WEBVTT 00:00:05.000 --> 00:00:09.320 Okay, today we're going to learn something called the derivative. 00:00:16.980 --> 00:00:20.380 This is what calculus is all about. 00:00:23.580 --> 00:00:29.200 The rest of your calculus career is going to depend on the derivative. 00:00:31.020 --> 00:00:32.140 I'll define it for you. 00:00:32.140 --> 00:00:37.860 But let's start with this function here. 00:00:40.380 --> 00:00:42.100 A nice little quadratic function. 00:00:44.600 --> 00:00:47.140 Okay, if we wanted to find the 00:00:49.260 --> 00:00:51.260 slope of the 00:00:52.580 --> 00:00:55.120 secant line, let's say between 00:00:55.120 --> 00:00:56.160 these two points. 00:01:01.060 --> 00:01:02.920 How do I find the slope of the secant line? 00:01:15.220 --> 00:01:32.040 Let's say, what is it, between... 00:01:32.040 --> 00:01:34.240 So between x equals one and three, for example. 00:01:35.660 --> 00:01:43.960 So this can be f of three minus f of one over three minus one 00:01:44.720 --> 00:01:49.720 or y two minus y one over x two minus x one. 00:01:53.060 --> 00:01:55.860 Which gives us, what is that? 00:01:56.520 --> 00:02:00.880 Three squared minus two times three minus one 00:02:00.880 --> 00:02:02.660 minus 00:02:17.880 --> 00:02:20.240 and what is that? 00:02:20.240 --> 00:02:30.860 9 minus 6 minus 1 minus 1 minus 2 minus 1 is that 0? 00:02:33.220 --> 00:02:34.220 Minus... 00:02:34.220 --> 00:02:36.880 Sorry, minus 1? 00:02:38.220 --> 00:02:39.220 Minus 2. 00:02:39.440 --> 00:02:44.640 Minus 2. 00:02:46.060 --> 00:02:47.600 Anyways, arithmetic and stuff. 00:02:54.040 --> 00:02:55.060 2 is that right? 00:02:55.600 --> 00:03:01.243 Ok, so the slope of that line is 2, and what if we wanted to find the equation of 00:03:01.243 --> 00:03:01.993 that 00:03:01.540 --> 00:03:02.700 line, what would we do? 00:03:06.520 --> 00:03:10.907 Just use point slope formula, doesn't matter if you use this point or this order 00:03:10.907 --> 00:03:11.657 there. 00:03:11.200 --> 00:03:25.771 So the equation of the secant line would be y minus, let's say 3, equals 2, the 00:03:25.771 --> 00:03:29.200 slope times x minus... 00:03:30.800 --> 00:03:31.840 Was that three? 00:03:33.380 --> 00:03:34.280 Sorry. 00:03:33.780 --> 00:03:34.680 Two. 00:03:35.500 --> 00:03:36.840 This should be a two. 00:03:37.500 --> 00:03:38.400 Two. 00:03:37.960 --> 00:03:38.860 This should be a three. 00:03:38.760 --> 00:03:39.660 There we go. 00:03:39.960 --> 00:03:40.860 Have you got it? 00:03:40.720 --> 00:03:41.620 Yeah. 00:03:41.420 --> 00:03:42.320 Okay. 00:03:44.560 --> 00:03:45.920 Um, that's easy. 00:03:46.760 --> 00:03:49.940 Well, we did this in like eighth grade. 00:03:50.620 --> 00:03:51.520 Yeah. 00:03:52.280 --> 00:03:54.540 Find equations of secant lines. 00:03:56.180 --> 00:04:05.600 So, we started calculus with this, from the first day. 00:04:06.460 --> 00:04:10.420 We wanted to know, not the slope of the secant line, but what? 00:04:11.600 --> 00:04:13.000 Slope of the tangent line. 00:04:17.500 --> 00:04:23.000 So our goal here is to find the slope of the tangent line at x equals 3. 00:04:23.000 --> 00:04:37.980 We don't have our pair. So first day of school or first day of our lesson we 00:04:37.980 --> 00:04:46.700 couldn't figure it out so how did we estimate it? We found secant lines like 00:04:46.700 --> 00:04:56.580 3 minus f of 2.9 over 3 minus 2.9. 00:04:58.160 --> 00:04:59.060 OK. 00:04:59.860 --> 00:05:00.920 And that got an estimate. 00:05:01.200 --> 00:05:02.440 How do we get a better estimate? 00:05:03.540 --> 00:05:04.440 2.9 by 3. 00:05:04.780 --> 00:05:07.540 And you choose a closer number, closer number. 00:05:08.620 --> 00:05:10.020 And then you want a better estimate, 00:05:10.120 --> 00:05:14.080 we choose a closer number, and so on. 00:05:26.260 --> 00:05:40.220 Let's call this... 00:05:40.220 --> 00:05:47.900 So this would be the slope of the secant line between two numbers between whatever 00:05:51.820 --> 00:05:55.780 For any chosen x, this gives the slope of the secant line between x and 3. 00:05:58.760 --> 00:06:07.080 So wherever X is this gives us the slope of the tangent line between X and 3 00:06:11.320 --> 00:06:14.480 And now that we learned something called limits 00:06:16.500 --> 00:06:18.480 We want to know 00:06:19.440 --> 00:06:24.200 What this is approaching as X approaches 3 00:06:25.060 --> 00:06:28.960 And we now have this tool called limits 00:06:29.960 --> 00:06:32.660 as X approaches 3 00:06:36.480 --> 00:06:39.700 Now all we got to do is if we were able to find this limit 00:06:40.220 --> 00:06:44.700 this is the slope of the tangent line and that is what we call the derivative 00:06:44.700 --> 00:06:52.660 which is the slope of the tangent line so in fact let's do this limit that's 00:06:52.660 --> 00:07:13.020 As x approaches 3, evaluate the limit of the secant slope. 00:07:13.020 --> 00:07:36.900 one is that it okay if we plug in three what do we get we try direct substitution 00:07:43.820 --> 00:07:45.920 You get 0 over 0. 00:07:47.400 --> 00:07:49.740 Right, you plug it in here, you get 2, 2 minus 2 is 0. 00:07:51.440 --> 00:07:53.640 Okay, so we gotta solve this limit. 00:07:54.880 --> 00:07:55.900 How are we gonna solve this limit? 00:07:58.540 --> 00:07:59.440 No. 00:08:02.800 --> 00:08:04.340 Yeah, let's multiply this out. 00:08:05.480 --> 00:08:06.400 First off, simplify. 00:08:18.100 --> 00:08:33.702 plus one. So that's two plus one is three. So negative x squared plus two x plus 00:08:33.702 --> 00:08:34.620 three. 00:08:34.620 --> 00:08:35.520 is that right? 00:08:41.460 --> 00:08:45.000 Probably in 3 you still get 0 over 0. So what are we gonna do next? 00:08:51.680 --> 00:08:54.680 So very first trick besides direct substitution 00:08:57.180 --> 00:08:59.180 Factor okay, we got factor this 00:09:00.320 --> 00:09:03.160 it's usually hard to factor with well not hard but 00:09:04.620 --> 00:09:11.780 Let me factor out a negative first. 00:09:30.440 --> 00:09:35.500 And what's this factor 2x minus 3x plus 1? 00:09:39.320 --> 00:09:41.200 Now these simplify. 00:09:53.620 --> 00:09:54.780 Now we just use direct substitution. 00:09:54.780 --> 00:10:10.260 plus 1 is 4. So we took the limit of the slope of the secant line and we ended up 00:10:10.260 --> 00:10:20.940 with 4 which is the slope of the tangent line. Okay so all we got to do is 00:10:21.200 --> 00:10:28.400 limits of the slope of the secant line and this gives us the slope of the tangent 00:10:28.400 --> 00:10:36.380 line so the slope of that line is 4 the equation of the tangent line would be y 00:10:36.380 --> 00:10:49.280 minus 2 is 4 times x minus 3. There's actually two different definitions of 00:10:49.280 --> 00:11:04.940 the derivative the my favorite one let's say we have any function here okay if we 00:11:04.940 --> 00:11:15.000 want to know the slope of that tangent line x equals 1 we need two points let 00:11:15.000 --> 00:11:23.100 Let me call some number right over here x plus some very small number h. 00:11:27.880 --> 00:11:41.020 The secant slope is f of x plus h minus f of x, divided by x plus h minus x. 00:11:41.020 --> 00:11:56.840 h minus x. Okay, so that's the slope of the tangent line there. h is the distance 00:11:56.840 --> 00:12:03.060 The distance between the x-values is h, and we let h approach zero. 00:12:03.060 --> 00:12:08.500 we're going to do is we want this distance to go to zero. So we take the 00:12:08.560 --> 00:12:34.687 limit as h approaches 0. And this simplifies, right? h approaches 0. Over h. This is 00:12:34.687 --> 00:12:36.320 the 00:12:36.320 --> 00:12:39.940 main definition we're going to use I'll show you the other one a slight 00:12:39.940 --> 00:12:44.740 difference this is the slope of this tangent line this is the derivative 00:12:46.480 --> 00:12:54.840 The other definition is the limit for this definition number two. 00:12:58.460 --> 00:12:59.860 The main notation uses f prime to represent the derivative of f. 00:12:59.860 --> 00:13:10.220 main notation we're going to use we put this apostrophe up here that stands for 00:13:10.220 --> 00:13:18.740 the derivative of F we sometimes raise F prime of X or just F prime that means 00:13:18.740 --> 00:13:23.860 the first derivative or the derivative the slope of the tangent line here this 00:13:23.860 --> 00:13:27.800 This value is the derivative at x equals a. 00:13:32.840 --> 00:13:36.220 okay this one is the derivative typically 00:13:36.220 --> 00:13:59.416 at, it's the derivative, at x equals a. This is the derivative at x. Okay, and I'll 00:13:59.416 --> 00:14:00.780 show 00:14:00.780 --> 00:14:04.060 in a minute why it's better. 00:14:07.000 --> 00:14:09.680 So let's do the same function again. 00:14:12.660 --> 00:14:13.880 What was it? 00:14:13.920 --> 00:14:15.840 x squared minus 2x? 00:14:16.360 --> 00:14:17.660 Minus 2x minus 1. 00:14:17.840 --> 00:14:18.740 Minus 1. 00:14:18.500 --> 00:14:19.400 Minus 1. 00:14:21.080 --> 00:14:29.720 This is the... 00:14:29.720 --> 00:14:37.684 The definition is the limit as h approaches zero of f of x plus h minus f of x, all 00:14:37.684 --> 00:14:38.480 over h. 00:14:39.360 --> 00:14:42.340 The reason this one I like better, well, for two reasons. 00:14:44.620 --> 00:14:54.132 One is when we try to simplify this, you end up having to factor out a binomial and 00:14:54.132 --> 00:14:54.882 factoring 00:14:54.660 --> 00:14:55.560 is hard. 00:14:56.600 --> 00:15:01.140 Okay here we're only going to have to factor out an H out of here. 00:15:01.360 --> 00:15:04.860 It's a lot easier when you factor out one term instead of a whole binomial. 00:15:07.100 --> 00:15:09.300 The other reason I'll show you in a minute. 00:15:10.300 --> 00:15:13.160 So let's go ahead and do that. Do this. 00:15:14.240 --> 00:15:15.260 The limit 00:15:17.620 --> 00:15:19.980 as h approaches zero. 00:15:23.440 --> 00:15:25.360 Here we plug in x plus 00:15:25.360 --> 00:15:27.500 h everywhere we see an x. 00:15:55.200 --> 00:16:03.460 x squared minus 2x minus 1 all over h. 00:16:12.060 --> 00:16:18.538 This is the slope of the secant line, and we let the distance h between the points 00:16:18.538 --> 00:16:19.300 approach zero. 00:16:19.300 --> 00:16:23.440 h, so the distance between those two ordered pairs goes to zero. 00:16:24.880 --> 00:16:28.884 Okay, now to solve this limit, we just have to multiply out a bunch of stuff and 00:16:28.884 --> 00:16:29.634 simplify, 00:16:29.500 --> 00:16:33.567 Eventually every numerator term has a factor of h, which cancels with the 00:16:33.567 --> 00:16:34.317 denominator. 00:16:33.880 --> 00:16:37.120 So let's go ahead and multiply that out. 00:17:04.680 --> 00:17:09.069 okay once you multiply it out a bunch of stuff is going to cancel out but not a 00:17:09.069 --> 00:17:09.819 bunch some stuff 00:17:15.040 --> 00:17:18.040 the x squared and the negative x squared is zero 00:17:21.840 --> 00:17:23.800 negative 2x and 2x is zero 00:17:28.400 --> 00:17:30.500 negative one positive one is zero 00:17:35.140 --> 00:17:42.820 Every term in the numerator has a factor of h, so factor out h. 00:17:46.580 --> 00:17:47.540 Let's do this. 00:18:02.420 --> 00:18:05.980 And now H divided by H is always 1. 00:18:05.980 --> 00:18:08.980 as long as we get... 00:18:10.480 --> 00:18:12.780 as h approaches zero 00:18:14.560 --> 00:18:17.400 After cancellation, the expression is 2x plus h minus 2. 00:18:17.980 --> 00:18:20.480 Now we use direct substitution 00:18:20.480 --> 00:18:23.540 Again, the only thing we plug in zero into 00:18:23.540 --> 00:18:25.740 is the h 00:18:28.560 --> 00:18:38.260 And we get 2x plus 0 minus 2 equals 2x minus 2. 00:18:46.140 --> 00:18:47.300 Okay, so 00:18:47.300 --> 00:18:51.280 But the derivative is the slope of the tangent line. 00:18:52.760 --> 00:18:53.980 Slopes are numbers. 00:18:55.000 --> 00:18:56.320 They're not functions. 00:19:00.280 --> 00:19:03.680 So how is this the slope of the tangent line? 00:19:04.840 --> 00:19:06.660 If it's not a number. 00:19:14.980 --> 00:19:17.000 Any thoughts on that? 00:19:23.760 --> 00:19:31.070 So this is not a slope because it's not a number, but I still call it the slope of 00:19:31.070 --> 00:19:31.840 the tangent 00:19:32.640 --> 00:19:34.620 Ok, yeah, this is very important. 00:19:34.760 --> 00:19:40.120 If we plug in a number for x, we get a slope, we get a number. 00:19:41.040 --> 00:19:52.980 So, for example, F prime of 3, we get 2 times 3 minus 2, which is 4. 00:19:53.900 --> 00:19:55.200 That is a slope. 00:19:55.740 --> 00:19:58.120 That is the slope of the tangent line x equals 3. 00:19:58.200 --> 00:20:00.180 This is what we did a few minutes ago. 00:20:03.040 --> 00:20:10.460 If I plug in 1, f prime of 1, we get 2 times 1 minus 2 is 0. 00:20:12.500 --> 00:20:15.340 And we get the slope of the tangent line. 00:20:16.040 --> 00:20:18.280 We get the derivative at x equals 1. 00:20:19.700 --> 00:20:30.280 If we plug in, let's say, zero, we get two times zero minus two is negative two. 00:20:31.580 --> 00:20:36.480 We get the slope of the tangent line right here. 00:20:37.740 --> 00:20:45.431 The definition is the limit as h approaches zero of f of x plus h minus f of x, all 00:20:45.431 --> 00:20:46.200 over h. 00:20:46.200 --> 00:20:50.780 0 f of x plus h minus fx over x plus h, this formula. 00:20:53.680 --> 00:20:58.000 Okay, we end up getting another function of x, 00:20:58.340 --> 00:21:01.800 which simply tells us all we do is plug in a number 00:21:01.800 --> 00:21:04.080 for x to give us the slope of the tangent line. 00:21:05.420 --> 00:21:08.720 Okay, so this gives us the slope of the tangent line, 00:21:09.720 --> 00:21:12.700 depending on the value of x, which is nice. 00:21:12.800 --> 00:21:14.000 Now we can find the slope anywhere. 00:21:14.720 --> 00:21:15.680 Just plug in a different x value, 00:21:15.680 --> 00:21:23.720 we get a different slope of the tangent line. If we use if we were to use the 00:21:23.720 --> 00:21:30.460 other formula the limit is X approaches we usually put a number in here like 3 00:21:31.600 --> 00:21:41.540 f of X minus f of 3 over X minus 3 this will give us a specific number this 00:21:41.540 --> 00:21:47.420 gives us 4 it does not tell us the slope of the tangent line anywhere okay so 00:21:47.420 --> 00:21:55.060 that's the main reason I like the other definition better okay let's talk about 00:21:55.700 --> 00:21:59.760 It gives us the instantaneous rate of change. 00:22:22.020 --> 00:22:33.520 One notation is f prime of x. 00:22:38.640 --> 00:22:42.680 Another one is dy dx. 00:22:49.820 --> 00:22:54.560 If we're given like a function f of x equals whatever, 00:22:56.260 --> 00:22:59.740 x squared plus 3 or something, we usually use this notation. 00:23:01.660 --> 00:23:09.860 If we were just given like y equals, we usually use this notation. 00:23:09.860 --> 00:23:14.620 Okay, the other notations would be 00:23:17.220 --> 00:23:21.060 D dx of f 00:23:22.380 --> 00:23:28.380 That also means the derivative of our function f with respect to our variable x 00:23:33.740 --> 00:23:36.420 That the only other one 00:23:40.320 --> 00:23:47.220 The expression dy over dx is another derivative notation, not an ordinary fraction. 00:23:47.220 --> 00:23:53.960 our function y and tells us with wit with respect to what variable we're 00:23:53.960 --> 00:23:57.860 eventually going to have multi very multiple variables so it's gonna be 00:23:57.860 --> 00:24:03.980 important which variable we're taking the derivative with respect to but we 00:24:03.980 --> 00:24:22.660 all three of these notations questions so far at Leibniz which he used this 00:24:22.660 --> 00:24:35.400 notation and I believe Renee Descartes used this notation but either one and 00:24:35.400 --> 00:24:40.780 we're gonna use them both all year round so let's do another one 00:24:40.780 --> 00:24:44.540 It's called g of x. Let's just keep it simple. Square root of x plus 1. 00:24:46.900 --> 00:24:57.860 Let's find the equation of the tangent line. 00:25:01.860 --> 00:25:07.980 Half tax equals 8. 00:25:19.240 --> 00:25:23.200 To find the equation of the tangent line, what do we have to find first? 00:25:29.980 --> 00:25:30.880 Uh oh. 00:25:32.220 --> 00:25:34.760 To find any equation, what do we need to know about the equation? 00:25:42.280 --> 00:25:45.994 We need to find the slope of the tangent line before we can find the equation of the 00:25:45.994 --> 00:25:46.744 tangent 00:25:46.200 --> 00:25:47.460 line, that's what I was asking for. 00:25:48.500 --> 00:25:50.480 So we need to find the slope of the tangent line. 00:25:50.480 --> 00:25:53.300 The slope of the tangent line is the derivative. 00:26:02.200 --> 00:26:06.420 We use g prime of x. 00:26:09.840 --> 00:26:10.867 Use g prime of x, the limit as h approaches zero of g of x plus h minus g of x, all 00:26:10.867 --> 00:26:11.617 over h. 00:26:12.520 --> 00:26:13.840 h approaches zero. 00:26:17.020 --> 00:26:25.665 g of x plus h minus g of x all over h. That is simply the definition of the 00:26:25.665 --> 00:26:26.415 derivative. 00:26:39.300 --> 00:26:43.040 Now we plug in this right here. 00:26:43.040 --> 00:26:49.284 Substitution gives the square root of x plus h plus 1, minus the square root of x 00:26:49.284 --> 00:26:51.120 plus 1, all over h. 00:26:55.000 --> 00:27:05.700 minus the square root of X plus H X plus one all over H. 00:27:11.540 --> 00:27:17.500 Every time we use the definition, you try direct substitution, you always get 0 over 00:27:17.500 --> 00:27:18.400 0. 00:27:18.920 --> 00:27:19.940 So we need to do some trick. 00:27:20.220 --> 00:27:22.340 What's going to be the first, what trick we're going to use here? 00:27:23.760 --> 00:27:24.740 Define this limit. 00:27:32.220 --> 00:27:35.880 Multiply numerator and denominator by the conjugate. 00:27:36.040 --> 00:27:51.380 conjugate which is what rad x plus h plus 1 minus plus rad x plus 1 over 00:27:55.560 --> 00:27:59.700 x plus x plus one. 00:27:59.700 --> 00:28:04.460 get this and this minus 00:28:31.340 --> 00:28:35.900 The middle terms cancel when multiplying conjugates. 00:28:35.900 --> 00:28:37.880 milli-turns. I always can't tell when you multiply 00:28:38.500 --> 00:28:41.880 by the derivative. That's zero. 00:28:58.780 --> 00:29:06.840 minus minus okay the x's cancel is zero the one and the negative one is zero 00:29:12.200 --> 00:29:15.900 And we're only left with H, right? 00:29:33.780 --> 00:29:35.580 Now the H is simplified. 00:29:38.900 --> 00:29:43.680 Direct substitution now gives 1 over 2 times the square root of x plus 1. 00:29:43.680 --> 00:29:44.580 Yeah. 00:29:58.100 --> 00:30:02.300 OK, so this is the slope of the secant line, 00:30:02.300 --> 00:30:04.740 but depending on the value of x. 00:30:06.660 --> 00:30:09.120 So what x value did we want to find today? 00:30:11.100 --> 00:30:28.780 At x equals 8, g prime of 8 is 1 over 2 times the square root of 9, or 1 sixth. 00:30:33.700 --> 00:30:46.360 This is the slope of the tangent line at x equals 8. 00:30:47.340 --> 00:30:52.240 and to find the equation now that we got the slope it's simply we always do point 00:30:52.240 --> 00:30:53.140 slope formula 00:31:00.740 --> 00:31:15.640 There it is. 00:31:17.040 --> 00:31:18.860 Let's see if that worked out. 00:31:55.080 --> 00:31:56.640 Okay, and there it is. 00:31:57.740 --> 00:31:59.780 That is the slope of the tangent line. 00:32:00.720 --> 00:32:06.460 That is the equation of the tangent line through the point 8 comma 3. 00:32:07.500 --> 00:32:08.400 Pretty cool. 00:32:08.560 --> 00:32:09.780 We need no calculators. 00:32:10.640 --> 00:32:17.119 We didn't have to keep plugging in values closer and closer and closer to 8 to the 00:32:17.119 --> 00:32:17.869 hard 00:32:19.060 --> 00:32:20.200 secant line formula. 00:32:20.960 --> 00:32:22.240 We were able to do it all with no calculator. 00:32:23.240 --> 00:32:24.700 We wanted any other point. 00:32:34.740 --> 00:32:38.380 What x value is this not defined at? 00:32:41.500 --> 00:32:44.920 Definitely not negative one, and then everything to the left of negative one. 00:32:49.480 --> 00:32:59.640 okay is our function defined at negative one it is okay but the slope of the 00:32:59.640 --> 00:33:06.240 tangent line is not defined the endpoints well this one's a little 00:33:06.240 --> 00:33:13.660 different but it will not have a derivative at endpoints because the 00:33:13.660 --> 00:33:21.060 limit as H approaches 0 from the left the function doesn't even exist okay so 00:33:21.060 --> 00:33:30.800 derivatives don't exist at endpoints the other problem is this is a vertical 00:33:30.800 --> 00:33:37.780 tangent line which the slope won't exist anyways okay let's do one more what time 00:33:38.080 --> 00:33:53.560 was V of T negative minus 1 squared plus 5. This gives us the volume, the amount of 00:33:53.560 --> 00:34:01.160 water in the tank, the volume of the tank in gallons. So this is in gallons. Yeah. 00:34:05.420 --> 00:34:13.860 So the question was, what's the instantaneous rate of flow at three hours? 00:34:15.900 --> 00:34:17.320 And t was in hours, right? 00:34:19.620 --> 00:34:24.980 We need to find the slope of the tangent line at t equals three. 00:34:32.760 --> 00:34:45.980 So, we need to find V prime of t, or also known as, well, dV dt, same thing. 00:34:49.800 --> 00:34:54.180 the limit as h approaches 0 00:34:57.580 --> 00:34:59.280 V of 00:35:00.600 --> 00:35:02.420 T plus 00:35:02.420 --> 00:35:07.640 H minus V of T all over H. 00:35:07.640 --> 00:35:08.540 Here. 00:35:10.120 --> 00:35:16.840 So it's going to be negative t plus h minus h. 00:35:16.860 --> 00:35:25.140 1 squared plus 5. 00:35:28.880 --> 00:35:35.680 Negative t minus 1 squared plus 5 all over h. 00:35:40.580 --> 00:35:42.780 Actually, before we finish this letter, 00:35:44.680 --> 00:35:49.620 There's going to be units for this answer. What are the units going to be? 00:35:50.420 --> 00:35:55.060 Gallons per hour. Why is that? 00:35:59.040 --> 00:36:05.200 This is the change in gallons. All these values are in gallons. 00:36:06.860 --> 00:36:10.920 And this is the change in x, this is going to be in hours. 00:36:13.060 --> 00:36:13.960 Okay. 00:36:14.800 --> 00:36:19.431 You can always look at this, it's always the units of this function divided by the 00:36:19.431 --> 00:36:20.181 units 00:36:19.720 --> 00:36:20.740 of this variable. 00:36:21.800 --> 00:36:24.140 So gallons per hour. 00:36:25.920 --> 00:36:27.760 Now we just have to multiply this out. 00:36:28.080 --> 00:36:29.000 This is going to be a little ugly. 00:36:32.440 --> 00:36:38.700 Expand t plus h minus 1, squared. 00:36:39.200 --> 00:36:41.320 What is that equal to? 00:36:41.780 --> 00:36:42.920 T squared. 00:36:44.460 --> 00:36:48.340 S t h minus t. 00:36:52.760 --> 00:37:03.780 plus TH plus H squared minus H, is that right? 00:37:05.260 --> 00:37:12.360 Minus T minus H, that's this guy right here. 00:37:18.340 --> 00:37:23.520 So we have negative t squared. 00:37:29.180 --> 00:37:47.860 THTH plus 2TH minus 2T minus 2H minus 1, is that right? 00:37:51.280 --> 00:38:07.880 Distribute the negatives and combine like terms. 00:38:07.880 --> 00:38:21.660 3t plus 1 minus 5 hopefully I did that right let me distribute this negative 00:38:21.660 --> 00:38:31.520 negative negative positive positive positive and we should have a bunch of 00:38:32.040 --> 00:38:53.720 here negative 2t squared t squared 2t minus 2t plus 1 that's not right 5 and 00:38:54.220 --> 00:38:55.120 Is that it? 00:38:56.320 --> 00:38:57.760 Where did I mess up? 00:38:58.040 --> 00:38:58.940 Did I? 00:39:04.480 --> 00:39:10.300 Uh, this should have been plus one right here, yeah? 00:39:12.200 --> 00:39:15.440 And then this becomes minus one. 00:39:15.840 --> 00:39:16.780 There we go. 00:39:16.780 --> 00:39:20.540 Now these cancel out. 00:39:30.080 --> 00:39:33.260 Is that all we got left? 00:39:34.940 --> 00:39:35.840 More rh. 00:39:35.620 --> 00:39:40.520 And we factor out an h. 00:39:51.980 --> 00:39:57.780 We get negative 2t plus 2. 00:40:10.600 --> 00:40:16.580 And then we wanted that V equals, or T equals 3. 00:40:19.240 --> 00:40:25.760 Negative 2 times 3 plus 2 times 6, negative 4. 00:40:25.760 --> 00:40:29.360 And again we said it's gallons per hour. 00:40:32.440 --> 00:40:41.440 The negative rate means the volume is decreasing at 4 gallons per hour. 00:40:41.440 --> 00:40:42.340 per hour. 00:40:43.980 --> 00:40:49.320 Okay, so all the derivatives for homework tonight. 00:40:49.320 --> 00:40:53.000 Alright, you gotta use either of the definitions. 00:40:56.380 --> 00:40:57.280 Can you use the power rule? 00:40:58.360 --> 00:40:59.840 I have no idea what the power rule is. 00:41:01.500 --> 00:41:03.040 We gotta use the definition. 00:41:05.060 --> 00:41:07.460 We will learn some shortcuts later on. 00:41:08.820 --> 00:41:10.640 With the homework we gotta use the definition. 00:41:10.640 --> 00:41:11.540 Okay, the uh...