1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:03 The following content is provided under a Creative 3 00:00:03 --> 00:00:04 Commons License. 4 00:00:04 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high quality educational resources for free. 6 00:00:09 --> 00:00:12 To make a donation or to view additional materials from 7 00:00:12 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:24 Professor: So, again welcome to 18.01. 10 00:00:24 --> 00:00:28 We're getting started today with what we're calling 11 00:00:28 --> 00:00:36 Unit One, a highly imaginative title. 12 00:00:36 --> 00:00:43 And it's differentiation. 13 00:00:43 --> 00:00:46 So, let me first tell you, briefly, what's in store in 14 00:00:46 --> 00:00:49 the next couple of weeks. 15 00:00:49 --> 00:01:02 The main topic today is what is a derivative. 16 00:01:02 --> 00:01:09 And, we're going to look at this from several different 17 00:01:09 --> 00:01:13 points of view, and the first one is the 18 00:01:13 --> 00:01:18 geometric interpretation. 19 00:01:18 --> 00:01:21 That's what we'll spend most of today on. 20 00:01:21 --> 00:01:32 And then, we'll also talk about a physical interpretation 21 00:01:32 --> 00:01:37 of what a derivative is. 22 00:01:37 --> 00:01:45 And then there's going to be something else which I guess is 23 00:01:45 --> 00:01:48 maybe the reason why Calculus is so fundamental, and why we 24 00:01:48 --> 00:01:53 always start with it in most science and engineering 25 00:01:53 --> 00:02:00 schools, which is the importance of derivatives, of 26 00:02:00 --> 00:02:09 this, to all measurements. 27 00:02:09 --> 00:02:11 So that means pretty much every place. 28 00:02:11 --> 00:02:18 That means in science, in engineering, in economics, 29 00:02:18 --> 00:02:23 in political science, etc. 30 00:02:23 --> 00:02:28 Polling, lots of commercial applications, just 31 00:02:28 --> 00:02:29 about everything. 32 00:02:29 --> 00:02:33 Now, that's what we'll be getting started with, and then 33 00:02:33 --> 00:02:39 there's another thing that we're gonna do in this unit, 34 00:02:39 --> 00:02:49 which is we're going to explain how to differentiate anything. 35 00:02:49 --> 00:03:01 So, how to differentiate any function you know. 36 00:03:01 --> 00:03:04 And that's kind of a tall order, but let me just 37 00:03:04 --> 00:03:06 give you an example. 38 00:03:06 --> 00:03:08 If you want to take the derivative - this we'll see 39 00:03:08 --> 00:03:11 today is the notation for the derivative of something - of 40 00:03:11 --> 00:03:15 some messy function like e ^ x arctanx. 41 00:03:15 --> 00:03:19 42 00:03:19 --> 00:03:25 We'll work this out by the end of this unit. 43 00:03:25 --> 00:03:26 All right? 44 00:03:26 --> 00:03:29 Anything you can think of, anything you can write down, 45 00:03:29 --> 00:03:32 we can differentiate it. 46 00:03:32 --> 00:03:37 All right, so that's what we're gonna do, and today as I said, 47 00:03:37 --> 00:03:41 we're gonna spend most of our time on this geometric 48 00:03:41 --> 00:03:44 interpretation. 49 00:03:44 --> 00:03:50 So let's begin with that. 50 00:03:50 --> 00:03:57 So here we go with the geometric interpretation 51 00:03:57 --> 00:04:01 of derivatives. 52 00:04:01 --> 00:04:11 And, what we're going to do is just ask the geometric problem 53 00:04:11 --> 00:04:26 of finding the tangent line to some graph of some 54 00:04:26 --> 00:04:31 function at some point. 55 00:04:31 --> 00:04:33 Which is to say (x0, y0). 56 00:04:33 --> 00:04:42 So that's the problem that we're addressing here. 57 00:04:42 --> 00:04:46 Alright, so here's our problem, and now let me 58 00:04:46 --> 00:04:49 show you the solution. 59 00:04:49 --> 00:04:58 So, well, let's graph the function. 60 00:04:58 --> 00:05:00 Here's it's graph. 61 00:05:00 --> 00:05:02 Here's some point. 62 00:05:02 --> 00:05:09 All right, maybe I should draw it just a bit lower. 63 00:05:09 --> 00:05:11 So here's a point P. 64 00:05:11 --> 00:05:17 Maybe it's above the point x0. x0, by the way, this 65 00:05:17 --> 00:05:19 was supposed to be an x0. 66 00:05:19 --> 00:05:26 That was some fixed place on the x-axis. 67 00:05:26 --> 00:05:35 And now, in order to perform this mighty feat, I will use 68 00:05:35 --> 00:05:36 another color of chalk. 69 00:05:36 --> 00:05:37 How about red? 70 00:05:37 --> 00:05:38 OK. 71 00:05:38 --> 00:05:42 So here it is. 72 00:05:42 --> 00:05:44 There's the tangent line, Well, not quite straight. 73 00:05:44 --> 00:05:45 Close enough. 74 00:05:45 --> 00:05:46 All right? 75 00:05:46 --> 00:05:49 I did it. 76 00:05:49 --> 00:05:50 That's the geometric problem. 77 00:05:50 --> 00:05:57 I achieved what I wanted to do, and it's kind of an interesting 78 00:05:57 --> 00:06:00 question, which unfortunately I can't solve for you in 79 00:06:00 --> 00:06:03 this class, which is, how did I do that? 80 00:06:03 --> 00:06:06 That is, how physically did I manage to know what to do 81 00:06:06 --> 00:06:07 to draw this tangent line? 82 00:06:07 --> 00:06:10 But that's what geometric problems are like. 83 00:06:10 --> 00:06:12 We visualize it. 84 00:06:12 --> 00:06:14 We can figure it out somewhere in our brains. 85 00:06:14 --> 00:06:15 It happens. 86 00:06:15 --> 00:06:20 And the task that we have now is to figure out how to do it 87 00:06:20 --> 00:06:27 analytically, to do it in a way that a machine could just as 88 00:06:27 --> 00:06:32 well as I did in drawing this tangent line. 89 00:06:32 --> 00:06:39 So, what did we learn in high school about what 90 00:06:39 --> 00:06:40 a tangent line is? 91 00:06:40 --> 00:06:44 Well, a tangent line has an equation, and any line through 92 00:06:44 --> 00:06:49 a point has the equation y - y0 is equal to m the 93 00:06:49 --> 00:06:52 slope, times x - x0. 94 00:06:52 --> 00:07:00 So here's the equation for that line, and now there are two 95 00:07:00 --> 00:07:03 pieces of information that we're going to need to work 96 00:07:03 --> 00:07:07 out what the line is. 97 00:07:07 --> 00:07:10 The first one is the point. 98 00:07:10 --> 00:07:13 That's that point P there. 99 00:07:13 --> 00:07:19 And to specify P, given x, we need to know the level of y, 100 00:07:19 --> 00:07:23 which is of course just f(x0). 101 00:07:23 --> 00:07:25 That's not a calculus problem, but anyway that's a very 102 00:07:25 --> 00:07:28 important part of the process. 103 00:07:28 --> 00:07:31 So that's the first thing we need to know. 104 00:07:31 --> 00:07:39 And the second thing we need to know is the slope. 105 00:07:39 --> 00:07:42 And that's this number m. 106 00:07:42 --> 00:07:45 And in calculus we have another name for it. 107 00:07:45 --> 00:07:48 We call it f prime of x0. 108 00:07:48 --> 00:07:51 Namely, the derivative of f. 109 00:07:51 --> 00:07:53 So that's the calculus part. 110 00:07:53 --> 00:07:55 That's the tricky part, and that's the part that we 111 00:07:55 --> 00:07:57 have to discuss now. 112 00:07:57 --> 00:08:01 So just to make that explicit here, I'm going to make a 113 00:08:01 --> 00:08:08 definition, which is that f '(x0) , which is known as the 114 00:08:08 --> 00:08:34 derivative, of f, at x0, is the slope of the tangent line to y 115 00:08:34 --> 00:08:47 = f (x) at the point, let's just call it p. 116 00:08:47 --> 00:08:50 All right? 117 00:08:50 --> 00:08:56 So, that's what it is, but still I haven't made any 118 00:08:56 --> 00:09:01 progress in figuring out any better how I drew that line. 119 00:09:01 --> 00:09:05 So I have to say something that's more concrete, because 120 00:09:05 --> 00:09:07 I want to be able to cook up what these numbers are. 121 00:09:07 --> 00:09:11 I have to figure out what this number m is. 122 00:09:11 --> 00:09:16 And one way of thinking about that, let me just try this, so 123 00:09:16 --> 00:09:19 I certainly am taking for granted that in sort of 124 00:09:19 --> 00:09:22 non-calculus part that I know what a line through a point is. 125 00:09:22 --> 00:09:24 So I know this equation. 126 00:09:24 --> 00:09:33 But another possibility might be, this line here, how do I 127 00:09:33 --> 00:09:34 know - well, unfortunately, I didn't draw it quite straight, 128 00:09:34 --> 00:09:36 but there it is - how do I know that this orange line is not a 129 00:09:36 --> 00:09:45 tangent line, but this other line is a tangent line? 130 00:09:45 --> 00:09:55 Well, it's actually not so obvious, but I'm gonna 131 00:09:55 --> 00:09:56 describe it a little bit. 132 00:09:56 --> 00:09:59 It's not really the fact, this thing crosses at some other 133 00:09:59 --> 00:10:02 place, which is this point Q. 134 00:10:02 --> 00:10:06 But it's not really the fact that the thing crosses at two 135 00:10:06 --> 00:10:07 place, because the line could be wiggly, the curve could be 136 00:10:07 --> 00:10:11 wiggly, and it could cross back and forth a number of times. 137 00:10:11 --> 00:10:17 That's not what distinguishes the tangent line. 138 00:10:17 --> 00:10:19 So I'm gonna have to somehow grasp this, and I'll 139 00:10:19 --> 00:10:23 first do it in language. 140 00:10:23 --> 00:10:29 And it's the following idea: it's that if you take this 141 00:10:29 --> 00:10:37 orange line, which is called a secant line, and you think of 142 00:10:37 --> 00:10:41 the point Q as getting closer and closer to P, then the slope 143 00:10:41 --> 00:10:45 of that line will get closer and closer to the slope 144 00:10:45 --> 00:10:47 of the red line. 145 00:10:47 --> 00:10:53 And if we draw it close enough, then that's gonna 146 00:10:53 --> 00:10:54 be the correct line. 147 00:10:54 --> 00:10:57 So that's really what I did, sort of in my brain when 148 00:10:57 --> 00:10:58 I drew that first line. 149 00:10:58 --> 00:11:01 And so that's the way I'm going to articulate it first. 150 00:11:01 --> 00:11:13 Now, so the tangent line is equal to the limit of 151 00:11:13 --> 00:11:24 so called secant lines PQ, as Q tends to P. 152 00:11:24 --> 00:11:31 And here we're thinking of P as being fixed and Q as variable. 153 00:11:31 --> 00:11:35 All right? 154 00:11:35 --> 00:11:39 Again, this is still the geometric discussion, but now 155 00:11:39 --> 00:11:42 we're gonna be able to put symbols and formulas 156 00:11:42 --> 00:11:43 to this computation. 157 00:11:43 --> 00:11:56 And we'll be able to work out formulas in any example. 158 00:11:56 --> 00:11:58 So let's do that. 159 00:11:58 --> 00:12:03 So first of all, I'm gonna write out these 160 00:12:03 --> 00:12:05 points P and Q again. 161 00:12:05 --> 00:12:11 So maybe we'll put P here and Q here. 162 00:12:11 --> 00:12:12 And I'm thinking of this line through them. 163 00:12:12 --> 00:12:16 I guess it was orange, so we'll leave it as orange. 164 00:12:16 --> 00:12:19 All right. 165 00:12:19 --> 00:12:24 And now I want to compute its slope. 166 00:12:24 --> 00:12:27 So this, gradually, we'll do this in two steps. 167 00:12:27 --> 00:12:30 And these steps will introduce us to the basic notations which 168 00:12:30 --> 00:12:33 are used throughout calculus, including multi-variable 169 00:12:33 --> 00:12:35 calculus, across the board. 170 00:12:35 --> 00:12:40 So the first notation that's used is you imagine here's 171 00:12:40 --> 00:12:45 the x-axis underneath, and here's the x0, the location 172 00:12:45 --> 00:12:47 directly below the point P. 173 00:12:47 --> 00:12:51 And we're traveling here a horizontal distance which 174 00:12:51 --> 00:12:53 is denoted by delta x. 175 00:12:53 --> 00:12:58 So that's delta x, so called. 176 00:12:58 --> 00:13:06 And we could also call it the change in x. 177 00:13:06 --> 00:13:09 So that's one thing we want to measure in order to get 178 00:13:09 --> 00:13:12 the slope of this line PQ. 179 00:13:12 --> 00:13:14 And the other thing is this height. 180 00:13:14 --> 00:13:18 So that's this distance here, which we denote delta f, 181 00:13:18 --> 00:13:21 which is the change in f. 182 00:13:21 --> 00:13:29 And then, the slope is just the ratio, delta f / delta x. 183 00:13:29 --> 00:13:39 So this is the slope of the secant. 184 00:13:39 --> 00:13:42 And the process I just described over here with this 185 00:13:42 --> 00:13:46 limit applies not just to the whole line itself, but also in 186 00:13:46 --> 00:13:48 particular to its slope. 187 00:13:48 --> 00:13:53 And the way we write that is the limit as delta x goes to 0. 188 00:13:53 --> 00:13:56 And that's going to be our slope. 189 00:13:56 --> 00:14:10 So this is slope of the tangent line. 190 00:14:10 --> 00:14:11 OK. 191 00:14:11 --> 00:14:22 Now, This is still a little general, and I want to work out 192 00:14:22 --> 00:14:28 a more usable form here, a better formula for this. 193 00:14:28 --> 00:14:33 And in order to do that, I'm gonna write delta f, the 194 00:14:33 --> 00:14:36 numerator more explicitly here. 195 00:14:36 --> 00:14:41 The change in f, so remember that the point P is 196 00:14:41 --> 00:14:43 the point (x0, f(x0)). 197 00:14:43 --> 00:14:51 All right, that's what we got for the formula for the point. 198 00:14:51 --> 00:14:55 And in order to compute these distances and in particular 199 00:14:55 --> 00:14:58 the vertical distance here, I'm gonna have to get a 200 00:14:58 --> 00:15:00 formula for Q as well. 201 00:15:00 --> 00:15:05 So if this horizontal distance is delta x, then this 202 00:15:05 --> 00:15:07 location is (x0 203 00:15:07 --> 00:15:11 delta x). 204 00:15:11 --> 00:15:14 And so the point above that point has a 205 00:15:14 --> 00:15:20 formula, which is (x0 206 00:15:20 --> 00:15:25 delta x, f(x0 and this is a mouthful, 207 00:15:25 --> 00:15:31 delta x)). 208 00:15:31 --> 00:15:33 All right, so there's the formula for the point Q. 209 00:15:33 --> 00:15:36 Here's the formula for the point P. 210 00:15:36 --> 00:15:47 And now I can write a different formula for the derivative, 211 00:15:47 --> 00:15:53 which is the following: so this f'(x0) , which is the same as 212 00:15:53 --> 00:16:01 m, is going to be the limit as delta x goes to 0 of the change 213 00:16:01 --> 00:16:07 in f, well the change in f is the value of f at the upper 214 00:16:07 --> 00:16:11 point here, which is (x0 215 00:16:11 --> 00:16:16 delta x), and minus its value at the lower point P, which is 216 00:16:16 --> 00:16:23 f(x0), divided by delta x. 217 00:16:23 --> 00:16:24 All right, so this is the formula. 218 00:16:24 --> 00:16:29 I'm going to put this in a little box, because this is by 219 00:16:29 --> 00:16:33 far the most important formula today, which we use to derive 220 00:16:33 --> 00:16:35 pretty much everything else. 221 00:16:35 --> 00:16:38 And this is the way that we're going to be able to 222 00:16:38 --> 00:16:46 compute these numbers. 223 00:16:46 --> 00:17:06 So let's do an example. 224 00:17:06 --> 00:17:13 This example, we'll call this example one. 225 00:17:13 --> 00:17:19 We'll take the function f(x) , which is 1/x . 226 00:17:19 --> 00:17:23 That's sufficiently complicated to have an interesting answer, 227 00:17:23 --> 00:17:27 and sufficiently straightforward that we can 228 00:17:27 --> 00:17:32 compute the derivative fairly quickly. 229 00:17:32 --> 00:17:36 So what is it that we're gonna do here? 230 00:17:36 --> 00:17:42 All we're going to do is we're going to plug in this formula 231 00:17:42 --> 00:17:44 here for that function. 232 00:17:44 --> 00:17:47 That's all we're going to do, and visually what we're 233 00:17:47 --> 00:17:53 accomplishing is somehow to take the hyperbola, and take a 234 00:17:53 --> 00:18:00 point on the hyperbola, and figure out some tangent line. 235 00:18:00 --> 00:18:03 That's what we're accomplishing when we do that. 236 00:18:03 --> 00:18:05 So we're accomplishing this geometrically but we'll be 237 00:18:05 --> 00:18:07 doing it algebraically. 238 00:18:07 --> 00:18:14 So first, we consider this difference delta f / delta x 239 00:18:14 --> 00:18:16 and write out its formula. 240 00:18:16 --> 00:18:18 So I have to have a place. 241 00:18:18 --> 00:18:21 So I'm gonna make it again above this point x0, which 242 00:18:21 --> 00:18:22 is the general point. 243 00:18:22 --> 00:18:25 We'll make the general calculation. 244 00:18:25 --> 00:18:30 So the value of f at the top, when we move to the right by 245 00:18:30 --> 00:18:35 f(x), so I just read off from this, read off from here. 246 00:18:35 --> 00:18:41 The formula, the first thing I get here is 1 / x0 247 00:18:41 --> 00:18:43 delta x. 248 00:18:43 --> 00:18:46 That's the left hand term. 249 00:18:46 --> 00:18:50 Minus 1 / x0, that's the right hand term. 250 00:18:50 --> 00:18:54 And then I have to divide that by delta x. 251 00:18:54 --> 00:18:57 OK, so here's our expression. 252 00:18:57 --> 00:19:00 And by the way this has a name. 253 00:19:00 --> 00:19:10 This thing is called a difference quotient. 254 00:19:10 --> 00:19:12 It's pretty complicated, because there's always a 255 00:19:12 --> 00:19:13 difference in the numerator. 256 00:19:13 --> 00:19:16 And in disguise, the denominator is a difference, 257 00:19:16 --> 00:19:19 because it's the difference between the value on the 258 00:19:19 --> 00:19:26 right side and the value on the left side here. 259 00:19:26 --> 00:19:34 OK, so now we're going to simplify it by some algebra. 260 00:19:34 --> 00:19:35 So let's just take a look. 261 00:19:35 --> 00:19:38 So this is equal to, let's continue on 262 00:19:38 --> 00:19:41 the next level here. 263 00:19:41 --> 00:19:45 This is equal to 1 / delta x times... 264 00:19:45 --> 00:19:49 All I'm going to do is put it over a common denominator. 265 00:19:49 --> 00:19:53 So the common denominator is (x0 266 00:19:53 --> 00:19:54 delta x)x0. 267 00:19:56 --> 00:19:59 And so in the numerator for the first expressions I 268 00:19:59 --> 00:20:03 have x0, and for the second expression I have x0 269 00:20:03 --> 00:20:05 delta x. 270 00:20:05 --> 00:20:08 So this is the same thing as I had in the numerator before, 271 00:20:08 --> 00:20:11 factoring out this denominator. 272 00:20:11 --> 00:20:17 And here I put that numerator into this more amenable form. 273 00:20:17 --> 00:20:20 And now there are two basic cancellations. 274 00:20:20 --> 00:20:33 The first one is that x0 and x0 cancel, so we have this. 275 00:20:33 --> 00:20:38 And then the second step is that these two expressions 276 00:20:38 --> 00:20:40 cancel, the numerator and the denominator. 277 00:20:40 --> 00:20:44 Now we have a cancellation that we can make use of. 278 00:20:44 --> 00:20:48 So we'll write that under here. 279 00:20:48 --> 00:20:53 And this is equals -1 / (x0 280 00:20:54 --> 00:20:56 delta x)x0. 281 00:20:57 --> 00:21:03 And then the very last step is to take the limit as delta 282 00:21:03 --> 00:21:09 x tends to 0, and now we can do it. 283 00:21:09 --> 00:21:10 Before we couldn't do it. 284 00:21:10 --> 00:21:11 Why? 285 00:21:11 --> 00:21:15 Because the numerator and the denominator gave us 0 / 0. 286 00:21:15 --> 00:21:17 But now that I've made this cancellation, I 287 00:21:17 --> 00:21:19 can pass to the limit. 288 00:21:19 --> 00:21:21 And all that happens is I set this delta x to 289 00:21:21 --> 00:21:22 0, and I get -1/x0^2. 290 00:21:25 --> 00:21:32 So that's the answer. 291 00:21:32 --> 00:21:34 All right, so in other words what I've shown - let me put 292 00:21:34 --> 00:21:36 it up here- is that f'(x0) = -1/x0^2. 293 00:21:52 --> 00:21:57 Now, let's look at the graph just a little bit to check this 294 00:21:57 --> 00:22:01 for plausibility, all right? 295 00:22:01 --> 00:22:04 What's happening here, is first of all it's negative. 296 00:22:04 --> 00:22:08 It's less than 0, which is a good thing. 297 00:22:08 --> 00:22:16 You see that slope there is negative. 298 00:22:16 --> 00:22:20 That's the simplest check that you could make. 299 00:22:20 --> 00:22:24 And the second thing that I would just like to point out is 300 00:22:24 --> 00:22:29 that as x goes to infinity, that as we go farther to the 301 00:22:29 --> 00:22:32 right, it gets less and less steep. 302 00:22:32 --> 00:22:46 So as x0 goes to infinity, less and less steep. 303 00:22:46 --> 00:22:49 So that's also consistent here, when x0 is very large, this is 304 00:22:49 --> 00:22:52 a smaller and smaller number in magnitude, although 305 00:22:52 --> 00:22:54 it's always negative. 306 00:22:54 --> 00:23:00 It's always sloping down. 307 00:23:00 --> 00:23:03 All right, so I've managed to fill the boards. 308 00:23:03 --> 00:23:06 So maybe I should stop for a question or two. 309 00:23:06 --> 00:23:06 Yes? 310 00:23:06 --> 00:23:11 Student: [INAUDIBLE] 311 00:23:11 --> 00:23:18 Professor: So the question is to explain again 312 00:23:18 --> 00:23:22 this limiting process. 313 00:23:22 --> 00:23:26 So the formula here is we have basically two numbers. 314 00:23:26 --> 00:23:29 So in other words, why is it that this expression, when 315 00:23:29 --> 00:23:33 delta x tends to 0, is equal to -1/x0^2 ? 316 00:23:33 --> 00:23:37 Let me illustrate it by sticking in a number for x0 317 00:23:37 --> 00:23:39 to make it more explicit. 318 00:23:39 --> 00:23:43 All right, so for instance, let me stick in here 319 00:23:43 --> 00:23:46 for x0 the number 3. 320 00:23:46 --> 00:23:48 Then it's -1 / (3 321 00:23:49 --> 00:23:50 delta x)3. 322 00:23:52 --> 00:23:54 That's the situation that we've got. 323 00:23:54 --> 00:23:56 And now the question is what happens as this number gets 324 00:23:56 --> 00:23:59 smaller and smaller and smaller, and gets to 325 00:23:59 --> 00:24:01 be practically 0? 326 00:24:01 --> 00:24:04 Well, literally what we can do is just plug in 0 327 00:24:04 --> 00:24:05 there, and you get (3 328 00:24:05 --> 00:24:07 0)3 in the denominator. 329 00:24:07 --> 00:24:08 Minus one in the numerator. 330 00:24:08 --> 00:24:13 So this tends to -1/9 (over 3^2). 331 00:24:16 --> 00:24:18 And that's what I'm saying in general with this 332 00:24:18 --> 00:24:21 extra number here. 333 00:24:21 --> 00:24:25 Other questions? 334 00:24:25 --> 00:24:25 Yes. 335 00:24:25 --> 00:24:34 Student: [INAUDIBLE] 336 00:24:34 --> 00:24:40 Professor: So the question is what happened between this 337 00:24:40 --> 00:24:43 step and this step, right? 338 00:24:43 --> 00:24:45 Explain this step here. 339 00:24:45 --> 00:24:48 Alright, so there were two parts to that. 340 00:24:48 --> 00:24:53 The first is this delta x which is sitting in the denominator, 341 00:24:53 --> 00:24:56 I factored all the way out front. 342 00:24:56 --> 00:24:58 And so what's in the parentheses is supposed to 343 00:24:58 --> 00:25:00 be the same as what's in 344 00:25:00 --> 00:25:03 the numerator of this other expression. 345 00:25:03 --> 00:25:07 And then, at the same time as doing that, I put that 346 00:25:07 --> 00:25:10 expression, which is the difference of two fractions, I 347 00:25:10 --> 00:25:12 expressed it with a common denominator. 348 00:25:12 --> 00:25:14 So in the denominator here, you see the product of 349 00:25:14 --> 00:25:17 the denominators of the two fractions. 350 00:25:17 --> 00:25:19 And then I just figured out what the numerator had 351 00:25:19 --> 00:25:22 to be without really... 352 00:25:22 --> 00:25:27 Other questions? 353 00:25:27 --> 00:25:32 OK. 354 00:25:32 --> 00:25:41 So I claim that on the whole, calculus gets a bad rap, that 355 00:25:41 --> 00:25:47 it's actually easier than most things. 356 00:25:47 --> 00:25:52 But there's a perception that it's harder. 357 00:25:52 --> 00:25:56 And so I really have a duty to give you the calculus 358 00:25:56 --> 00:25:59 made harder story here. 359 00:25:59 --> 00:26:03 So we have to make things harder, because that's our job. 360 00:26:03 --> 00:26:06 And this is actually what most people do in calculus, and it's 361 00:26:06 --> 00:26:09 the reason why calculus has a bad reputation. 362 00:26:09 --> 00:26:15 So the secret is that when people ask problems in 363 00:26:15 --> 00:26:19 calculus, they generally ask them in context. 364 00:26:19 --> 00:26:22 And there are many, many other things going on. 365 00:26:22 --> 00:26:25 And so the little piece of the problem which is calculus is 366 00:26:25 --> 00:26:28 actually fairly routine and has to be isolated and 367 00:26:28 --> 00:26:28 gotten through. 368 00:26:28 --> 00:26:32 But all the rest of it, relies on everything else you learned 369 00:26:32 --> 00:26:36 in mathematics up to this stage, from grade school 370 00:26:36 --> 00:26:36 through high school. 371 00:26:36 --> 00:26:39 So that's the complication. 372 00:26:39 --> 00:26:41 So now we're going to do a little bit of 373 00:26:41 --> 00:26:49 calculus made hard. 374 00:26:49 --> 00:26:53 By talking about a word problem. 375 00:26:53 --> 00:26:57 We only have one sort of word problem that we can pose, 376 00:26:57 --> 00:27:03 because all we've talked about is this geometry point of view. 377 00:27:03 --> 00:27:06 So far those are the only kinds of word problems we can pose. 378 00:27:06 --> 00:27:08 So what we're gonna do is just pose such a problem. 379 00:27:08 --> 00:27:30 So find the areas of triangles, enclosed by the axes and 380 00:27:30 --> 00:27:39 the tangent to y = 1/x. 381 00:27:39 --> 00:27:43 OK, so that's a geometry problem. 382 00:27:43 --> 00:27:46 And let me draw a picture of it. 383 00:27:46 --> 00:27:52 It's practically the same as the picture for example one. 384 00:27:52 --> 00:27:54 We only consider the first quadrant. 385 00:27:54 --> 00:27:55 Here's our shape. 386 00:27:55 --> 00:28:00 All right, it's the hyperbola. 387 00:28:00 --> 00:28:03 And here's maybe one of our tangent lines, which is 388 00:28:03 --> 00:28:05 coming in like this. 389 00:28:05 --> 00:28:12 And then we're trying to find this area here. 390 00:28:12 --> 00:28:14 Right, so there's our problem. 391 00:28:14 --> 00:28:16 So why does it have to do with calculus? 392 00:28:16 --> 00:28:18 It has to do with calculus because there's a tangent line 393 00:28:18 --> 00:28:22 in it, so we're gonna need to do some calculus to 394 00:28:22 --> 00:28:24 answer this question. 395 00:28:24 --> 00:28:30 But as you'll see, the calculus is the easy part. 396 00:28:30 --> 00:28:34 So let's get started with this problem. 397 00:28:34 --> 00:28:37 First of all, I'm gonna label a few things. 398 00:28:37 --> 00:28:40 And one important thing to remember of course, is that 399 00:28:40 --> 00:28:41 the curve is y = 1/x. 400 00:28:42 --> 00:28:44 That's perfectly reasonable to do. 401 00:28:44 --> 00:28:49 And also, we're gonna calculate the areas of the triangles, and 402 00:28:49 --> 00:28:51 you could ask yourself, in terms of what? 403 00:28:51 --> 00:28:54 Well, we're gonna have to pick a point and give it a name. 404 00:28:54 --> 00:28:56 And since we need a number, we're gonna have to do 405 00:28:56 --> 00:28:56 more than geometry. 406 00:28:56 --> 00:28:59 We're gonna have to do some of this analysis just 407 00:28:59 --> 00:29:01 as we've done before. 408 00:29:01 --> 00:29:04 So I'm gonna pick a point and, consistent with the labeling 409 00:29:04 --> 00:29:08 we've done before, I'm gonna to call it (x0, y0). 410 00:29:08 --> 00:29:13 So that's almost half the battle, having notations, x and 411 00:29:13 --> 00:29:18 y for the variables, and x0 and y0, for the specific point. 412 00:29:18 --> 00:29:25 Now, once you see that you have these labellings, I hope it's 413 00:29:25 --> 00:29:28 reasonable to do the following. 414 00:29:28 --> 00:29:31 So first of all, this is the point x0, and over 415 00:29:31 --> 00:29:33 here is the point y0. 416 00:29:33 --> 00:29:37 That's something that we're used to in graphs. 417 00:29:37 --> 00:29:40 And in order to figure out the area of this triangle, it's 418 00:29:40 --> 00:29:43 pretty clear that we should find the base, which is that we 419 00:29:43 --> 00:29:45 should find this location here. 420 00:29:45 --> 00:29:48 And we should find the height, so we need to 421 00:29:48 --> 00:29:55 find that value there. 422 00:29:55 --> 00:29:58 Let's go ahead and do it. 423 00:29:58 --> 00:30:02 So how are we going to do this? 424 00:30:02 --> 00:30:14 Well, so let's just take a look. 425 00:30:14 --> 00:30:17 So what is it that we need to do? 426 00:30:17 --> 00:30:21 I claim that there's only one calculus step, and I'm gonna 427 00:30:21 --> 00:30:25 put a star here for this tangent line. 428 00:30:25 --> 00:30:28 I have to understand what the tangent line is. 429 00:30:28 --> 00:30:30 Once I've figured out what the tangent line is, the rest of 430 00:30:30 --> 00:30:33 the problem is no longer calculus. 431 00:30:33 --> 00:30:35 It's just that slope that we need. 432 00:30:35 --> 00:30:38 So what's the formula for the tangent line? 433 00:30:38 --> 00:30:45 Put that over here. it's going to be y - y0 is equal to, 434 00:30:45 --> 00:30:48 and here's the magic number, we already calculated it. 435 00:30:48 --> 00:30:50 It's in the box over there. 436 00:30:50 --> 00:30:58 It's -1/x0^2 ( x - x0). 437 00:30:58 --> 00:31:12 So this is the only bit of calculus in this problem. 438 00:31:12 --> 00:31:15 But now we're not done. 439 00:31:15 --> 00:31:16 We have to finish it. 440 00:31:16 --> 00:31:19 We have to figure out all the rest of these quantities so 441 00:31:19 --> 00:31:27 we can figure out the area. 442 00:31:27 --> 00:31:31 All right. 443 00:31:31 --> 00:31:40 So how do we do that? 444 00:31:40 --> 00:31:44 Well, to find this point, this has a name. 445 00:31:44 --> 00:31:52 We're gonna find the so called x-intercept. 446 00:31:52 --> 00:31:54 That's the first thing we're going to do. 447 00:31:54 --> 00:31:58 So to do that, what we need to do is to find where 448 00:31:58 --> 00:32:02 this horizontal line meets that diagonal line. 449 00:32:02 --> 00:32:10 And the equation for the x-intercept is y = 0. 450 00:32:10 --> 00:32:13 So we plug in y = 0, that's this horizontal line, 451 00:32:13 --> 00:32:15 and we find this point. 452 00:32:15 --> 00:32:18 So let's do that into star. 453 00:32:18 --> 00:32:22 We get 0 minus, oh one other thing we need to know. 454 00:32:22 --> 00:32:28 We know that y0 is f(x0) , and f(x) is 1/x , so 455 00:32:28 --> 00:32:31 this thing is 1/x0. 456 00:32:31 --> 00:32:33 457 00:32:33 --> 00:32:36 And that's equal to -1/x0^2. 458 00:32:38 --> 00:32:41 And here's x, and here's x0. 459 00:32:41 --> 00:32:49 All right, so in order to find this x value, I have to plug in 460 00:32:49 --> 00:32:53 one equation into the other. 461 00:32:53 --> 00:32:59 So this simplifies a bit. 462 00:32:59 --> 00:33:01 This is -x/x0^2. 463 00:33:03 --> 00:33:07 And this is plus 1/x0 because the x0 and 464 00:33:07 --> 00:33:10 x0^2 cancel somewhat. 465 00:33:10 --> 00:33:15 And so if I put this on the other side, I get x / 466 00:33:15 --> 00:33:20 x0^2 is equal to 2 / x0. 467 00:33:20 --> 00:33:27 And if I then multiply through - so that's what this implies - 468 00:33:27 --> 00:33:39 and if I multiply through by x0^2 I get x = 2x0. 469 00:33:39 --> 00:33:51 OK, so I claim that this point weve just calculated it's 2x0. 470 00:33:51 --> 00:33:57 Now, I'm almost done. 471 00:33:57 --> 00:34:00 I need to get the other one. 472 00:34:00 --> 00:34:03 I need to get this one up here. 473 00:34:03 --> 00:34:06 Now I'm gonna use a very big shortcut to do that. 474 00:34:06 --> 00:34:27 So the shortcut to the y-intercept is to use symmetry. 475 00:34:27 --> 00:34:30 All right, I claim I can stare at this and I can look at that, 476 00:34:30 --> 00:34:33 and I know the formula for the y-intercept. 477 00:34:33 --> 00:34:40 It's equal to 2y0. 478 00:34:40 --> 00:34:40 All right. 479 00:34:40 --> 00:34:42 That's what that one is. 480 00:34:42 --> 00:34:44 So this one is 2y0. 481 00:34:44 --> 00:34:48 And the reason I know this is the following: so here's the 482 00:34:48 --> 00:34:52 symmetry of the situation, which is not completely direct. 483 00:34:52 --> 00:34:56 It's a kind of mirror symmetry around the diagonal. 484 00:34:56 --> 00:35:05 It involves the exchange of (x, y) with (y, x); so trading 485 00:35:05 --> 00:35:06 the roles of x and y. 486 00:35:06 --> 00:35:09 So the symmetry that I'm using is that any formula I get that 487 00:35:09 --> 00:35:13 involves x's and y's, if I trade all the x's and replace 488 00:35:13 --> 00:35:16 them by y's and trade all the y's and replace them by x's, 489 00:35:16 --> 00:35:18 then I'll have a correct formula on the other ways. 490 00:35:18 --> 00:35:21 So if everywhere I see a y I make it an x, and everywhere I 491 00:35:21 --> 00:35:24 see an x I make it a y, the switch will take place. 492 00:35:24 --> 00:35:27 So why is that? 493 00:35:27 --> 00:35:30 That's just an accident of this equation. 494 00:35:30 --> 00:35:46 That's because, so the symmetry explained... is that the 495 00:35:46 --> 00:35:48 equation is y= 1 / x. 496 00:35:48 --> 00:35:53 But that's the same thing as xy = 1, if I multiply through by 497 00:35:53 --> 00:35:58 x, which is the same thing as x = 1/y. 498 00:35:58 --> 00:36:05 So here's where the x and the y get reversed. 499 00:36:05 --> 00:36:13 OK now if you don't trust this explanation, you can also get 500 00:36:13 --> 00:36:28 the y-intercept by plugging x = 0 into the equation star. 501 00:36:28 --> 00:36:29 OK? 502 00:36:29 --> 00:36:34 We plugged y = 0 in and we got the x value. 503 00:36:34 --> 00:36:43 And you can do the same thing analogously the other way. 504 00:36:43 --> 00:36:47 All right so I'm almost done with the geometry problem, 505 00:36:47 --> 00:36:58 and let's finish it off now. 506 00:36:58 --> 00:37:00 Well, let me hold off for one second before I finish it off. 507 00:37:00 --> 00:37:05 What I'd like to say is just make one more tiny remark. 508 00:37:05 --> 00:37:09 And this is the hardest part of calculus in my opinion. 509 00:37:09 --> 00:37:15 So the hardest part of calculus is that we call it one variable 510 00:37:15 --> 00:37:21 calculus, but we're perfectly happy to deal with four 511 00:37:21 --> 00:37:25 variables at a time or five, or any number. 512 00:37:25 --> 00:37:29 In this problem, I had an x, a y, an x0 and a y0. 513 00:37:29 --> 00:37:32 That's already four different things that have various 514 00:37:32 --> 00:37:35 relationships between them. 515 00:37:35 --> 00:37:37 Of course the manipulations we do with them are algebraic, and 516 00:37:37 --> 00:37:41 when we're doing the derivatives we just consider 517 00:37:41 --> 00:37:43 what's known as one variable calculus. 518 00:37:43 --> 00:37:45 But really there are millions of variable floating 519 00:37:45 --> 00:37:46 around potentially. 520 00:37:46 --> 00:37:49 So that's what makes things complicated, and that's 521 00:37:49 --> 00:37:51 something that you have to get used to. 522 00:37:51 --> 00:37:53 Now there's something else which is more subtle, and that 523 00:37:53 --> 00:37:58 I think many people who teach the subject or use the subject 524 00:37:58 --> 00:38:01 aren't aware, because they've already entered into the 525 00:38:01 --> 00:38:04 language and they're so comfortable with it that they 526 00:38:04 --> 00:38:06 don't even notice this confusion. 527 00:38:06 --> 00:38:10 There's something deliberately sloppy about the way we 528 00:38:10 --> 00:38:12 deal with these variables. 529 00:38:12 --> 00:38:14 The reason is very simple. 530 00:38:14 --> 00:38:16 There are already four variables here. 531 00:38:16 --> 00:38:20 I don't wanna create six names for variables or 532 00:38:20 --> 00:38:23 eight names for variables. 533 00:38:23 --> 00:38:26 But really in this problem there were about eight. 534 00:38:26 --> 00:38:29 I just slipped them by you. 535 00:38:29 --> 00:38:30 So why is that? 536 00:38:30 --> 00:38:35 Well notice that the first time that I got a formula for y0 537 00:38:35 --> 00:38:39 here, it was this point. 538 00:38:39 --> 00:38:44 And so the formula for y0, which I plugged in right here, 539 00:38:44 --> 00:38:50 was from the equation of the curve. y0 = 1 / x0. 540 00:38:50 --> 00:38:55 The second time I did it, I did not use y = 1 / x. 541 00:38:55 --> 00:39:01 I used this equation here, so this is not y = 1/x. 542 00:39:01 --> 00:39:03 That's the wrong thing to do. 543 00:39:03 --> 00:39:06 It's an easy mistake to make if the formulas are all a blur to 544 00:39:06 --> 00:39:09 you and you're not paying attention to where they 545 00:39:09 --> 00:39:11 are on the diagram. 546 00:39:11 --> 00:39:16 You see that x-intercept calculation there involved 547 00:39:16 --> 00:39:21 where this horizontal line met this diagonal line, and y = 0 548 00:39:21 --> 00:39:25 represented this line here. 549 00:39:25 --> 00:39:31 So the sloppines is that y means two different things. 550 00:39:31 --> 00:39:34 And we do this constantly because it's way, way more 551 00:39:34 --> 00:39:37 complicated not to do it. 552 00:39:37 --> 00:39:40 It's much more convenient for us to allow ourselves the 553 00:39:40 --> 00:39:44 flexibility to change the role that this letter plays in 554 00:39:44 --> 00:39:47 the middle of a computation. 555 00:39:47 --> 00:39:50 And similarly, later on, if I had done this by this more 556 00:39:50 --> 00:39:54 straightforward method, for the y-intercept, I would 557 00:39:54 --> 00:39:55 have set x equal to 0. 558 00:39:55 --> 00:39:59 That would have been this vertical line, which is x = 0. 559 00:39:59 --> 00:40:03 But I didn't change the letter x when I did that, because 560 00:40:03 --> 00:40:06 that would be a waste for us. 561 00:40:06 --> 00:40:09 So this is one of the main confusions that happens. 562 00:40:09 --> 00:40:13 If you can keep yourself straight, you're a lot better 563 00:40:13 --> 00:40:21 off, and as I say this is one of the complexities. 564 00:40:21 --> 00:40:24 All right, so now let's finish off the problem. 565 00:40:24 --> 00:40:30 Let me finally get this area here. 566 00:40:30 --> 00:40:33 So, actually I'll just finish it off right here. 567 00:40:33 --> 00:40:41 So the area of the triangle is, well it's the base 568 00:40:41 --> 00:40:42 times the height. 569 00:40:42 --> 00:40:46 The base is 2x0 the height is 2y0, and a half of that. 570 00:40:46 --> 00:40:54 So it's 1/2( 2x0)(2y0) , which is (2x0)(y0), which 571 00:40:54 --> 00:40:57 is, lo and behold, 2. 572 00:40:57 --> 00:41:00 So the amusing thing in this case is that it actually didn't 573 00:41:00 --> 00:41:02 matter what x0 and y0 are. 574 00:41:02 --> 00:41:05 We get the same answer every time. 575 00:41:05 --> 00:41:10 That's just an accident of the function 1 / x. 576 00:41:10 --> 00:41:19 It happens to be the function with that property. 577 00:41:19 --> 00:41:23 All right, so we have some more business today, 578 00:41:23 --> 00:41:24 some serious business. 579 00:41:24 --> 00:41:30 So let me continue. 580 00:41:30 --> 00:41:41 So, first of all, I want to give you a few more notations. 581 00:41:41 --> 00:41:50 And these are just other notations that people use 582 00:41:50 --> 00:41:51 to refer to derivatives. 583 00:41:51 --> 00:41:54 And the first one is the following: we already 584 00:41:54 --> 00:41:56 wrote y = f(x). 585 00:41:56 --> 00:42:00 And so when we write delta y, that means the same 586 00:42:00 --> 00:42:01 thing as delta f. 587 00:42:01 --> 00:42:04 That's a typical notation. 588 00:42:04 --> 00:42:13 And previously we wrote f' for the derivative, so 589 00:42:13 --> 00:42:20 this is Newton's notation for the derivative. 590 00:42:20 --> 00:42:22 But there are other notations. 591 00:42:22 --> 00:42:28 And one of them is df/dx, and another one is dy/ dx, meaning 592 00:42:28 --> 00:42:29 exactly the same thing. 593 00:42:29 --> 00:42:34 And sometimes we let the function slip down below 594 00:42:34 --> 00:42:40 so that becomes d / dx (f) and d/ dx(y) . 595 00:42:40 --> 00:42:43 So these are all notations that are used for the 596 00:42:43 --> 00:42:49 derivative, and these were initiated by Leibniz. 597 00:42:49 --> 00:42:55 And these notations are used interchangeably, sometimes 598 00:42:55 --> 00:42:56 practically together. 599 00:42:56 --> 00:42:59 They both turn out to be extremely useful. 600 00:42:59 --> 00:43:04 This one omits - notice that this thing omits- the 601 00:43:04 --> 00:43:07 underlying base point, x0. 602 00:43:07 --> 00:43:09 That's one of the nuisances. 603 00:43:09 --> 00:43:11 It doesn't give you all the information. 604 00:43:11 --> 00:43:17 But there are lots of situations like that where 605 00:43:17 --> 00:43:20 people leave out some of the important information, and 606 00:43:20 --> 00:43:23 you have to fill it in from context. 607 00:43:23 --> 00:43:28 So that's another couple of notations. 608 00:43:28 --> 00:43:33 So now I have one more calculation for you today. 609 00:43:33 --> 00:43:35 I carried out this calculation of the derivative of 610 00:43:35 --> 00:43:45 the function 1 / x. 611 00:43:45 --> 00:43:48 I wanna take care of some other powers. 612 00:43:48 --> 00:43:59 So let's do that. 613 00:43:59 --> 00:44:07 So Example 2 is going to be the function f(x) = x^n. 614 00:44:08 --> 00:44:14 n = 1, 2, 3; one of these guys. 615 00:44:14 --> 00:44:18 And now what we're trying to figure out is the derivative 616 00:44:18 --> 00:44:21 with respect to x of x^n in our new notation, what 617 00:44:21 --> 00:44:27 this is equal to. 618 00:44:27 --> 00:44:32 So again, we're going to form this expression, 619 00:44:32 --> 00:44:35 delta f / delta x. 620 00:44:35 --> 00:44:38 And we're going to make some algebraic simplification. 621 00:44:38 --> 00:44:41 So what we plug in for delta f is ((x 622 00:44:41 --> 00:44:46 delta x)^n - x^n)/delta x. 623 00:44:48 --> 00:44:50 Now before, let me just stick this in then 624 00:44:50 --> 00:44:52 I'm gonna erase it. 625 00:44:52 --> 00:44:56 Before, I wrote x0 here and x0 there. 626 00:44:56 --> 00:45:00 But now I'm going to get rid of it, because in this particular 627 00:45:00 --> 00:45:01 calculation, it's a nuisance. 628 00:45:01 --> 00:45:04 I don't have an x floating around, which means something 629 00:45:04 --> 00:45:06 different from the x0. 630 00:45:06 --> 00:45:08 And I just don't wanna have to keep on writing 631 00:45:08 --> 00:45:10 all those symbols. 632 00:45:10 --> 00:45:13 It's a waste of blackboard energy. 633 00:45:13 --> 00:45:17 There's a total amount of energy, and I've already filled 634 00:45:17 --> 00:45:21 up so many blackboards that, there's just a limited amount. 635 00:45:21 --> 00:45:23 Plus, I'm trying to conserve chalk. 636 00:45:23 --> 00:45:25 Anyway, no 0's. 637 00:45:25 --> 00:45:28 So think of x as fixed. 638 00:45:28 --> 00:45:40 In this case, delta x moves and x is fixed in this calculation. 639 00:45:40 --> 00:45:42 All right now, in order to simplify this, in order to 640 00:45:42 --> 00:45:45 understand algebraically what's going on, I need to understand 641 00:45:45 --> 00:45:48 what the nth power of a sum is. 642 00:45:48 --> 00:45:50 And that's a famous formula. 643 00:45:50 --> 00:45:52 We only need a little tiny bit of it, called the 644 00:45:52 --> 00:45:56 binomial theorem. 645 00:45:56 --> 00:46:08 So, the binomial theorem which is in your text and explained 646 00:46:08 --> 00:46:15 in an appendix, says that if you take the sum of two guys 647 00:46:15 --> 00:46:18 and you take them to the nth power, that of course is (x 648 00:46:18 --> 00:46:24 delta x) multiplied by itself n times. 649 00:46:24 --> 00:46:29 And so the first term is x^n, that's when all of 650 00:46:29 --> 00:46:31 the n factors come in. 651 00:46:31 --> 00:46:35 And then, you could have this factor of delta x 652 00:46:35 --> 00:46:36 and all the rest x's. 653 00:46:36 --> 00:46:39 So at least one term of the form (x^(n-1))delta x. 654 00:46:39 --> 00:46:41 655 00:46:41 --> 00:46:43 And how many times does that happen? 656 00:46:43 --> 00:46:46 Well, it happens when there's a factor from here, from the next 657 00:46:46 --> 00:46:48 factor, and so on, and so on, and so on. 658 00:46:48 --> 00:46:54 There's a total of n possible times that that happens. 659 00:46:54 --> 00:46:59 And now the great thing is that, with this alone, all the 660 00:46:59 --> 00:47:06 rest of the terms are junk that we won't have to worry about. 661 00:47:06 --> 00:47:11 So to be more specific, there's a very careful 662 00:47:11 --> 00:47:12 notation for the junk. 663 00:47:12 --> 00:47:14 The junk is what's called big O((delta x)^2). 664 00:47:14 --> 00:47:18 665 00:47:18 --> 00:47:27 What that means is that these are terms of order, so with 666 00:47:27 --> 00:47:33 (delta x)^2, (delta x)^3 or higher. 667 00:47:34 --> 00:47:38 All right, that's how. 668 00:47:38 --> 00:47:42 Very exciting, higher order terms. 669 00:47:42 --> 00:47:47 OK, so this is the only algebra that we need to do, and now 670 00:47:47 --> 00:47:50 we just need to combine it together to get our result. 671 00:47:50 --> 00:47:54 So, now I'm going to just carry out the cancellations 672 00:47:54 --> 00:48:02 that we need. 673 00:48:02 --> 00:48:03 So here we go. 674 00:48:03 --> 00:48:10 We have delta f / delta x, which remember was 1 / delta 675 00:48:10 --> 00:48:22 x times this, which is this times, now this is (x^n 676 00:48:22 --> 00:48:27 nx^(n-1) delta x 677 00:48:27 --> 00:48:35 this junk term) - x^n. 678 00:48:35 --> 00:48:38 So that's what we have so far based on our 679 00:48:38 --> 00:48:41 previous calculations. 680 00:48:41 --> 00:48:48 Now, I'm going to do the main cancellation, which is this. 681 00:48:48 --> 00:48:49 All right. 682 00:48:49 --> 00:48:55 So, that's 1/delta x( nx^(n-1) delta x 683 00:48:55 --> 00:49:01 this term here). 684 00:49:01 --> 00:49:05 And now I can divide in by delta x. 685 00:49:05 --> 00:49:08 So I get nx^(n-1) 686 00:49:08 --> 00:49:09 now it's O(delta x). 687 00:49:10 --> 00:49:13 There's at least one factor of delta x not two factors of 688 00:49:13 --> 00:49:17 delta x, because I have to cancel one of them. 689 00:49:17 --> 00:49:19 And now I can just take the limit. 690 00:49:19 --> 00:49:22 And the limit this term is gonna be 0. 691 00:49:22 --> 00:49:25 That's why I called it junk originally, 692 00:49:25 --> 00:49:26 because it disappears. 693 00:49:26 --> 00:49:31 And in math, junk is something that goes away. 694 00:49:31 --> 00:49:36 So this tends to, as delta x goes to 0, nx ^ (n-1). 695 00:49:37 --> 00:49:43 And so what I've shown you is that d/dx of x to the n minus - 696 00:49:43 --> 00:49:47 sorry -n, is equal to nx^(n-1). 697 00:49:47 --> 00:49:51 698 00:49:51 --> 00:49:54 So now this is gonna be super important to you right on your 699 00:49:54 --> 00:49:57 problem set in every possible way, and I want to tell you one 700 00:49:57 --> 00:50:00 thing, one way in which it's very important. 701 00:50:00 --> 00:50:02 One way that extends it immediately. 702 00:50:02 --> 00:50:10 So this thing extends to polynomials. 703 00:50:10 --> 00:50:13 We get quite a lot out of this one calculation. 704 00:50:13 --> 00:50:18 Namely, if I take d / dx of something like (x^3 705 00:50:18 --> 00:50:25 5x^10) that's gonna be equal to 3x^2, that's applying 706 00:50:25 --> 00:50:27 this rule to x^3. 707 00:50:27 --> 00:50:33 And then here, I'll get 5*10 so 50x^9. 708 00:50:35 --> 00:50:38 So this is the type of thing that we get out of it, and 709 00:50:38 --> 00:50:50 we're gonna make more hay with that next time. 710 00:50:50 --> 00:50:50 Question. 711 00:50:50 --> 00:50:50 Yes. 712 00:50:50 --> 00:50:51 I turned myself off. 713 00:50:51 --> 00:50:52 Yes? 714 00:50:52 --> 00:50:56 Student: [INAUDIBLE] 715 00:50:56 --> 00:51:01 Professor: The question was the binomial theorem only works 716 00:51:01 --> 00:51:04 when delta x goes to 0. 717 00:51:04 --> 00:51:07 No, the binomial theorem is a general formula which also 718 00:51:07 --> 00:51:10 specifies exactly what the junk is. 719 00:51:10 --> 00:51:11 It's very much more detailed. 720 00:51:11 --> 00:51:13 But we only needed this part. 721 00:51:13 --> 00:51:18 We didn't care what all these crazy terms were. 722 00:51:18 --> 00:51:24 It's junk for our purposes now, because we don't happen to need 723 00:51:24 --> 00:51:27 any more than those first two terms. 724 00:51:27 --> 00:51:29 Yes, because delta x goes to 0. 725 00:51:29 --> 00:51:32 OK, see you next time.