1 00:00:05,000 --> 00:00:09,320 Okay, today we're going to learn something called the derivative. 2 00:00:16,980 --> 00:00:20,380 This is what calculus is all about. 3 00:00:23,580 --> 00:00:29,200 The rest of your calculus career is going to depend on the derivative. 4 00:00:31,020 --> 00:00:32,140 I'll define it for you. 5 00:00:32,140 --> 00:00:37,860 But let's start with this function here. 6 00:00:40,380 --> 00:00:42,100 A nice little quadratic function. 7 00:00:44,600 --> 00:00:47,140 Okay, if we wanted to find the 8 00:00:49,260 --> 00:00:51,260 slope of the 9 00:00:52,580 --> 00:00:55,120 secant line, let's say between 10 00:00:55,120 --> 00:00:56,160 these two points. 11 00:01:01,060 --> 00:01:02,920 How do I find the slope of the secant line? 12 00:01:15,220 --> 00:01:32,040 Let's say, what is it, between... 13 00:01:32,040 --> 00:01:34,240 So between x equals one and three, for example. 14 00:01:35,660 --> 00:01:43,960 So this can be f of three minus f of one over three minus one 15 00:01:44,720 --> 00:01:49,720 or y two minus y one over x two minus x one. 16 00:01:53,060 --> 00:01:55,860 Which gives us, what is that? 17 00:01:56,520 --> 00:02:00,880 Three squared minus two times three minus one 18 00:02:00,880 --> 00:02:02,660 minus 19 00:02:17,880 --> 00:02:20,240 and what is that? 20 00:02:20,240 --> 00:02:30,860 9 minus 6 minus 1 minus 1 minus 2 minus 1 is that 0? 21 00:02:33,220 --> 00:02:34,220 Minus... 22 00:02:34,220 --> 00:02:36,880 Sorry, minus 1? 23 00:02:38,220 --> 00:02:39,220 Minus 2. 24 00:02:39,440 --> 00:02:44,640 Minus 2. 25 00:02:46,060 --> 00:02:47,600 Anyways, arithmetic and stuff. 26 00:02:54,040 --> 00:02:55,060 2 is that right? 27 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 28 00:03:01,243 --> 00:03:01,993 that 29 00:03:01,540 --> 00:03:02,700 line, what would we do? 30 00:03:06,520 --> 00:03:10,907 Just use point slope formula, doesn't matter if you use this point or this order 31 00:03:10,907 --> 00:03:11,657 there. 32 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 33 00:03:25,771 --> 00:03:29,200 slope times x minus... 34 00:03:30,800 --> 00:03:31,840 Was that three? 35 00:03:33,380 --> 00:03:34,280 Sorry. 36 00:03:33,780 --> 00:03:34,680 Two. 37 00:03:35,500 --> 00:03:36,840 This should be a two. 38 00:03:37,500 --> 00:03:38,400 Two. 39 00:03:37,960 --> 00:03:38,860 This should be a three. 40 00:03:38,760 --> 00:03:39,660 There we go. 41 00:03:39,960 --> 00:03:40,860 Have you got it? 42 00:03:40,720 --> 00:03:41,620 Yeah. 43 00:03:41,420 --> 00:03:42,320 Okay. 44 00:03:44,560 --> 00:03:45,920 Um, that's easy. 45 00:03:46,760 --> 00:03:49,940 Well, we did this in like eighth grade. 46 00:03:50,620 --> 00:03:51,520 Yeah. 47 00:03:52,280 --> 00:03:54,540 Find equations of secant lines. 48 00:03:56,180 --> 00:04:05,600 So, we started calculus with this, from the first day. 49 00:04:06,460 --> 00:04:10,420 We wanted to know, not the slope of the secant line, but what? 50 00:04:11,600 --> 00:04:13,000 Slope of the tangent line. 51 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. 52 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 53 00:04:37,980 --> 00:04:46,700 couldn't figure it out so how did we estimate it? We found secant lines like 54 00:04:46,700 --> 00:04:56,580 3 minus f of 2.9 over 3 minus 2.9. 55 00:04:58,160 --> 00:04:59,060 OK. 56 00:04:59,860 --> 00:05:00,920 And that got an estimate. 57 00:05:01,200 --> 00:05:02,440 How do we get a better estimate? 58 00:05:03,540 --> 00:05:04,440 2.9 by 3. 59 00:05:04,780 --> 00:05:07,540 And you choose a closer number, closer number. 60 00:05:08,620 --> 00:05:10,020 And then you want a better estimate, 61 00:05:10,120 --> 00:05:14,080 we choose a closer number, and so on. 62 00:05:26,260 --> 00:05:40,220 Let's call this... 63 00:05:40,220 --> 00:05:47,900 So this would be the slope of the secant line between two numbers between whatever 64 00:05:51,820 --> 00:05:55,780 For any chosen x, this gives the slope of the secant line between x and 3. 65 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 66 00:06:11,320 --> 00:06:14,480 And now that we learned something called limits 67 00:06:16,500 --> 00:06:18,480 We want to know 68 00:06:19,440 --> 00:06:24,200 What this is approaching as X approaches 3 69 00:06:25,060 --> 00:06:28,960 And we now have this tool called limits 70 00:06:29,960 --> 00:06:32,660 as X approaches 3 71 00:06:36,480 --> 00:06:39,700 Now all we got to do is if we were able to find this limit 72 00:06:40,220 --> 00:06:44,700 this is the slope of the tangent line and that is what we call the derivative 73 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 74 00:06:52,660 --> 00:07:13,020 As x approaches 3, evaluate the limit of the secant slope. 75 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 76 00:07:43,820 --> 00:07:45,920 You get 0 over 0. 77 00:07:47,400 --> 00:07:49,740 Right, you plug it in here, you get 2, 2 minus 2 is 0. 78 00:07:51,440 --> 00:07:53,640 Okay, so we gotta solve this limit. 79 00:07:54,880 --> 00:07:55,900 How are we gonna solve this limit? 80 00:07:58,540 --> 00:07:59,440 No. 81 00:08:02,800 --> 00:08:04,340 Yeah, let's multiply this out. 82 00:08:05,480 --> 00:08:06,400 First off, simplify. 83 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 84 00:08:33,702 --> 00:08:34,620 three. 85 00:08:34,620 --> 00:08:35,520 is that right? 86 00:08:41,460 --> 00:08:45,000 Probably in 3 you still get 0 over 0. So what are we gonna do next? 87 00:08:51,680 --> 00:08:54,680 So very first trick besides direct substitution 88 00:08:57,180 --> 00:08:59,180 Factor okay, we got factor this 89 00:09:00,320 --> 00:09:03,160 it's usually hard to factor with well not hard but 90 00:09:04,620 --> 00:09:11,780 Let me factor out a negative first. 91 00:09:30,440 --> 00:09:35,500 And what's this factor 2x minus 3x plus 1? 92 00:09:39,320 --> 00:09:41,200 Now these simplify. 93 00:09:53,620 --> 00:09:54,780 Now we just use direct substitution. 94 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 95 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 96 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 97 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 98 00:10:36,380 --> 00:10:49,280 minus 2 is 4 times x minus 3. There's actually two different definitions of 99 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 100 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 101 00:11:15,000 --> 00:11:23,100 Let me call some number right over here x plus some very small number h. 102 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. 103 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 104 00:11:56,840 --> 00:12:03,060 The distance between the x-values is h, and we let h approach zero. 105 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 106 00:12:08,560 --> 00:12:34,687 limit as h approaches 0. And this simplifies, right? h approaches 0. Over h. This is 107 00:12:34,687 --> 00:12:36,320 the 108 00:12:36,320 --> 00:12:39,940 main definition we're going to use I'll show you the other one a slight 109 00:12:39,940 --> 00:12:44,740 difference this is the slope of this tangent line this is the derivative 110 00:12:46,480 --> 00:12:54,840 The other definition is the limit for this definition number two. 111 00:12:58,460 --> 00:12:59,860 The main notation uses f prime to represent the derivative of f. 112 00:12:59,860 --> 00:13:10,220 main notation we're going to use we put this apostrophe up here that stands for 113 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 114 00:13:18,740 --> 00:13:23,860 the first derivative or the derivative the slope of the tangent line here this 115 00:13:23,860 --> 00:13:27,800 This value is the derivative at x equals a. 116 00:13:32,840 --> 00:13:36,220 okay this one is the derivative typically 117 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 118 00:13:59,416 --> 00:14:00,780 show 119 00:14:00,780 --> 00:14:04,060 in a minute why it's better. 120 00:14:07,000 --> 00:14:09,680 So let's do the same function again. 121 00:14:12,660 --> 00:14:13,880 What was it? 122 00:14:13,920 --> 00:14:15,840 x squared minus 2x? 123 00:14:16,360 --> 00:14:17,660 Minus 2x minus 1. 124 00:14:17,840 --> 00:14:18,740 Minus 1. 125 00:14:18,500 --> 00:14:19,400 Minus 1. 126 00:14:21,080 --> 00:14:29,720 This is the... 127 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 128 00:14:37,684 --> 00:14:38,480 over h. 129 00:14:39,360 --> 00:14:42,340 The reason this one I like better, well, for two reasons. 130 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 131 00:14:54,132 --> 00:14:54,882 factoring 132 00:14:54,660 --> 00:14:55,560 is hard. 133 00:14:56,600 --> 00:15:01,140 Okay here we're only going to have to factor out an H out of here. 134 00:15:01,360 --> 00:15:04,860 It's a lot easier when you factor out one term instead of a whole binomial. 135 00:15:07,100 --> 00:15:09,300 The other reason I'll show you in a minute. 136 00:15:10,300 --> 00:15:13,160 So let's go ahead and do that. Do this. 137 00:15:14,240 --> 00:15:15,260 The limit 138 00:15:17,620 --> 00:15:19,980 as h approaches zero. 139 00:15:23,440 --> 00:15:25,360 Here we plug in x plus 140 00:15:25,360 --> 00:15:27,500 h everywhere we see an x. 141 00:15:55,200 --> 00:16:03,460 x squared minus 2x minus 1 all over h. 142 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 143 00:16:18,538 --> 00:16:19,300 approach zero. 144 00:16:19,300 --> 00:16:23,440 h, so the distance between those two ordered pairs goes to zero. 145 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 146 00:16:28,884 --> 00:16:29,634 simplify, 147 00:16:29,500 --> 00:16:33,567 Eventually every numerator term has a factor of h, which cancels with the 148 00:16:33,567 --> 00:16:34,317 denominator. 149 00:16:33,880 --> 00:16:37,120 So let's go ahead and multiply that out. 150 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 151 00:17:09,069 --> 00:17:09,819 bunch some stuff 152 00:17:15,040 --> 00:17:18,040 the x squared and the negative x squared is zero 153 00:17:21,840 --> 00:17:23,800 negative 2x and 2x is zero 154 00:17:28,400 --> 00:17:30,500 negative one positive one is zero 155 00:17:35,140 --> 00:17:42,820 Every term in the numerator has a factor of h, so factor out h. 156 00:17:46,580 --> 00:17:47,540 Let's do this. 157 00:18:02,420 --> 00:18:05,980 And now H divided by H is always 1. 158 00:18:05,980 --> 00:18:08,980 as long as we get... 159 00:18:10,480 --> 00:18:12,780 as h approaches zero 160 00:18:14,560 --> 00:18:17,400 After cancellation, the expression is 2x plus h minus 2. 161 00:18:17,980 --> 00:18:20,480 Now we use direct substitution 162 00:18:20,480 --> 00:18:23,540 Again, the only thing we plug in zero into 163 00:18:23,540 --> 00:18:25,740 is the h 164 00:18:28,560 --> 00:18:38,260 And we get 2x plus 0 minus 2 equals 2x minus 2. 165 00:18:46,140 --> 00:18:47,300 Okay, so 166 00:18:47,300 --> 00:18:51,280 But the derivative is the slope of the tangent line. 167 00:18:52,760 --> 00:18:53,980 Slopes are numbers. 168 00:18:55,000 --> 00:18:56,320 They're not functions. 169 00:19:00,280 --> 00:19:03,680 So how is this the slope of the tangent line? 170 00:19:04,840 --> 00:19:06,660 If it's not a number. 171 00:19:14,980 --> 00:19:17,000 Any thoughts on that? 172 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 173 00:19:31,070 --> 00:19:31,840 the tangent 174 00:19:32,640 --> 00:19:34,620 Ok, yeah, this is very important. 175 00:19:34,760 --> 00:19:40,120 If we plug in a number for x, we get a slope, we get a number. 176 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. 177 00:19:53,900 --> 00:19:55,200 That is a slope. 178 00:19:55,740 --> 00:19:58,120 That is the slope of the tangent line x equals 3. 179 00:19:58,200 --> 00:20:00,180 This is what we did a few minutes ago. 180 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. 181 00:20:12,500 --> 00:20:15,340 And we get the slope of the tangent line. 182 00:20:16,040 --> 00:20:18,280 We get the derivative at x equals 1. 183 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. 184 00:20:31,580 --> 00:20:36,480 We get the slope of the tangent line right here. 185 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 186 00:20:45,431 --> 00:20:46,200 over h. 187 00:20:46,200 --> 00:20:50,780 0 f of x plus h minus fx over x plus h, this formula. 188 00:20:53,680 --> 00:20:58,000 Okay, we end up getting another function of x, 189 00:20:58,340 --> 00:21:01,800 which simply tells us all we do is plug in a number 190 00:21:01,800 --> 00:21:04,080 for x to give us the slope of the tangent line. 191 00:21:05,420 --> 00:21:08,720 Okay, so this gives us the slope of the tangent line, 192 00:21:09,720 --> 00:21:12,700 depending on the value of x, which is nice. 193 00:21:12,800 --> 00:21:14,000 Now we can find the slope anywhere. 194 00:21:14,720 --> 00:21:15,680 Just plug in a different x value, 195 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 196 00:21:23,720 --> 00:21:30,460 other formula the limit is X approaches we usually put a number in here like 3 197 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 198 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 199 00:21:47,420 --> 00:21:55,060 that's the main reason I like the other definition better okay let's talk about 200 00:21:55,700 --> 00:21:59,760 It gives us the instantaneous rate of change. 201 00:22:22,020 --> 00:22:33,520 One notation is f prime of x. 202 00:22:38,640 --> 00:22:42,680 Another one is dy dx. 203 00:22:49,820 --> 00:22:54,560 If we're given like a function f of x equals whatever, 204 00:22:56,260 --> 00:22:59,740 x squared plus 3 or something, we usually use this notation. 205 00:23:01,660 --> 00:23:09,860 If we were just given like y equals, we usually use this notation. 206 00:23:09,860 --> 00:23:14,620 Okay, the other notations would be 207 00:23:17,220 --> 00:23:21,060 D dx of f 208 00:23:22,380 --> 00:23:28,380 That also means the derivative of our function f with respect to our variable x 209 00:23:33,740 --> 00:23:36,420 That the only other one 210 00:23:40,320 --> 00:23:47,220 The expression dy over dx is another derivative notation, not an ordinary fraction. 211 00:23:47,220 --> 00:23:53,960 our function y and tells us with wit with respect to what variable we're 212 00:23:53,960 --> 00:23:57,860 eventually going to have multi very multiple variables so it's gonna be 213 00:23:57,860 --> 00:24:03,980 important which variable we're taking the derivative with respect to but we 214 00:24:03,980 --> 00:24:22,660 all three of these notations questions so far at Leibniz which he used this 215 00:24:22,660 --> 00:24:35,400 notation and I believe Renee Descartes used this notation but either one and 216 00:24:35,400 --> 00:24:40,780 we're gonna use them both all year round so let's do another one 217 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. 218 00:24:46,900 --> 00:24:57,860 Let's find the equation of the tangent line. 219 00:25:01,860 --> 00:25:07,980 Half tax equals 8. 220 00:25:19,240 --> 00:25:23,200 To find the equation of the tangent line, what do we have to find first? 221 00:25:29,980 --> 00:25:30,880 Uh oh. 222 00:25:32,220 --> 00:25:34,760 To find any equation, what do we need to know about the equation? 223 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 224 00:25:45,994 --> 00:25:46,744 tangent 225 00:25:46,200 --> 00:25:47,460 line, that's what I was asking for. 226 00:25:48,500 --> 00:25:50,480 So we need to find the slope of the tangent line. 227 00:25:50,480 --> 00:25:53,300 The slope of the tangent line is the derivative. 228 00:26:02,200 --> 00:26:06,420 We use g prime of x. 229 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 230 00:26:10,867 --> 00:26:11,617 over h. 231 00:26:12,520 --> 00:26:13,840 h approaches zero. 232 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 233 00:26:25,665 --> 00:26:26,415 derivative. 234 00:26:39,300 --> 00:26:43,040 Now we plug in this right here. 235 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 236 00:26:49,284 --> 00:26:51,120 plus 1, all over h. 237 00:26:55,000 --> 00:27:05,700 minus the square root of X plus H X plus one all over H. 238 00:27:11,540 --> 00:27:17,500 Every time we use the definition, you try direct substitution, you always get 0 over 239 00:27:17,500 --> 00:27:18,400 0. 240 00:27:18,920 --> 00:27:19,940 So we need to do some trick. 241 00:27:20,220 --> 00:27:22,340 What's going to be the first, what trick we're going to use here? 242 00:27:23,760 --> 00:27:24,740 Define this limit. 243 00:27:32,220 --> 00:27:35,880 Multiply numerator and denominator by the conjugate. 244 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 245 00:27:55,560 --> 00:27:59,700 x plus x plus one. 246 00:27:59,700 --> 00:28:04,460 get this and this minus 247 00:28:31,340 --> 00:28:35,900 The middle terms cancel when multiplying conjugates. 248 00:28:35,900 --> 00:28:37,880 milli-turns. I always can't tell when you multiply 249 00:28:38,500 --> 00:28:41,880 by the derivative. That's zero. 250 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 251 00:29:12,200 --> 00:29:15,900 And we're only left with H, right? 252 00:29:33,780 --> 00:29:35,580 Now the H is simplified. 253 00:29:38,900 --> 00:29:43,680 Direct substitution now gives 1 over 2 times the square root of x plus 1. 254 00:29:43,680 --> 00:29:44,580 Yeah. 255 00:29:58,100 --> 00:30:02,300 OK, so this is the slope of the secant line, 256 00:30:02,300 --> 00:30:04,740 but depending on the value of x. 257 00:30:06,660 --> 00:30:09,120 So what x value did we want to find today? 258 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. 259 00:30:33,700 --> 00:30:46,360 This is the slope of the tangent line at x equals 8. 260 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 261 00:30:52,240 --> 00:30:53,140 slope formula 262 00:31:00,740 --> 00:31:15,640 There it is. 263 00:31:17,040 --> 00:31:18,860 Let's see if that worked out. 264 00:31:55,080 --> 00:31:56,640 Okay, and there it is. 265 00:31:57,740 --> 00:31:59,780 That is the slope of the tangent line. 266 00:32:00,720 --> 00:32:06,460 That is the equation of the tangent line through the point 8 comma 3. 267 00:32:07,500 --> 00:32:08,400 Pretty cool. 268 00:32:08,560 --> 00:32:09,780 We need no calculators. 269 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 270 00:32:17,119 --> 00:32:17,869 hard 271 00:32:19,060 --> 00:32:20,200 secant line formula. 272 00:32:20,960 --> 00:32:22,240 We were able to do it all with no calculator. 273 00:32:23,240 --> 00:32:24,700 We wanted any other point. 274 00:32:34,740 --> 00:32:38,380 What x value is this not defined at? 275 00:32:41,500 --> 00:32:44,920 Definitely not negative one, and then everything to the left of negative one. 276 00:32:49,480 --> 00:32:59,640 okay is our function defined at negative one it is okay but the slope of the 277 00:32:59,640 --> 00:33:06,240 tangent line is not defined the endpoints well this one's a little 278 00:33:06,240 --> 00:33:13,660 different but it will not have a derivative at endpoints because the 279 00:33:13,660 --> 00:33:21,060 limit as H approaches 0 from the left the function doesn't even exist okay so 280 00:33:21,060 --> 00:33:30,800 derivatives don't exist at endpoints the other problem is this is a vertical 281 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 282 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 283 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. 284 00:34:05,420 --> 00:34:13,860 So the question was, what's the instantaneous rate of flow at three hours? 285 00:34:15,900 --> 00:34:17,320 And t was in hours, right? 286 00:34:19,620 --> 00:34:24,980 We need to find the slope of the tangent line at t equals three. 287 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. 288 00:34:49,800 --> 00:34:54,180 the limit as h approaches 0 289 00:34:57,580 --> 00:34:59,280 V of 290 00:35:00,600 --> 00:35:02,420 T plus 291 00:35:02,420 --> 00:35:07,640 H minus V of T all over H. 292 00:35:07,640 --> 00:35:08,540 Here. 293 00:35:10,120 --> 00:35:16,840 So it's going to be negative t plus h minus h. 294 00:35:16,860 --> 00:35:25,140 1 squared plus 5. 295 00:35:28,880 --> 00:35:35,680 Negative t minus 1 squared plus 5 all over h. 296 00:35:40,580 --> 00:35:42,780 Actually, before we finish this letter, 297 00:35:44,680 --> 00:35:49,620 There's going to be units for this answer. What are the units going to be? 298 00:35:50,420 --> 00:35:55,060 Gallons per hour. Why is that? 299 00:35:59,040 --> 00:36:05,200 This is the change in gallons. All these values are in gallons. 300 00:36:06,860 --> 00:36:10,920 And this is the change in x, this is going to be in hours. 301 00:36:13,060 --> 00:36:13,960 Okay. 302 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 303 00:36:19,431 --> 00:36:20,181 units 304 00:36:19,720 --> 00:36:20,740 of this variable. 305 00:36:21,800 --> 00:36:24,140 So gallons per hour. 306 00:36:25,920 --> 00:36:27,760 Now we just have to multiply this out. 307 00:36:28,080 --> 00:36:29,000 This is going to be a little ugly. 308 00:36:32,440 --> 00:36:38,700 Expand t plus h minus 1, squared. 309 00:36:39,200 --> 00:36:41,320 What is that equal to? 310 00:36:41,780 --> 00:36:42,920 T squared. 311 00:36:44,460 --> 00:36:48,340 S t h minus t. 312 00:36:52,760 --> 00:37:03,780 plus TH plus H squared minus H, is that right? 313 00:37:05,260 --> 00:37:12,360 Minus T minus H, that's this guy right here. 314 00:37:18,340 --> 00:37:23,520 So we have negative t squared. 315 00:37:29,180 --> 00:37:47,860 THTH plus 2TH minus 2T minus 2H minus 1, is that right? 316 00:37:51,280 --> 00:38:07,880 Distribute the negatives and combine like terms. 317 00:38:07,880 --> 00:38:21,660 3t plus 1 minus 5 hopefully I did that right let me distribute this negative 318 00:38:21,660 --> 00:38:31,520 negative negative positive positive positive and we should have a bunch of 319 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 320 00:38:54,220 --> 00:38:55,120 Is that it? 321 00:38:56,320 --> 00:38:57,760 Where did I mess up? 322 00:38:58,040 --> 00:38:58,940 Did I? 323 00:39:04,480 --> 00:39:10,300 Uh, this should have been plus one right here, yeah? 324 00:39:12,200 --> 00:39:15,440 And then this becomes minus one. 325 00:39:15,840 --> 00:39:16,780 There we go. 326 00:39:16,780 --> 00:39:20,540 Now these cancel out. 327 00:39:30,080 --> 00:39:33,260 Is that all we got left? 328 00:39:34,940 --> 00:39:35,840 More rh. 329 00:39:35,620 --> 00:39:40,520 And we factor out an h. 330 00:39:51,980 --> 00:39:57,780 We get negative 2t plus 2. 331 00:40:10,600 --> 00:40:16,580 And then we wanted that V equals, or T equals 3. 332 00:40:19,240 --> 00:40:25,760 Negative 2 times 3 plus 2 times 6, negative 4. 333 00:40:25,760 --> 00:40:29,360 And again we said it's gallons per hour. 334 00:40:32,440 --> 00:40:41,440 The negative rate means the volume is decreasing at 4 gallons per hour. 335 00:40:41,440 --> 00:40:42,340 per hour. 336 00:40:43,980 --> 00:40:49,320 Okay, so all the derivatives for homework tonight. 337 00:40:49,320 --> 00:40:53,000 Alright, you gotta use either of the definitions. 338 00:40:56,380 --> 00:40:57,280 Can you use the power rule? 339 00:40:58,360 --> 00:40:59,840 I have no idea what the power rule is. 340 00:41:01,500 --> 00:41:03,040 We gotta use the definition. 341 00:41:05,060 --> 00:41:07,460 We will learn some shortcuts later on. 342 00:41:08,820 --> 00:41:10,640 With the homework we gotta use the definition. 343 00:41:10,640 --> 00:41:11,540 Okay, the uh...