point two. 00:22:36.570 --> 00:22:39.930 So we got point two. It's really point one eight. 00:22:40.710 --> 00:22:41.750 It's a pretty good approximation. 00:22:42.450 --> 00:22:46.370 The farther we get away from our a value, in this case it was one. 00:22:46.910 --> 00:22:47.710 So the worse approximation. 00:22:49.330 --> 00:22:51.190 The closer we get, the better approximation. 00:22:52.250 --> 00:22:54.010 There is a way to calculate the error. 00:22:54.750 --> 00:22:58.870 We don't need to talk about that, but you don't need that for this. 00:23:01.350 --> 00:23:03.770 Actually going back here. 00:23:08.430 --> 00:23:14.843 Without actually plugging in the calculator, how can we tell if this is going to be 00:23:14.843 --> 00:23:19.170 an overestimate or an underestimate of the actual value? 00:23:20.530 --> 00:23:21.790 We should discuss that. 00:23:22.270 --> 00:23:25.957 Looking at the graph, how can we tell if it's going to be an overestimate or an 00:23:25.957 --> 00:23:26.610 underestimate? 00:23:29.850 --> 00:23:30.410 What's that? 00:23:32.530 --> 00:23:37.890 The tangent line is higher, greater than the function. 00:23:38.090 --> 00:23:40.430 So it's definitely going to be an overestimate. 00:23:40.810 --> 00:23:44.170 Let's say this is one point two. 00:23:44.170 --> 00:23:51.690 And we're going to learn later, because this is curving down, called concave down. 00:23:54.590 --> 00:23:58.190 We're going to be able to tell whether it's an overestimate or an underestimate. 00:23:58.630 --> 00:24:00.570 Next question, approximate. 00:24:06.470 --> 00:24:12.710 The square root of nine point three. 00:24:15.910 --> 00:24:17.430 Here, I just give you a value. 00:24:21.930 --> 00:24:23.750 So we're going to have to come up with two things. 00:24:23.910 --> 00:24:30.470 We're going to have to come up with a function and an A value where we can easily 00:24:30.470 --> 00:24:33.710 find the function value at that A value. 00:24:37.430 --> 00:24:39.310 So what function are we going to come up with? 00:24:43.730 --> 00:24:45.510 Let's go with the square root of X. 00:24:47.810 --> 00:24:48.950 And what A value? 00:24:49.110 --> 00:24:52.450 What value do we know the square root of nearby? 00:24:52.990 --> 00:24:53.490 Nine point three? 00:24:54.210 --> 00:24:54.730 Nine. 00:24:56.490 --> 00:25:02.950 So F prime of X is, what is that? 00:25:03.230 --> 00:25:07.370 One half X to negative one half. 00:25:08.430 --> 00:25:11.210 One over two square root x. 00:25:15.730 --> 00:25:17.750 F prime of nine. 00:25:21.770 --> 00:25:24.090 Is that one sixth? 00:25:28.290 --> 00:25:31.830 And F of nine. 00:25:33.510 --> 00:25:35.450 Square root of nine is three. 00:25:39.790 --> 00:25:42.430 So our function, square root x, is approximately 00:25:43.650 --> 00:25:55.590 f prime of nine times x minus nine plus f of nine. 00:26:05.070 --> 00:26:07.810 There's our linear approximation. 00:26:09.290 --> 00:26:10.370 Our linearization. 00:26:12.270 --> 00:26:13.490 Tangent line approximation. 00:26:14.670 --> 00:26:15.750 Square root of X. 00:26:18.430 --> 00:26:20.550 And now we plug in nine point three. 00:26:22.630 --> 00:26:24.830 Is that what I said nine point three? 00:26:24.850 --> 00:26:25.210 Yes. 00:26:25.210 --> 00:26:25.510 Okay. 00:26:46.550 --> 00:26:48.310 Point three. 00:26:50.510 --> 00:26:53.710 What is that? 00:26:54.410 --> 00:26:55.630 Tangent line. 00:26:59.550 --> 00:27:03.390 Three over sixty. 00:27:07.790 --> 00:27:08.990 One twentieth. Is that right? 00:27:14.570 --> 00:27:18.230 Change that to decimals. That's three point oh five. 00:27:23.870 --> 00:27:25.830 So that's a pretty good approximate. 00:27:25.970 --> 00:27:33.750 Well, I don't know how good it is, but we'll do that with some very simple math. 00:27:33.750 --> 00:27:39.950 Obviously all we got to know what derivatives are. All our rules. 00:27:40.510 --> 00:27:46.570 At least the power rule. Let's see, what is the square root of 9.3? 00:27:51.570 --> 00:27:54.370 What do we get? Three point oh five. 00:27:55.210 --> 00:27:57.130 Three point oh four nine. 00:27:57.970 --> 00:27:59.210 Pretty good. 00:28:05.390 --> 00:28:09.606 So linearization has everything to do with finding the equation of the tangent line 00:28:09.606 --> 00:28:10.470 to some function. 00:28:11.510 --> 00:28:14.350 Finding the derivative and plugging in the values. 00:28:14.530 --> 00:28:19.690 Let me draw you something real quick and then I'll explain it. 00:28:28.450 --> 00:28:29.810 This is some function. 00:28:34.490 --> 00:28:35.970 And it's some X value. 00:28:39.750 --> 00:28:45.330 This would be the tangent line. 00:28:51.690 --> 00:28:52.290 Okay. 00:28:54.950 --> 00:28:58.230 And if you have some other X value. 00:29:21.550 --> 00:29:24.130 Let me do different colors here. 00:29:24.690 --> 00:29:28.230 This right here is what's known as delta X. 00:29:28.370 --> 00:29:29.550 The change in X from A. 00:29:30.290 --> 00:29:31.590 This is our delta X right here. 00:29:34.270 --> 00:29:36.490 And this is our actual change in Y. 00:29:37.930 --> 00:29:38.510 Delta Y. 00:29:40.490 --> 00:29:40.930 Okay. 00:29:41.610 --> 00:29:42.570 Does this make sense so far? 00:29:43.650 --> 00:29:43.870 Okay. 00:29:44.410 --> 00:29:45.310 We have two different different directions. 00:29:45.310 --> 00:29:52.450 One is called dy and the other is called dx. 00:29:56.390 --> 00:30:00.270 This delta X is the same as our DX. 00:30:03.110 --> 00:30:11.510 Our differential DY is this distance right here. 00:30:13.030 --> 00:30:13.610 DY. 00:30:15.430 --> 00:30:23.793 It's the change from our equation of the tangent line as opposed to just the actual 00:30:23.793 --> 00:30:26.110 change in the function. 00:30:27.770 --> 00:30:29.510 That's what the differentials are. 00:30:30.270 --> 00:30:31.730 You don't really need to know that. 00:30:33.090 --> 00:30:33.230 Okay. 00:30:33.310 --> 00:30:36.110 But this was just a brief explanation. 00:30:36.950 --> 00:30:38.190 You don't have to memorize that. 00:30:38.190 --> 00:30:40.910 But that's what they are. 00:30:43.850 --> 00:30:45.170 And it's very simple. 00:30:46.690 --> 00:30:52.490 dy over dx equals f prime of x. 00:30:52.570 --> 00:30:53.730 Those mean the same thing, right? 00:30:55.570 --> 00:30:55.890 Yes. 00:30:59.710 --> 00:31:03.350 This is not a fraction even though it looks like a fraction, right? 00:31:04.770 --> 00:31:05.770 What is this? 00:31:05.770 --> 00:31:09.670 This is the derivative of this function Y with respect to X. 00:31:10.650 --> 00:31:16.510 But all you have to do is pretend it is a fraction and you multiply. 00:31:16.850 --> 00:31:18.090 And that's not really what's happening. 00:31:18.910 --> 00:31:20.350 Multiply both sides by DX. 00:31:22.650 --> 00:31:24.030 It looks like these cancel. 00:31:26.950 --> 00:31:27.890 And we get this. 00:31:32.570 --> 00:31:33.630 And this is it. 00:31:33.630 --> 00:31:41.610 Our differential dy is equal to the derivative times the differential dx. 00:31:49.530 --> 00:31:51.930 And we just have to know this. 00:31:52.110 --> 00:31:53.350 This is going to be very important later. 00:31:55.270 --> 00:31:55.350 Okay. 00:31:55.830 --> 00:31:57.170 So quick example. 00:32:04.470 --> 00:32:05.750 Sine of x. 00:32:06.610 --> 00:32:11.570 Find the differential dy. 00:32:14.130 --> 00:32:22.450 All we do is we take the derivative, which is cosine of X, right? 00:32:25.910 --> 00:32:29.750 And the derivative is the same thing as DY DX. 00:32:29.750 --> 00:32:32.150 We set it equal to cosine of x. 00:32:35.110 --> 00:32:37.810 Just pretend that you're multiplying by DX. 00:32:41.170 --> 00:32:42.450 And we get our differentials. 00:32:42.670 --> 00:32:46.290 dy is cosine of x times dx. 00:32:51.210 --> 00:32:57.170 dy is the derivative of the function times our differential dx. 00:32:58.070 --> 00:33:00.430 That's all you got to know how to do. 00:33:01.910 --> 00:33:08.470 We could find this DY here for that function we just did. 00:33:08.930 --> 00:33:14.150 dy for this function is cosine of x times dx. 00:33:16.230 --> 00:33:20.610 And that will give you the change right there. 00:33:21.430 --> 00:33:22.070 You don't really need that. 00:33:22.070 --> 00:33:28.635 All you got to do is you find the derivative, which is DY DX, multiplied by DX, even 00:33:28.635 --> 00:33:31.370 though that's not what's happening. 00:33:32.710 --> 00:33:33.410 And that's your answer. 00:33:34.990 --> 00:33:35.470 Okay. 00:33:36.370 --> 00:33:37.910 What if I want to find DX? 00:33:38.270 --> 00:33:39.110 What is it? 00:33:44.530 --> 00:33:46.070 We divide both sides by cosine. 00:33:48.970 --> 00:33:52.430 One over cosine times DY. 00:33:56.630 --> 00:33:58.610 And sometimes we write it like this. 00:33:58.750 --> 00:34:00.850 dy over cosine of x. 00:34:02.010 --> 00:34:02.330 Okay. 00:34:02.670 --> 00:34:04.030 This is going to be very important later. 00:34:05.570 --> 00:34:09.590 All you need to know is dy equals the derivative times dx. 00:34:10.590 --> 00:34:10.770 Okay.