1 00:00:00 --> 00:00:00,11 2 00:00:00,11 --> 00:00:02,32 The following content is provided under a Creative 3 00:00:02,32 --> 00:00:03,15 Commons license. 4 00:00:03,15 --> 00:00:06,57 Your support will help MIT OpenCourseWare continue to 5 00:00:06,57 --> 00:00:09,96 offer high quality educational resources for free. 6 00:00:09,96 --> 00:00:13,17 To make a donation, or to view additional materials from 7 00:00:13,17 --> 00:00:15,91 hundreds of MIT courses visit MIT OpenCourseWare 8 00:00:15,91 --> 00:00:22,24 at ocw.mit.edu. 9 00:00:22,24 --> 00:00:25,23 Professor: I am Haynes Miller, I am substituting for 10 00:00:25,23 --> 00:00:26,51 David Jerison today. 11 00:00:26,51 --> 00:00:41,88 So you have a substitute teacher today. 12 00:00:41,88 --> 00:00:45,08 So I haven't been here in this class with you so I'm not 13 00:00:45,08 --> 00:00:47,58 completely sure where you are. 14 00:00:47,58 --> 00:00:53,59 I think just been talking about differentiation and you've 15 00:00:53,59 --> 00:00:56,84 got some examples of differentiation like these 16 00:00:56,84 --> 00:00:59,86 basic examples: the derivative of x^n is nx^(x-1). 17 00:00:59,86 --> 00:01:03,35 18 00:01:03,35 --> 00:01:05,42 But I think maybe you've spent some time computing the 19 00:01:05,42 --> 00:01:11,23 derivative of the sine function as well, recently. 20 00:01:11,23 --> 00:01:16,51 And I think you have some rules for extending these 21 00:01:16,51 --> 00:01:18,95 calculations as well. 22 00:01:18,95 --> 00:01:24,19 For instance, I think you know that if you differentiate a 23 00:01:24,19 --> 00:01:27,77 constant times a function, what do you get? 24 00:01:27,77 --> 00:01:32,59 Student: [INAUDIBLE]. 25 00:01:32,59 --> 00:01:36,67 Professor: The constant comes outside like this. 26 00:01:36,67 --> 00:01:40,03 Or I could write (cu)' = cu'. 27 00:01:40,03 --> 00:01:42,55 28 00:01:42,55 --> 00:01:45,51 That's this rule, multiplying by a constant, and I 29 00:01:45,51 --> 00:01:58,87 think you also know about differentiating a sum. 30 00:01:58,87 --> 00:02:00,3 Or I could write this as (u 31 00:02:00,3 --> 00:02:03,65 v)' = u' + v'. 32 00:02:03,65 --> 00:02:06,87 33 00:02:06,87 --> 00:02:10,83 So I'm going to be using those but today I'll talk about a 34 00:02:10,83 --> 00:02:14,14 collection of other rules about how to deal with a product of 35 00:02:14,14 --> 00:02:18,18 functions, a quotient of functions, and, best of all, 36 00:02:18,18 --> 00:02:20,34 composition of functions. 37 00:02:20,34 --> 00:02:22,03 And then at the end, I'll have something to say 38 00:02:22,03 --> 00:02:23,48 about higher derivatives. 39 00:02:23,48 --> 00:02:26,67 So that's the story for today. 40 00:02:26,67 --> 00:02:29,12 That's the program. 41 00:02:29,12 --> 00:02:43,36 So let's begin by talking about the product rule. 42 00:02:43,36 --> 00:02:45,855 So the product rule tells you how to differentiate a product 43 00:02:45,855 --> 00:02:48,04 of functions, and I'll just give you the rule, 44 00:02:48,04 --> 00:02:49,27 first of all. 45 00:02:49,27 --> 00:02:51,6 The rule is it's u'v + uv'. 46 00:02:51,6 --> 00:02:57,28 47 00:02:57,28 --> 00:02:58,5 It's a little bit funny. 48 00:02:58,5 --> 00:03:02,32 Differentiating a product gives you a sum. 49 00:03:02,32 --> 00:03:06,73 But let's see how that works out in a particular example. 50 00:03:06,73 --> 00:03:10,3 For example, suppose that I wanted to differentiate 51 00:03:10,3 --> 00:03:11,28 the product. 52 00:03:11,28 --> 00:03:14,67 Well, the product of these two basic examples that 53 00:03:14,67 --> 00:03:15,61 we just talked about. 54 00:03:15,61 --> 00:03:18,06 I'm going to use the same variable in both cases 55 00:03:18,06 --> 00:03:20,79 instead of different ones like I did here. 56 00:03:20,79 --> 00:03:23,23 So the derivative of (x^n)sin x. 57 00:03:23,23 --> 00:03:28,43 58 00:03:28,43 --> 00:03:30,3 So this is a new thing. 59 00:03:30,3 --> 00:03:36,12 We couldn't do this without using the product rule. 60 00:03:36,12 --> 00:03:39,67 So the first function is x^n and the second one is sin x. 61 00:03:39,67 --> 00:03:41,74 And we're going to apply this rule. 62 00:03:41,74 --> 00:03:49,45 So u is x^n. u' is, according to the rule, nx^(n - 1). 63 00:03:49,45 --> 00:03:56,05 And then I take v and write it down the way it is, sine of x. 64 00:03:56,05 --> 00:03:57,32 And then I do it the other way. 65 00:03:57,32 --> 00:04:02,45 I take u the way it is, that's x^n, and multiply it by 66 00:04:02,45 --> 00:04:05,26 the derivative of v, v'. 67 00:04:05,26 --> 00:04:09,235 We just saw v' is cosine of x. 68 00:04:09,235 --> 00:04:11,52 So that's it. 69 00:04:11,52 --> 00:04:14,98 Obviously, you can differentiate longer products, 70 00:04:14,98 --> 00:04:20,56 products of more things by doing it one at a time. 71 00:04:20,56 --> 00:04:22,7 Let's see why this is true. 72 00:04:22,7 --> 00:04:25,73 I want to try to show you why the product rule holds. 73 00:04:25,73 --> 00:04:31,63 So you have a standard way of trying to understand this, and 74 00:04:31,63 --> 00:04:34,71 it involves looking at the change in the function that 75 00:04:34,71 --> 00:04:37,35 you're interested in differentiating. 76 00:04:37,35 --> 00:04:41,64 So I should look at how much the product uv changes when 77 00:04:41,64 --> 00:04:44,63 x changes a little bit. 78 00:04:44,63 --> 00:04:47,12 Well, so how do compute the change? 79 00:04:47,12 --> 00:04:51,48 Well, I write down the value of the function at some 80 00:04:51,48 --> 00:04:53,48 new value of x, (x 81 00:04:53,48 --> 00:04:56,16 delta x). 82 00:04:56,16 --> 00:04:58,41 Well, I better write down the whole new value 83 00:04:58,41 --> 00:05:01,48 of the function, and the function is uv. 84 00:05:01,48 --> 00:05:05,23 So the whole new value looks like this. 85 00:05:05,23 --> 00:05:06,19 It's u (x 86 00:05:06,19 --> 00:05:09,45 delta x)v(x + delta x). 87 00:05:09,45 --> 00:05:10,96 That's the new value. 88 00:05:10,96 --> 00:05:13,2 But what's the change in the product? 89 00:05:13,2 --> 00:05:18,27 Well, I better subtract off what the old value was, 90 00:05:18,27 --> 00:05:20,92 which is u(x) v(x). 91 00:05:20,92 --> 00:05:25,05 Okay, according to the rule we're trying to prove, I 92 00:05:25,05 --> 00:05:27,75 have to get u' involved. 93 00:05:27,75 --> 00:05:31,42 So I want to involve the change in u alone, by itself. 94 00:05:31,42 --> 00:05:32,99 Let's just try that. 95 00:05:32,99 --> 00:05:36,62 I see part of the formula for the change in u right there. 96 00:05:36,62 --> 00:05:40,26 Let's see if we can get the rest of it in place. 97 00:05:40,26 --> 00:05:42,66 So the change in x = ((u(x) 98 00:05:42,66 --> 00:05:46,33 + delta x) - u(x)). 99 00:05:46,33 --> 00:05:49,312 That's the change in x [Correction:___c 100 00:05:49,312 --> 00:05:50,08 hange___in___u]. 101 00:05:50,08 --> 00:05:54,1 This part of it occurs up here, multiplied by v (x 102 00:05:54,1 --> 00:05:57,89 delta x), so let's put that in too. 103 00:05:57,89 --> 00:06:00,44 Now this equality sign isn't very good right now. 104 00:06:00,44 --> 00:06:05,65 I've got this product here so far, but I've introduced 105 00:06:05,65 --> 00:06:06,69 something I don't like. 106 00:06:06,69 --> 00:06:09,05 I've introduced u (v ( x 107 00:06:09,05 --> 00:06:09,8 delta x)), right? 108 00:06:09,8 --> 00:06:12,01 Minus that. 109 00:06:12,01 --> 00:06:16,37 So the next thing I'm gonna do is correct that little 110 00:06:16,37 --> 00:06:24,62 defect by adding in u(x) (v (x + delta x)). 111 00:06:24,62 --> 00:06:28,88 Okay, now I cancelled off what was wrong with this line. 112 00:06:28,88 --> 00:06:30,81 But I'm still not quite there, because I haven't 113 00:06:30,81 --> 00:06:32,67 put this in yet. 114 00:06:32,67 --> 00:06:38,42 So I better subtract off uv, and then I'll be home. 115 00:06:38,42 --> 00:06:41,2 But I'm going to do that in a clever way, because I noticed 116 00:06:41,2 --> 00:06:43,9 that I already have a u here. 117 00:06:43,9 --> 00:06:47,15 So I'm gonna take this factor of u and make it 118 00:06:47,15 --> 00:06:48,56 the same as this factor. 119 00:06:48,56 --> 00:06:52,51 So I get u(x) times this, minus u(x) times that. 120 00:06:52,51 --> 00:06:57,18 That's the same thing as u times the difference. 121 00:06:57,18 --> 00:07:00,18 So that was a little bit strange, but when you stand 122 00:07:00,18 --> 00:07:02,86 back and look at it, you can see multiplied out, the 123 00:07:02,86 --> 00:07:04,28 middle terms cancel. 124 00:07:04,28 --> 00:07:07,34 And you get the right answer. 125 00:07:07,34 --> 00:07:09,615 Well I like that because it's involved the change 126 00:07:09,615 --> 00:07:12,24 in u and the change in v. 127 00:07:12,24 --> 00:07:16,81 So this is equal to (delta u) v (x 128 00:07:16,81 --> 00:07:26,07 delta x) - u(x) times the change in v. 129 00:07:26,07 --> 00:07:27,51 Well, I'm almost there. 130 00:07:27,51 --> 00:07:30,66 The next step in computing the derivative is take difference 131 00:07:30,66 --> 00:07:42,91 quotient, divide this by delta x. 132 00:07:42,91 --> 00:07:51,89 So, (delta (uv)) / (delta x) is well, I'll say (delta 133 00:07:51,89 --> 00:07:55,58 u / delta x) v ( x 134 00:07:55,58 --> 00:08:03,15 delta x). 135 00:08:03,15 --> 00:08:10 Have I made a mistake here? 136 00:08:10 --> 00:08:13,34 This plus magically became a minus on the way down here, 137 00:08:13,34 --> 00:08:18,81 so I better fix that. 138 00:08:18,81 --> 00:08:23,26 Plus u(delta v / (delta x). 139 00:08:23,26 --> 00:08:27,95 This u is this u over here. 140 00:08:27,95 --> 00:08:30,55 So I've just divided this formula by delta x, and now I 141 00:08:30,55 --> 00:08:35,72 can take the limit as goes to 0, so this is as 142 00:08:35,72 --> 00:08:42,07 delta x goes to 0. 143 00:08:42,07 --> 00:08:46,86 This becomes the definition of the derivative, and on this 144 00:08:46,86 --> 00:08:54,62 side, I get du/dx times... now what happens to this quantity 145 00:08:54,62 --> 00:09:02,75 when delta x goes to 0? 146 00:09:02,75 --> 00:09:05,71 So this quantity is getting closer and closer to x. 147 00:09:05,71 --> 00:09:09 So what happens to the value of v? 148 00:09:09 --> 00:09:10,75 It becomes equal to x(v). 149 00:09:10,75 --> 00:09:13,02 That uses continuity of v. 150 00:09:13,02 --> 00:09:15,35 So, v (x 151 00:09:15,35 --> 00:09:22,59 delta x) goes to v(x) by continuity. 152 00:09:22,59 --> 00:09:27,07 So this gives me times v, and then I have u(n delta v/ v 153 00:09:27,07 --> 00:09:30,68 delta x) gives me dv/dx. 154 00:09:30,68 --> 00:09:31,93 And that's the formula. 155 00:09:31,93 --> 00:09:33,53 That's the formula as I wrote it down at the 156 00:09:33,53 --> 00:09:35,66 beginning over here. 157 00:09:35,66 --> 00:09:39,17 The derivative of a product is given by this sum. 158 00:09:39,17 --> 00:09:46,29 Yeah? 159 00:09:46,29 --> 00:09:46,871 Student: How did you get from the first line to the second 160 00:09:46,871 --> 00:09:49,3 of the long equation? 161 00:09:49,3 --> 00:09:51,39 Professor: From here to here? 162 00:09:51,39 --> 00:09:53,46 Student: Yes. 163 00:09:53,46 --> 00:09:55,89 Professor: So maybe it's easiest to work backwards and 164 00:09:55,89 --> 00:09:59,8 verify that what I wrote down is correct here. 165 00:09:59,8 --> 00:10:03,48 So, if you look there's a u(v (x 166 00:10:03,48 --> 00:10:05,13 delta x)) there. 167 00:10:05,13 --> 00:10:07,84 And there's also one here. 168 00:10:07,84 --> 00:10:09,94 And they occur with opposite signs. 169 00:10:09,94 --> 00:10:11,49 So they cancel. 170 00:10:11,49 --> 00:10:13,71 What's left is u ( x 171 00:10:13,71 --> 00:10:15,09 delta x) v (x 172 00:10:15,09 --> 00:10:20,53 delta x) - (uv). 173 00:10:20,53 --> 00:10:29,12 And that's just what I started with. 174 00:10:29,12 --> 00:10:33,92 Student: They cancel right? 175 00:10:33,92 --> 00:10:37,55 Professor: I cancelled out this term and this term, and 176 00:10:37,55 --> 00:10:39,7 what's left is the ends. 177 00:10:39,7 --> 00:10:41,49 Any other questions? 178 00:10:41,49 --> 00:10:49,66 Student: [INAUDIBLE]. 179 00:10:49,66 --> 00:10:55,64 Professor: Well, I just calculated what delta uv is, 180 00:10:55,64 --> 00:10:58,08 and now I'm gonna divide that by delta x on my way to 181 00:10:58,08 --> 00:11:00,25 computing the derivative. 182 00:11:00,25 --> 00:11:07,76 And so I copied down the right hand side and divided delta x. 183 00:11:07,76 --> 00:11:10,7 I just decided to divide the delta u by delta x 184 00:11:10,7 --> 00:11:16,23 and delta v by delta x. 185 00:11:16,23 --> 00:11:16,99 Good. 186 00:11:16,99 --> 00:11:22,49 Anything else? 187 00:11:22,49 --> 00:11:24,26 So we have the product rule here. 188 00:11:24,26 --> 00:11:26,98 The rule for differentiating a product of two functions. 189 00:11:26,98 --> 00:11:28,55 This is making us stronger. 190 00:11:28,55 --> 00:11:30,24 There are many more functions you can find 191 00:11:30,24 --> 00:11:31,42 derivatives of now. 192 00:11:31,42 --> 00:11:33,58 How about quotients? 193 00:11:33,58 --> 00:11:36,19 Let's find out how to differentiate a quotient 194 00:11:36,19 --> 00:11:47,83 of two functions. 195 00:11:47,83 --> 00:11:49,97 Well again, I'll write down what the answer is and then 196 00:11:49,97 --> 00:11:52,37 we'll try to verify it. 197 00:11:52,37 --> 00:11:55,38 So there's a quotient. 198 00:11:55,38 --> 00:11:56,15 Let me write this down. 199 00:11:56,15 --> 00:11:58,97 There's a quotient of two functions. 200 00:11:58,97 --> 00:12:00,34 And here's the rule for it. 201 00:12:00,34 --> 00:12:02,82 I always have to think about this and hope that I get 202 00:12:02,82 --> 00:12:09,14 it right. u'v - uv' / v^2. 203 00:12:09,14 --> 00:12:11,26 This may be the craziest rule you'll see in this 204 00:12:11,26 --> 00:12:14,33 course, but there it is. 205 00:12:14,33 --> 00:12:18,21 And I'll try to show you why that's true and see an example. 206 00:12:18,21 --> 00:12:18,93 Yeah there was a hand? 207 00:12:18,93 --> 00:12:27,3 Student: [INAUDIBLE] 208 00:12:27,3 --> 00:12:31,45 Professor: What letters look the same? u and 209 00:12:31,45 --> 00:12:33,04 v look the same? 210 00:12:33,04 --> 00:12:37,04 I'll try to make them look more different. 211 00:12:37,04 --> 00:12:39,34 The v's have points on the bottom. u's have little 212 00:12:39,34 --> 00:12:41,16 round things on the bottom. 213 00:12:41,16 --> 00:12:44,98 What's the new value of u? 214 00:12:44,98 --> 00:12:50,99 The value of u at (x 215 00:12:50,99 --> 00:12:53,53 delta x) is (u 216 00:12:53,53 --> 00:12:55,55 delta u), right? 217 00:12:55,55 --> 00:12:56,4 That's what delta u is. 218 00:12:56,4 --> 00:13:01,07 It's the change in u when x gets replaced by (x 219 00:13:01,07 --> 00:13:02,41 delta x). 220 00:13:02,41 --> 00:13:07,64 And the change in v, the new value v, is (v 221 00:13:07,64 --> 00:13:09,7 delta v). 222 00:13:09,7 --> 00:13:13,13 So this is the new value of u divided by the new value of v. 223 00:13:13,13 --> 00:13:16,13 That's the beginning. 224 00:13:16,13 --> 00:13:21,25 And then I subtract off the old values, which are - u/v. 225 00:13:21,25 --> 00:13:26,1 This'll be easier to work out when I write it out this way. 226 00:13:26,1 --> 00:13:27,75 So now, we'll cross multiply, as I said. 227 00:13:27,75 --> 00:13:33,35 So I get (uv 228 00:13:33,35 --> 00:13:38,89 (delta u)v) minus, now I cross multiply this way, 229 00:13:38,89 --> 00:13:46,33 you get (uv - u(delta v)). 230 00:13:46,33 --> 00:13:48,14 And I divide all this by (v 231 00:13:48,14 --> 00:13:49,98 delta v)u [Correction:___( v___+___delta___v)v]. 232 00:13:49,98 --> 00:13:52,84 233 00:13:52,84 --> 00:13:57,37 Okay, now the reason I like to do it this way is that you see 234 00:13:57,37 --> 00:14:00,73 the cancellation happening here. uv and uv occur twice 235 00:14:00,73 --> 00:14:02,19 and so I can cancel them. 236 00:14:02,19 --> 00:14:04,52 And I will, and I'll answer these questions in a minute. 237 00:14:04,52 --> 00:14:06,26 Audience: [INAUDIBLE]. 238 00:14:06,26 --> 00:14:11,34 Professor: Ooo, that's a v. 239 00:14:11,34 --> 00:14:14,03 All right. 240 00:14:14,03 --> 00:14:15,57 Good, anything else? 241 00:14:15,57 --> 00:14:16,75 That's what all hands were. 242 00:14:16,75 --> 00:14:17,88 Good. 243 00:14:17,88 --> 00:14:20,87 All right, so I cancel these and what I'm left with then 244 00:14:20,87 --> 00:14:28,2 is (delta u)v - u(delta v) and all this is over (v 245 00:14:28,2 --> 00:14:31,53 delta v)v. 246 00:14:31,53 --> 00:14:33,36 Ok, there's the difference. 247 00:14:33,36 --> 00:14:36,6 There's the change in the quotient. 248 00:14:36,6 --> 00:14:39,58 The change in this function is given by this formula. 249 00:14:39,58 --> 00:14:42,42 And now to compute the derivative, I want to divide by 250 00:14:42,42 --> 00:14:45 delta x, and take the limit. 251 00:14:45 --> 00:14:51,65 So let's write that down, delta(u/v)/delta x 252 00:14:51,65 --> 00:14:56,82 is this formula here divided by delta x. 253 00:14:56,82 --> 00:15:00,42 And again, I'm going to put the delta x under these 254 00:15:00,42 --> 00:15:02,33 delta u and delta v. 255 00:15:02,33 --> 00:15:02,78 Okay? 256 00:15:02,78 --> 00:15:06,08 I'm gonna put delta x in the denominator, but I can think 257 00:15:06,08 --> 00:15:09,92 of that as dividing into this factor and this factor. 258 00:15:09,92 --> 00:15:16,98 So this is ((delta u/ delta x)v) - u(delta v/delta x)). 259 00:15:16,98 --> 00:15:21,16 260 00:15:21,16 --> 00:15:23,71 And all that is divided by the same denominator, (v 261 00:15:23,71 --> 00:15:24,28 delta v)v. 262 00:15:24,28 --> 00:15:28,4 263 00:15:28,4 --> 00:15:28,97 Right? 264 00:15:28,97 --> 00:15:33,01 Put the delta x up in the numerator there. 265 00:15:33,01 --> 00:15:37,83 Next up, take the limit as delta x goes to 0. 266 00:15:37,83 --> 00:15:43,47 I get, by definition, the derivative of (u/v). 267 00:15:43,47 --> 00:15:46,81 And on the right hand side, well, this is the derivative 268 00:15:46,81 --> 00:15:51,3 u(du/ dx) right? 269 00:15:51,3 --> 00:15:51,49 Times v. 270 00:15:51,49 --> 00:16:00,42 See and then u times, and here it's the derivative (dv/ dx). 271 00:16:00,42 --> 00:16:04,25 Now what about the denominator? 272 00:16:04,25 --> 00:16:09,56 So when delta x goes to 0, v stays the same, 273 00:16:09,56 --> 00:16:10,72 v stays the same. 274 00:16:10,72 --> 00:16:13,48 What happens to this delta v? 275 00:16:13,48 --> 00:16:17,97 It goes to 0, again, because v is continuous. 276 00:16:17,97 --> 00:16:23,94 So again, delta v goes to 0 with delta x because they're 277 00:16:23,94 --> 00:16:25,36 continuous and you just get (v*v). 278 00:16:25,36 --> 00:16:28,18 279 00:16:28,18 --> 00:16:31,02 I think that's the formula I wrote down over there. 280 00:16:31,02 --> 00:16:31,7 (du/dx)v - u(dv/dx). 281 00:16:31,7 --> 00:16:35,51 282 00:16:35,51 --> 00:16:40,77 And all divided by the square of the old denominator. 283 00:16:40,77 --> 00:16:42,16 Well, that's it. 284 00:16:42,16 --> 00:16:43,54 That's the quotient rule. 285 00:16:43,54 --> 00:16:44,52 Weird formula. 286 00:16:44,52 --> 00:16:46,16 Let's see an application. 287 00:16:46,16 --> 00:16:51,07 Let's see an example. 288 00:16:51,07 --> 00:16:54,68 So the example I'm going to give is pretty simple. 289 00:16:54,68 --> 00:16:58,1 I'm going to take the numerator to be just 1. 290 00:16:58,1 --> 00:17:02,79 So I'm gonna take u = 1. 291 00:17:02,79 --> 00:17:10,71 So now I'm differentiating 1 / v, the reciprocal of a 292 00:17:10,71 --> 00:17:14,43 function; 1 over a function. 293 00:17:14,43 --> 00:17:16,88 Here's a copy of my rule. 294 00:17:16,88 --> 00:17:23,72 What's du/ dx in that case? u is a constant, so that 295 00:17:23,72 --> 00:17:27,05 term is 0 in this rule. 296 00:17:27,05 --> 00:17:28,7 I don't have to worry about this. 297 00:17:28,7 --> 00:17:31,65 I get a minus. 298 00:17:31,65 --> 00:17:36,8 And then u = 1, and dv/ dx. 299 00:17:36,8 --> 00:17:38,82 Well, v is whatever v is. 300 00:17:38,82 --> 00:17:40,79 I'll write dv/dx as v'. 301 00:17:40,79 --> 00:17:43,91 302 00:17:43,91 --> 00:17:45,52 And then I get a v^2 in the denominator. 303 00:17:45,52 --> 00:17:50,07 So that's the rule. 304 00:17:50,07 --> 00:17:51,38 I could write it as (v^-2)v'. 305 00:17:51,38 --> 00:17:56,84 306 00:17:56,84 --> 00:17:59,3 (-v'/v^2). 307 00:17:59,3 --> 00:18:03,73 That's the derivative of 1 / v. 308 00:18:03,73 --> 00:18:12,11 How about sub-example of that? 309 00:18:12,11 --> 00:18:15,84 I'm going to take the special case where u = 1 again. 310 00:18:15,84 --> 00:18:16,77 And v = x^n. 311 00:18:16,77 --> 00:18:21 312 00:18:21 --> 00:18:25,63 And I'm gonna use the rule that we developed earlier about 313 00:18:25,63 --> 00:18:29,08 the derivative of x^n. 314 00:18:29,08 --> 00:18:40,4 So what do I get here? d / dx (1/x^n) is, I'm plugging into 315 00:18:40,4 --> 00:18:45,26 this formula here with v = x^n. 316 00:18:45,26 --> 00:18:51,58 So I get minus, uh, v^-2. 317 00:18:51,58 --> 00:18:57,25 If v = x^n, v^-2 is, by the rule of exponents, x^-2n. 318 00:18:57,25 --> 00:19:01,43 319 00:19:01,43 --> 00:19:05,55 And then v' is the derivative of x^n, which is (nx)^(n-1). 320 00:19:05,55 --> 00:19:10,15 321 00:19:10,15 --> 00:19:12,01 Ok, so let's put these together. 322 00:19:12,01 --> 00:19:13,55 There's several powers of x here. 323 00:19:13,55 --> 00:19:14,94 I can put them together. 324 00:19:14,94 --> 00:19:21,045 I get -nx ^ ((- 2n) 325 00:19:21,045 --> 00:19:22,33 (n - 1)). 326 00:19:22,33 --> 00:19:24,05 One of these n's cancels. 327 00:19:24,05 --> 00:19:29,26 And what I'm left with is ((-n) - 1). 328 00:19:29,26 --> 00:19:32,55 So we've computed the derivative of 1 / x^n, 329 00:19:32,55 --> 00:19:39,21 which I could also write as x^-n, right? 330 00:19:39,21 --> 00:19:42,64 So I've computed the derivative of negative powers of x. 331 00:19:42,64 --> 00:19:46,56 And this is the formula that I get. 332 00:19:46,56 --> 00:19:52,405 If you think of this - n as a unit, as a thing to itself, it 333 00:19:52,405 --> 00:19:54,31 occurs here in the exponent. 334 00:19:54,31 --> 00:19:59,89 It occurs here, and it occurs here. 335 00:19:59,89 --> 00:20:01,82 So how does that compare with the formula 336 00:20:01,82 --> 00:20:04,12 that we had up here? 337 00:20:04,12 --> 00:20:09,32 The derivative of a power of x is that power times x to 338 00:20:09,32 --> 00:20:12,3 one less than that power. 339 00:20:12,3 --> 00:20:16,01 That's exactly the same as the rule that I wrote down here. 340 00:20:16,01 --> 00:20:19,27 But the power here happens to be a negative number, and the 341 00:20:19,27 --> 00:20:22,5 same negative number shows up as a coefficient and 342 00:20:22,5 --> 00:20:23,96 there in the exponent. 343 00:20:23,96 --> 00:20:24,28 Yeah? 344 00:20:24,28 --> 00:20:30,44 Student: [INAUDIBLE]. 345 00:20:30,44 --> 00:20:34,93 Professor: How did I do this? 346 00:20:34,93 --> 00:20:49,15 Student: [INAUDIBLE]. 347 00:20:49,15 --> 00:20:55,99 Professor: Where did that x^-2n come from? 348 00:20:55,99 --> 00:20:59,9 So I'm applying this rule. 349 00:20:59,9 --> 00:21:04,44 So the denominator in the quotient rule is v^2. 350 00:21:04,44 --> 00:21:11,19 And v was x^n, so the denominator is x^2n. 351 00:21:11,19 --> 00:21:12,65 And I decided to write it as x^-2n. 352 00:21:12,65 --> 00:21:19,01 353 00:21:19,01 --> 00:21:22,08 So the green comments there... 354 00:21:22,08 --> 00:21:26,27 What they say is that I can enlarge this rule. 355 00:21:26,27 --> 00:21:31,35 This exact same rule is true for negative values of n, as 356 00:21:31,35 --> 00:21:36,31 well as positive values of n. 357 00:21:36,31 --> 00:21:42,43 So there's something new in your list of rules that you 358 00:21:42,43 --> 00:21:46,67 can apply, of values of the derivative. 359 00:21:46,67 --> 00:21:48,69 That standard rule is true for negative as well 360 00:21:48,69 --> 00:21:51,12 as positive exponents. 361 00:21:51,12 --> 00:21:57,29 And that comes out of a quotient rule. 362 00:21:57,29 --> 00:21:59,02 Okay, so we've done two rules. 363 00:21:59,02 --> 00:22:04,65 I've talked about the product rule and the quotient rule. 364 00:22:04,65 --> 00:22:05,67 What's next? 365 00:22:05,67 --> 00:22:07,15 Let's see the chain rule. 366 00:22:07,15 --> 00:22:22,22 So this is a composition rule. 367 00:22:22,22 --> 00:22:25,02 So the kind of thing that I have in mind, composition of 368 00:22:25,02 --> 00:22:28,21 functions is about substitution. 369 00:22:28,21 --> 00:22:30,36 So the kind of function that I have in mind is, for 370 00:22:30,36 --> 00:22:31,73 instance, y = (sin t)^10. 371 00:22:31,73 --> 00:22:39,7 372 00:22:39,7 --> 00:22:43,04 That's a new one. 373 00:22:43,04 --> 00:22:44,46 We haven't seen how to differentiate that 374 00:22:44,46 --> 00:22:46,59 before, I think. 375 00:22:46,59 --> 00:22:50,6 This kind of power of a trig function happens very often. 376 00:22:50,6 --> 00:22:53,54 You've seen them happen, as well, I'm sure, already. 377 00:22:53,54 --> 00:22:58,02 And there's a little notational switch that people use. 378 00:22:58,02 --> 00:22:59,32 They'll write sin^10(t). 379 00:22:59,32 --> 00:23:02,91 380 00:23:02,91 --> 00:23:05,63 But remember that when you write sin^10(t), what you mean 381 00:23:05,63 --> 00:23:10,44 is take the sine of t, and then take the tenth power of that. 382 00:23:10,44 --> 00:23:13,59 It's the meaning of sin^10(t). 383 00:23:13,59 --> 00:23:21,05 So the method of dealing with this kind of composition of 384 00:23:21,05 --> 00:23:33,19 functions is to use new variable names. 385 00:23:33,19 --> 00:23:36,83 What I mean is, I can think of this (sin t)^10. 386 00:23:36,83 --> 00:23:39,71 387 00:23:39,71 --> 00:23:42,07 I can think of it it as a two step process. 388 00:23:42,07 --> 00:23:44,16 First of all, I compute the sine of t. 389 00:23:44,16 --> 00:23:47,45 And let's call the result x. 390 00:23:47,45 --> 00:23:50,15 There's the new variable name. 391 00:23:50,15 --> 00:23:53,34 And then, I express y in terms of x. 392 00:23:53,34 --> 00:23:58,07 So y says take this and raise it to the tenth power. 393 00:23:58,07 --> 00:23:59,36 In other words, y = x^10. 394 00:23:59,36 --> 00:24:03,4 395 00:24:03,4 --> 00:24:06,94 And then you plug x = sine of t into that, and you get the 396 00:24:06,94 --> 00:24:10,59 formula for what y is in terms of t. 397 00:24:10,59 --> 00:24:14,55 So it's good practice to introduce new letters when 398 00:24:14,55 --> 00:24:17,06 they're convenient, and this is one place where 399 00:24:17,06 --> 00:24:21,82 it's very convenient. 400 00:24:21,82 --> 00:24:25,07 So let's find a rule for differentiating a composition, 401 00:24:25,07 --> 00:24:27,77 a function that can be expressed by doing one function 402 00:24:27,77 --> 00:24:30,27 and then applying another function. 403 00:24:30,27 --> 00:24:32,88 And here's the rule. 404 00:24:32,88 --> 00:24:35,05 Well, maybe I'll actually derive this rule first, and 405 00:24:35,05 --> 00:24:37,42 then you'll see what it is. 406 00:24:37,42 --> 00:24:40,6 In fact, the rule is very simple to derive. 407 00:24:40,6 --> 00:24:43,89 So this is a proof first, and then we'll write down the rule. 408 00:24:43,89 --> 00:24:51,11 I'm interested in delta y / delta t. y is a function of 409 00:24:51,11 --> 00:24:53,76 x. x is a function of t. 410 00:24:53,76 --> 00:24:56,95 And I'm interested in how y changes with respect to 411 00:24:56,95 --> 00:25:00,85 t, with respect to the original variable t. 412 00:25:00,85 --> 00:25:05,41 Well, because of that intermediate variable, I can 413 00:25:05,41 --> 00:25:12,67 write this as ((delta y / delta x) (delta x/ delta t)). 414 00:25:12,67 --> 00:25:15,33 It cancels, right? 415 00:25:15,33 --> 00:25:17,6 The delta x cancels. 416 00:25:17,6 --> 00:25:23,1 The change in that immediate variable cancels out. 417 00:25:23,1 --> 00:25:26,12 This is just basic algebra. 418 00:25:26,12 --> 00:25:29,93 But what happens when I let delta t get small? 419 00:25:29,93 --> 00:25:34,37 Well this give me dy /dt. 420 00:25:34,37 --> 00:25:42,22 On the right hand side, I get (dy/dx) (dx/dt). 421 00:25:42,22 --> 00:25:44,43 So students will often remember this rule. 422 00:25:44,43 --> 00:25:46,85 This is the rule, by saying that you can cancel 423 00:25:46,85 --> 00:25:49,08 out for the dx's. 424 00:25:49,08 --> 00:25:51,86 And that's not so far from the truth. 425 00:25:51,86 --> 00:25:55,16 That's a good way to think of it. 426 00:25:55,16 --> 00:26:01,41 In other words, this is the so-called chain rule. 427 00:26:01,41 --> 00:26:14,54 And it says that differentiation of a 428 00:26:14,54 --> 00:26:26,69 composition is a product. 429 00:26:26,69 --> 00:26:34,91 It's just the product of the two derivatives. 430 00:26:34,91 --> 00:26:39,57 So that's how you differentiate a composite of two functions. 431 00:26:39,57 --> 00:26:42,07 And let's just do an example. 432 00:26:42,07 --> 00:26:44,69 Let's do this example. 433 00:26:44,69 --> 00:26:48,82 Let's see how that comes out. 434 00:26:48,82 --> 00:26:55,25 So let's differentiate, what did I say? 435 00:26:55,25 --> 00:26:56,53 (sin t)^10. 436 00:26:56,53 --> 00:26:59,4 437 00:26:59,4 --> 00:27:01,856 Okay, there's an inside function and an 438 00:27:01,856 --> 00:27:03,13 outside function. 439 00:27:03,13 --> 00:27:07,91 The inside function is x as a function of t. 440 00:27:07,91 --> 00:27:12,08 This is the inside function, and this is 441 00:27:12,08 --> 00:27:19,17 the outside function. 442 00:27:19,17 --> 00:27:22,59 So the rule says, first of all let's differentiate 443 00:27:22,59 --> 00:27:23,55 the outside function. 444 00:27:23,55 --> 00:27:25,37 Take dy/ dx. 445 00:27:25,37 --> 00:27:29,2 Differentiate it with respect to that variable x. 446 00:27:29,2 --> 00:27:31,02 The outside function is the 10th power. 447 00:27:31,02 --> 00:27:34,64 What's it's derivative? 448 00:27:34,64 --> 00:27:42,44 So I get 10x^9. 449 00:27:42,44 --> 00:27:50,31 In this account, I'm using this newly introduced 450 00:27:50,31 --> 00:27:53,99 variable named x. 451 00:27:53,99 --> 00:27:58,15 So the derivative of the outside function is 10x^9. 452 00:27:58,15 --> 00:28:01,04 And then here's the inside function, and the next thing I 453 00:28:01,04 --> 00:28:03,13 want to do is differentiate it. 454 00:28:03,13 --> 00:28:07,73 So what's dx /dt, d/dt (sine t), the derivative of sine t? 455 00:28:07,73 --> 00:28:11,83 All right, that's cosine t. 456 00:28:11,83 --> 00:28:13,16 That's what the chain gives you. 457 00:28:13,16 --> 00:28:18,155 This is correct, but since we were the ones to introduce 458 00:28:18,155 --> 00:28:21,99 this notation x here, that wasn't given to us in the 459 00:28:21,99 --> 00:28:24,56 original problem here. 460 00:28:24,56 --> 00:28:27,98 The last step in this process should be to put back, to 461 00:28:27,98 --> 00:28:32,44 substitute back in what x is in terms of t. 462 00:28:32,44 --> 00:28:35,32 So x = sin t. 463 00:28:35,32 --> 00:28:44,79 So that tells me that I get 10(sin(t))^9, that's 464 00:28:44,79 --> 00:28:47,86 x^9, times the cos(t). 465 00:28:47,86 --> 00:28:50,86 Or the same thing is sin^9(t)cos(t). 466 00:28:50,86 --> 00:28:56,04 467 00:28:56,04 --> 00:28:59,54 So there's an application of the chain rule. 468 00:28:59,54 --> 00:29:02,29 You know, people often wonder where the name 469 00:29:02,29 --> 00:29:03,17 chain rule comes from. 470 00:29:03,17 --> 00:29:06,34 I was just wondering about that myself. 471 00:29:06,34 --> 00:29:15,23 So is it because it chains you down? 472 00:29:15,23 --> 00:29:18,07 Is it like a chain fence? 473 00:29:18,07 --> 00:29:19,59 I decided what it is. 474 00:29:19,59 --> 00:29:24,22 It's because by using it, you burst the chains of 475 00:29:24,22 --> 00:29:26,41 differentiation, and you can differentiate many more 476 00:29:26,41 --> 00:29:28,04 functions using it. 477 00:29:28,04 --> 00:29:32,915 So when you want to think of the chain rule, just think 478 00:29:32,915 --> 00:29:35,64 of that chain there. 479 00:29:35,64 --> 00:29:47,96 It lets you burst free. 480 00:29:47,96 --> 00:30:04,83 Let me give you another application of the chain rule. 481 00:30:04,83 --> 00:30:16,27 Ready for this one? 482 00:30:16,27 --> 00:30:17,97 So I'd like to differentiate the sin(10t). 483 00:30:17,97 --> 00:30:25,76 484 00:30:25,76 --> 00:30:27,44 Again, this is the composite of two functions. 485 00:30:27,44 --> 00:30:30,22 What's the inside function? 486 00:30:30,22 --> 00:30:36,76 Okay, so I think I'll introduce this new notation. x = 10t, and 487 00:30:36,76 --> 00:30:38,26 the outside function is the sine. 488 00:30:38,26 --> 00:30:41,32 So y = sin x. 489 00:30:41,32 --> 00:30:46,66 So now the chain rule says dy/ dt is... 490 00:30:46,66 --> 00:30:47,92 Okay, let's see. 491 00:30:47,92 --> 00:30:50,71 I take the derivative of the outside function, 492 00:30:50,71 --> 00:30:54,24 and what's that? 493 00:30:54,24 --> 00:30:58,52 Sine' and we can substitute because we know what sine' is. 494 00:30:58,52 --> 00:31:06,47 So I get cosine of whatever, x, and then times what? 495 00:31:06,47 --> 00:31:11,4 Now I differentiate the inside function, which is just 10. 496 00:31:11,4 --> 00:31:16,38 So I could write this as 10cos of what? 497 00:31:16,38 --> 00:31:17,36 10t, x = 10t. 498 00:31:17,36 --> 00:31:20,26 499 00:31:20,26 --> 00:31:26,55 Now, once you get used to this, this middle variable, you don't 500 00:31:26,55 --> 00:31:33,19 have to give a name for it. 501 00:31:33,19 --> 00:31:35,81 You can just to think about it in your mind without actually 502 00:31:35,81 --> 00:31:44,89 writing it down, d/dt (sin(10t)). 503 00:31:44,89 --> 00:31:47,98 504 00:31:47,98 --> 00:31:50,28 I'll just do it again without introducing this middle 505 00:31:50,28 --> 00:31:52,24 variable explicitly. 506 00:31:52,24 --> 00:31:54,53 Think about it. 507 00:31:54,53 --> 00:31:58,1 I first of all differentiate the outside function, 508 00:31:58,1 --> 00:31:59,74 and I get cosine. 509 00:31:59,74 --> 00:32:03,17 But I don't change the thing that I'm plugging into it. 510 00:32:03,17 --> 00:32:08,56 It's still x that I'm plugging into it. x is 10t. 511 00:32:08,56 --> 00:32:11,47 So let's just write 10t and not worry about the name 512 00:32:11,47 --> 00:32:12,72 of that extra variable. 513 00:32:12,72 --> 00:32:15,51 If it confuses you, introduce the new variable. 514 00:32:15,51 --> 00:32:18,18 And do it carefully and slowly like this. 515 00:32:18,18 --> 00:32:20,95 But, quite quickly, I think you'll get to be able to keep 516 00:32:20,95 --> 00:32:23,24 that step in your mind. 517 00:32:23,24 --> 00:32:24,16 I'm not quite done yet. 518 00:32:24,16 --> 00:32:27,08 I haven't differentiated the inside function, the 519 00:32:27,08 --> 00:32:29,19 derivative of 10t = 10. 520 00:32:29,19 --> 00:32:33,25 So you get, again, the same result. 521 00:32:33,25 --> 00:32:36,42 A little short cut that you'll get used to. 522 00:32:36,42 --> 00:32:39,62 Really and truly, once you have the chain rule, the world 523 00:32:39,62 --> 00:32:41,11 is yours to conquer. 524 00:32:41,11 --> 00:32:46,73 It puts you in a very, very powerful position. 525 00:32:46,73 --> 00:32:50,21 Okay, well let's see. 526 00:32:50,21 --> 00:32:51,31 What have I covered today? 527 00:32:51,31 --> 00:32:57,37 I've talked about product rule, quotient rule, composition. 528 00:32:57,37 --> 00:32:58,7 I should tell you something about higher 529 00:32:58,7 --> 00:33:00,67 derivatives, as well. 530 00:33:00,67 --> 00:33:10,44 So let's do that. 531 00:33:10,44 --> 00:33:12,15 This is a simple story. 532 00:33:12,15 --> 00:33:14,95 Higher is kind of a strange word. 533 00:33:14,95 --> 00:33:32,95 It just means differentiate over and over again. 534 00:33:32,95 --> 00:33:34,6 All right, so let's see. 535 00:33:34,6 --> 00:33:39,55 If we have a function u or u(x), please allow me to just 536 00:33:39,55 --> 00:33:45,01 write it as briefly as u. 537 00:33:45,01 --> 00:33:49,33 Well, this is a sort of notational thing. 538 00:33:49,33 --> 00:33:51,78 I can differentiate it and get u'. 539 00:33:51,78 --> 00:33:54,79 540 00:33:54,79 --> 00:33:55,9 That's a new function. 541 00:33:55,9 --> 00:33:57,68 Like if you started with the sine, that's 542 00:33:57,68 --> 00:34:00,76 gonna be the cosine. 543 00:34:00,76 --> 00:34:03,57 A new function, so I can differentiate it again. 544 00:34:03,57 --> 00:34:05,01 And the notation for the differentiating 545 00:34:05,01 --> 00:34:07,47 of it again, is u''. 546 00:34:07,47 --> 00:34:12,93 So u'' is just u' differentiated again. 547 00:34:12,93 --> 00:34:21,38 For example, if u = sin x, so u' = cos(x). 548 00:34:21,38 --> 00:34:24,31 Has Professor Gerison talked about what the 549 00:34:24,31 --> 00:34:26,58 derivative of cosine is? 550 00:34:26,58 --> 00:34:28,22 What is it? 551 00:34:28,22 --> 00:34:33,02 Ha, ok so u'' = - sin x. 552 00:34:33,02 --> 00:34:36,81 553 00:34:36,81 --> 00:34:38,93 Let me go on. 554 00:34:38,93 --> 00:34:42,97 What do you suppose u''' means? 555 00:34:42,97 --> 00:34:46,42 I guess it's the derivative of u''. 556 00:34:46,42 --> 00:34:53,05 It's called the third derivative. 557 00:34:53,05 --> 00:34:56,21 And u'' is called the second derivative. 558 00:34:56,21 --> 00:34:59 And it's (u')' differentiated again. 559 00:34:59 --> 00:35:03,68 So to compute u''' in this example, what do I do? 560 00:35:03,68 --> 00:35:05,34 I differentiate that again. 561 00:35:05,34 --> 00:35:08,46 There's a constant term, - 1, constant factor. 562 00:35:08,46 --> 00:35:09,95 That comes out. 563 00:35:09,95 --> 00:35:13,5 The derivative of sine is what? 564 00:35:13,5 --> 00:35:17,93 Okay, so u''' = - cos x. 565 00:35:17,93 --> 00:35:18,69 Let's do it again. 566 00:35:18,69 --> 00:35:21,89 Now after a while, you get tired of writing these things. 567 00:35:21,89 --> 00:35:24,65 And so maybe I'll use the notation u^(4). 568 00:35:24,65 --> 00:35:27,29 That's the fourth derivative. 569 00:35:27,29 --> 00:35:29,49 That's u''''. 570 00:35:29,49 --> 00:35:33,44 Or it's (u''')' the fourth derivative. 571 00:35:33,44 --> 00:35:37,97 And what is that in this example? 572 00:35:37,97 --> 00:35:42,01 Okay, the cosine has derivative -sin , like you told me. 573 00:35:42,01 --> 00:35:45,883 And that -sin cancels with that sine, and all together, I get 574 00:35:45,883 --> 00:35:47,64 sin x. 575 00:35:47,64 --> 00:35:48,94 That's pretty bizarre. 576 00:35:48,94 --> 00:35:51,32 When I differentiate the function sine of x four 577 00:35:51,32 --> 00:35:56,92 times, I get back to the sine of x again. 578 00:35:56,92 --> 00:36:00,29 That's the way it is. 579 00:36:00,29 --> 00:36:03,15 Now this notation, prime prime prime prime, 580 00:36:03,15 --> 00:36:03,99 and things like that. 581 00:36:03,99 --> 00:36:13,65 There are different variants of that notation. 582 00:36:13,65 --> 00:36:24,07 For example, that's another notation. 583 00:36:24,07 --> 00:36:29,97 Well, you've used the notation du/ dx before. u' could 584 00:36:29,97 --> 00:36:35,73 also be denoted du/ dx. 585 00:36:35,73 --> 00:36:40,485 I think we've already here, today, used this way 586 00:36:40,485 --> 00:36:43,23 of rewriting du/ dx. 587 00:36:43,23 --> 00:36:47,66 I think when I was talking about d/dt(uv) and so on, I 588 00:36:47,66 --> 00:36:52,36 pulled that d / dt outside and put whatever function 589 00:36:52,36 --> 00:36:55,01 you're differentiating over to the right. 590 00:36:55,01 --> 00:36:57,43 So that's just a notational switch. 591 00:36:57,43 --> 00:36:58,11 It looks good. 592 00:36:58,11 --> 00:37:06,26 It looks like good algebra doesn't it? 593 00:37:06,26 --> 00:37:09,86 But what it's doing is regarding this notation 594 00:37:09,86 --> 00:37:12,41 as an operator. 595 00:37:12,41 --> 00:37:16,92 It's something you apply to a function to get a new function. 596 00:37:16,92 --> 00:37:18,81 I apply it to the sine function, and I get 597 00:37:18,81 --> 00:37:20,68 the cosine function. 598 00:37:20,68 --> 00:37:24,22 I apply it to x^2, and I get 2x. 599 00:37:24,22 --> 00:37:31,32 This thing here, that symbol, represents an operator, which 600 00:37:31,32 --> 00:37:40,34 you apply to a function. 601 00:37:40,34 --> 00:37:44,86 And the operator says, take the function and differentiate it. 602 00:37:44,86 --> 00:37:47,91 So further notation that people often use, is they give a 603 00:37:47,91 --> 00:37:49,46 different name to that operator. 604 00:37:49,46 --> 00:37:52,27 And they'll write capital D for it. 605 00:37:52,27 --> 00:38:02,98 So this is just using capital D for the symbol d/dx. 606 00:38:02,98 --> 00:38:05,05 So in terms of that notation, let's see. 607 00:38:05,05 --> 00:38:20,44 Let's write down what higher derivatives look like. 608 00:38:20,44 --> 00:38:21,87 So let's see. 609 00:38:21,87 --> 00:38:23,09 That's what u' is. 610 00:38:23,09 --> 00:38:24,36 How about u''? 611 00:38:24,36 --> 00:38:28,89 Let's write that in terms of the d/dx notation. 612 00:38:28,89 --> 00:38:31,71 Well I'm supposed to differentiate u' right? 613 00:38:31,71 --> 00:38:40,92 So that's d/dx applied to the function du/ dx. 614 00:38:40,92 --> 00:38:43,03 Differentiate the derivative. 615 00:38:43,03 --> 00:38:47,24 That's what I've done. 616 00:38:47,24 --> 00:38:54,35 Or I could write that as d/dx applied to d/dx applied to u. 617 00:38:54,35 --> 00:38:57,85 Just pulling that u outside. 618 00:38:57,85 --> 00:38:59,57 So I'm doing d/dx twice. 619 00:38:59,57 --> 00:39:01,59 I'm doing that operator twice. 620 00:39:01,59 --> 00:39:08,03 I could write that as (d/dx)^2 applied to u. 621 00:39:08,03 --> 00:39:15,17 Differentiate twice, and do it to the function u. 622 00:39:15,17 --> 00:39:23,13 Or, I can write it as, now this is a strange one. 623 00:39:23,13 --> 00:39:33,33 I could also write as like that. 624 00:39:33,33 --> 00:39:36,63 It's getting stranger and stranger, isn't it? 625 00:39:36,63 --> 00:39:40,77 This is definitely just a kind of abuse of notation. 626 00:39:40,77 --> 00:39:46,03 But people will go even further and write (d^2)u/dx^2. 627 00:39:46,03 --> 00:39:50,5 628 00:39:50,5 --> 00:39:52,19 So this is the strangest one. 629 00:39:52,19 --> 00:39:57,34 This identity quality is the strangest one, because you 630 00:39:57,34 --> 00:40:01,33 may think that you're taking d of the quantity x^2. 631 00:40:01,33 --> 00:40:03,93 But that's not what's intended. 632 00:40:03,93 --> 00:40:08,24 This is not d(x^2). 633 00:40:08,24 --> 00:40:12,75 What's intended is the quantity dx^2. 634 00:40:12,75 --> 00:40:15,79 In this notation, which is very common, what's intended by the 635 00:40:15,79 --> 00:40:18,25 denominator is the quantity dx^2. 636 00:40:18,25 --> 00:40:23,63 It's part of this second differentiation operator. 637 00:40:23,63 --> 00:40:26,9 So I've written a bunch of equalities down here, and the 638 00:40:26,9 --> 00:40:29,28 only content to them is that these are all different 639 00:40:29,28 --> 00:40:32,32 notations for the same thing. 640 00:40:32,32 --> 00:40:34,94 You'll see this notation very commonly. 641 00:40:34,94 --> 00:40:37,95 So for instance the third derivative is 642 00:40:37,95 --> 00:40:47,54 (d^3)u/dx^3, and so on. 643 00:40:47,54 --> 00:40:47,83 Sorry? 644 00:40:47,83 --> 00:40:59,49 Student: [INAUDIBLE]. 645 00:40:59,49 --> 00:40:59,88 Professor: Yes, absolutely. 646 00:40:59,88 --> 00:41:05,5 Or an equally good notation is to write the operator (D^3)u. 647 00:41:05,5 --> 00:41:09,4 648 00:41:09,4 --> 00:41:11,65 Absolutely. 649 00:41:11,65 --> 00:41:13,96 So I guess I should also write over here when I was talking 650 00:41:13,96 --> 00:41:17,56 about d^2, the second derivative, another notation 651 00:41:17,56 --> 00:41:20,82 is do the operator capital D twice. 652 00:41:20,82 --> 00:41:22,82 Let's see an example of how this can be applied. 653 00:41:22,82 --> 00:41:23,69 I'll answer this question. 654 00:41:23,69 --> 00:41:32,8 Student: [INAUDIBLE]. 655 00:41:32,8 --> 00:41:34,86 Professor: Yeah, so the question is whether the fourth 656 00:41:34,86 --> 00:41:37,54 derivative always gives you the original function back, 657 00:41:37,54 --> 00:41:38,88 like what happened here. 658 00:41:38,88 --> 00:41:39,58 No. 659 00:41:39,58 --> 00:41:43,47 That's very, very special to sines and cosines. 660 00:41:43,47 --> 00:41:45,2 All right? 661 00:41:45,2 --> 00:41:47,85 And, in fact, let's see an example of that. 662 00:41:47,85 --> 00:41:50,92 I'll do a calculation. 663 00:41:50,92 --> 00:42:06,13 Let's calculate the nth derivative of x^n. 664 00:42:06,13 --> 00:42:13,19 Okay, n is a number, like 1, 2, 3, 4. 665 00:42:13,19 --> 00:42:13,72 Here we go. 666 00:42:13,72 --> 00:42:15,36 Let's do this. 667 00:42:15,36 --> 00:42:17,65 So, let's do this bit by bit. 668 00:42:17,65 --> 00:42:22,5 What's the first derivative of x^n? 669 00:42:22,5 --> 00:42:24,09 So everybody knows this. 670 00:42:24,09 --> 00:42:27,83 I'm just using a new notation, this capital D notation. 671 00:42:27,83 --> 00:42:30,52 So it's n x ^ (n -1). 672 00:42:30,52 --> 00:42:33,64 Now you know know, by the way, n could be a negative number 673 00:42:33,64 --> 00:42:38,372 for that, but for now, for this application, I wanna take n to 674 00:42:38,372 --> 00:42:43,07 be 1, 2, 3, and so on; one of those numbers. 675 00:42:43,07 --> 00:42:44,55 Ok, we did one derivative. 676 00:42:44,55 --> 00:42:49,53 Let's compute the second derivative of x ^ n. 677 00:42:49,53 --> 00:42:53,805 Well there's this n constant that comes out, and then the 678 00:42:53,805 --> 00:42:59,98 exponent comes down, and it gets reduced by 1. 679 00:42:59,98 --> 00:43:01,19 All right? 680 00:43:01,19 --> 00:43:03,78 Should I do one more? 681 00:43:03,78 --> 00:43:07,6 D^3 (x^n) = n(n-1). 682 00:43:07,6 --> 00:43:09,41 That's the constant from here. 683 00:43:09,41 --> 00:43:13,42 Times that exponent, (n - 2), times 1 less, (n - 684 00:43:13,42 --> 00:43:15,74 3) is the new exponent. 685 00:43:15,74 --> 00:43:26,43 Well, I keep on going until I come to a new blackboard. 686 00:43:26,43 --> 00:43:29,49 Now, I think I'm going to stop when I get to the n minus first 687 00:43:29,49 --> 00:43:35,37 derivative, so we can see what's likely to happen. 688 00:43:35,37 --> 00:43:40,68 So when I took the third derivative, I had the n 689 00:43:40,68 --> 00:43:43,31 minus third power of x. 690 00:43:43,31 --> 00:43:44,94 And when I took the second derivative, I had the 691 00:43:44,94 --> 00:43:45,76 second power of x. 692 00:43:45,76 --> 00:43:49,6 So, I think what'll happen when I have the n minus first 693 00:43:49,6 --> 00:43:53,51 derivative is I'll have the first power of x left over. 694 00:43:53,51 --> 00:43:55,39 The powers of x keep coming down. 695 00:43:55,39 --> 00:43:59,35 And what I've done it n - 1 times, I get the first power. 696 00:43:59,35 --> 00:44:04,23 And then I get a big constant out in front here times more 697 00:44:04,23 --> 00:44:06,84 and more and more of these smaller and smaller 698 00:44:06,84 --> 00:44:08,5 integers that come down. 699 00:44:08,5 --> 00:44:10,63 What's the last integer that came down before 700 00:44:10,63 --> 00:44:17,46 I got x^1 here? 701 00:44:17,46 --> 00:44:19,39 Well, let's see. 702 00:44:19,39 --> 00:44:23,09 It's just 2, because this x^1 occurred as the 703 00:44:23,09 --> 00:44:24,34 derivative of x^2. 704 00:44:24,34 --> 00:44:27,8 And the coefficient in front of that is 2. 705 00:44:27,8 --> 00:44:29,73 So that's what you get. 706 00:44:29,73 --> 00:44:35,14 The numbers n( n - 1)...2)x^1. 707 00:44:35,14 --> 00:44:40,76 And now we can differentiate one more time and calculate 708 00:44:40,76 --> 00:44:42,77 what (D^n)(x^n) is. 709 00:44:42,77 --> 00:44:46,49 So I get the same number, n(n-1)... 710 00:44:46,49 --> 00:44:49,68 and so on and so on, times 2. 711 00:44:49,68 --> 00:44:52,5 And then I guess I'll say times 1. 712 00:44:52,5 --> 00:44:55,79 Times, what's the derivative of x ^ 1? 713 00:44:55,79 --> 00:44:58,64 1, so times 1. 714 00:44:58,64 --> 00:45:01,26 Time 1, times 1. 715 00:45:01,26 --> 00:45:10,49 Where this one means the constant function 1. 716 00:45:10,49 --> 00:45:14,07 Does anyone know what this number is called? 717 00:45:14,07 --> 00:45:15,11 That has a name. 718 00:45:15,11 --> 00:45:19,72 It's called n factorial. 719 00:45:19,72 --> 00:45:21,4 And it's written n! 720 00:45:21,4 --> 00:45:24,24 721 00:45:24,24 --> 00:45:28,83 And we just used an example of mathematical induction. 722 00:45:28,83 --> 00:45:34,26 So the end result is (D^n) (x^n) = n! 723 00:45:34,26 --> 00:45:37,75 constant. 724 00:45:37,75 --> 00:45:42,46 Okay that's a neat fact. 725 00:45:42,46 --> 00:45:45,55 Final question for the lecture is what's D^n 726 00:45:45,55 --> 00:45:49,73 1 applied to x ^ n? 727 00:45:49,73 --> 00:45:50,85 Ha. 728 00:45:50,85 --> 00:45:54,34 Excellent. 729 00:45:54,34 --> 00:45:56,62 It's the derivative of a constant. 730 00:45:56,62 --> 00:45:58,18 So it's 0. 731 00:45:58,18 --> 00:45:58,44 Okay. 732 00:45:58,44 --> 00:46:00,03 Thank you. 733 00:46:00,03 --> 00:46:02,245