1 00:00:00 --> 00:00:00,04 2 00:00:00,04 --> 00:00:02,32 The following content is provided under a Creative 3 00:00:02,32 --> 00:00:03,59 Commons license. 4 00:00:03,59 --> 00:00:06,56 Your support will help MIT OpenCourseWare continue to 5 00:00:06,56 --> 00:00:09,99 offer high quality educational resources for free. 6 00:00:09,99 --> 00:00:12,765 To make a donation, or to view additional material from 7 00:00:12,765 --> 00:00:16,12 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16,12 --> 00:00:21,7 at ocw.mit.edu. 9 00:00:21,7 --> 00:00:25,93 PROFESSOR: Right now, we're finishing up with the first 10 00:00:25,93 --> 00:00:31,89 unit, and I'd like to continue in this lecture, lecture seven, 11 00:00:31,89 --> 00:00:45,51 with some final remarks about exponents. 12 00:00:45,51 --> 00:00:49,95 So what I'd like to do is just review something that I did 13 00:00:49,95 --> 00:00:53,19 quickly last time, and make a few philosophical 14 00:00:53,19 --> 00:00:53,92 remarks about it. 15 00:00:53,92 --> 00:00:57,88 I think that the steps involved were maybe a little tricky, and 16 00:00:57,88 --> 00:01:00,68 so I'd like to go through it one more time. 17 00:01:00,68 --> 00:01:03,02 Remember, that we were talking about this 18 00:01:03,02 --> 00:01:05,37 number ak, which is (1 19 00:01:05,37 --> 00:01:05,51 1/k)^k. 20 00:01:05,51 --> 00:01:08,27 21 00:01:08,27 --> 00:01:11,82 And what we showed was that the limit as k goes to 22 00:01:11,82 --> 00:01:16,92 infinity of ak was e. 23 00:01:16,92 --> 00:01:21,29 So the first thing that I'd like to do is just explain the 24 00:01:21,29 --> 00:01:27,98 proof a little bit more clearly than I did last time with fewer 25 00:01:27,98 --> 00:01:31,83 symbols, or at least with this abbreviation of the symbol 26 00:01:31,83 --> 00:01:35,05 here, to show you what we actually did. 27 00:01:35,05 --> 00:01:43,79 So I'll just remind you of what we did last time, and the first 28 00:01:43,79 --> 00:01:46,95 observation was to check, rather than the limit of 29 00:01:46,95 --> 00:01:49,55 this function, but to take the ln first. 30 00:01:49,55 --> 00:01:52,17 And this is typically what's done when you have an 31 00:01:52,17 --> 00:01:54,81 exponential, when you have an exponent. 32 00:01:54,81 --> 00:01:59,31 And what we found was that the limit here was 1 33 00:01:59,31 --> 00:02:03,5 as k goes to infinity. 34 00:02:03,5 --> 00:02:05,16 So last time, this is what we did. 35 00:02:05,16 --> 00:02:08,49 And I just wanted to be careful and show you exactly 36 00:02:08,49 --> 00:02:09,83 what the next step is. 37 00:02:09,83 --> 00:02:14,36 If you exponentiate this fact; you take e to this power, 38 00:02:14,36 --> 00:02:21,23 that's going to tend to e ^ 1, which is just e. 39 00:02:21,23 --> 00:02:26,57 And then, we just observe that this is the same as ak. 40 00:02:26,57 --> 00:02:32,18 So the basic ingredient here is that e ^ ln a = a. 41 00:02:32,18 --> 00:02:36,99 That's because the ln function is the inverse of the 42 00:02:36,99 --> 00:02:38,03 exponential function. 43 00:02:38,03 --> 00:02:38,86 Yes, question? 44 00:02:38,86 --> 00:02:54,19 STUDENT: [INAUDIBLE] 45 00:02:54,19 --> 00:02:59,42 PROFESSOR: So the question was, wouldn't the log of this be 0 46 00:02:59,42 --> 00:03:01,53 because ak is tending to 1. 47 00:03:01,53 --> 00:03:03,83 But ak isn't tending to 1. 48 00:03:03,83 --> 00:03:06,64 Who said it was? 49 00:03:06,64 --> 00:03:08,89 If you take the logarithm, which is what we did last 50 00:03:08,89 --> 00:03:12,88 time, logarithm of ak is indeed k(ln(1 + 1/k)). 51 00:03:12,88 --> 00:03:17,03 52 00:03:17,03 --> 00:03:18,52 That does not tend to 0. 53 00:03:18,52 --> 00:03:22,61 This part of it tends to 0, and this part tends to infinity. 54 00:03:22,61 --> 00:03:26,46 And they balance each other, 0 times infinity. 55 00:03:26,46 --> 00:03:29,18 We don't really know yet from this expression, in fact we did 56 00:03:29,18 --> 00:03:32,63 some cleverness with limits and derivatives, to figure 57 00:03:32,63 --> 00:03:33,23 out this limit. 58 00:03:33,23 --> 00:03:34,28 It was a very subtle thing. 59 00:03:34,28 --> 00:03:37,56 It turned out to be 1. 60 00:03:37,56 --> 00:03:38,85 All right? 61 00:03:38,85 --> 00:03:41,74 Now, the thing that I'd like to say - I'm sorry I'm going to 62 00:03:41,74 --> 00:03:46,1 erase this aside here - but you need to go back to your notes 63 00:03:46,1 --> 00:03:49,12 and remember that this is what we did last time. 64 00:03:49,12 --> 00:03:51,97 Because I want to have room for the next comment that I want to 65 00:03:51,97 --> 00:03:54,42 make on this little blackboard here. 66 00:03:54,42 --> 00:03:59,66 What we just derived was this property here, but I made a 67 00:03:59,66 --> 00:04:03 claim yesterday, and I just want to emphasize it again so 68 00:04:03 --> 00:04:07,83 that we realized what it is that we're doing. 69 00:04:07,83 --> 00:04:08,91 I looked at this backwards. 70 00:04:08,91 --> 00:04:11,24 One way you can think of this is we're evaluating this 71 00:04:11,24 --> 00:04:13 limit and getting an answer. 72 00:04:13 --> 00:04:16,74 But all equalities can be read both directions. 73 00:04:16,74 --> 00:04:21,3 And we can write it the other way: e = the limit, as k goes 74 00:04:21,3 --> 00:04:26,38 to infinity, of this expression here. 75 00:04:26,38 --> 00:04:28,64 So that's just the same thing. 76 00:04:28,64 --> 00:04:31,32 And if we read it backwards, what we're saying is that this 77 00:04:31,32 --> 00:04:35,7 limit is a formula for e. 78 00:04:35,7 --> 00:04:38,44 So this is very typical of mathematics. 79 00:04:38,44 --> 00:04:40,7 You want to always reverse your perspective all the time. 80 00:04:40,7 --> 00:04:45,17 Equations work both ways, and in this case, we have two 81 00:04:45,17 --> 00:04:46,3 different things here. 82 00:04:46,3 --> 00:04:50,55 This e was what we defined as the base, which when you graph 83 00:04:50,55 --> 00:04:54,51 e ^ x, you get slope 1 at 0. 84 00:04:54,51 --> 00:04:56,85 And then it turns out to be equal to this limit, which we 85 00:04:56,85 --> 00:04:59,06 can calculate numerically. 86 00:04:59,06 --> 00:05:02,1 If you do this on your calculators, you, of course, 87 00:05:02,1 --> 00:05:05,53 will have a way of programming in this number and 88 00:05:05,53 --> 00:05:07,29 evaluating it for each k. 89 00:05:07,29 --> 00:05:10,88 And you'll have another button available to evaluate this one. 90 00:05:10,88 --> 00:05:13,39 So another way of saying it is it that there's a relationship 91 00:05:13,39 --> 00:05:14,64 between these two things. 92 00:05:14,64 --> 00:05:19,21 And all of Calculus is a matter of getting these relationships. 93 00:05:19,21 --> 00:05:21,86 So we can look at these things in several different ways. 94 00:05:21,86 --> 00:05:25,17 And indeed, that's what we're going to be doing at least at 95 00:05:25,17 --> 00:05:27,04 the end of today in talking about derivatives. 96 00:05:27,04 --> 00:05:30,23 A lot of times when we talk about derivative, we're trying 97 00:05:30,23 --> 00:05:34,78 to look at them from several perspectives at once. 98 00:05:34,78 --> 00:05:37,58 Okay, so I have to keep on going with exponents, because 99 00:05:37,58 --> 00:05:40,13 I have one loose end. 100 00:05:40,13 --> 00:05:44,49 One loose end that I did not cover yet. 101 00:05:44,49 --> 00:05:49,07 There's one very important formula that's left, and it's 102 00:05:49,07 --> 00:05:51,92 the derivative of the powers. 103 00:05:51,92 --> 00:05:54,51 We actually didn't do this - well we did it for 104 00:05:54,51 --> 00:05:57,12 rational numbers r. 105 00:05:57,12 --> 00:06:00,07 So this is the formula here. 106 00:06:00,07 --> 00:06:06,2 But now we're going to check this for all real numbers, r. 107 00:06:06,2 --> 00:06:09,2 So including all the irrational ones as well. 108 00:06:09,2 --> 00:06:14,2 This is also good practice for using base e and using 109 00:06:14,2 --> 00:06:16,48 logarithmic differentiation. 110 00:06:16,48 --> 00:06:21,63 So let me do this by our two methods that we can use to 111 00:06:21,63 --> 00:06:26,16 handle exponential type problems. 112 00:06:26,16 --> 00:06:32,07 So method one was base e. 113 00:06:32,07 --> 00:06:34,78 So if I just rewrite this base e again, that's just this 114 00:06:34,78 --> 00:06:44,53 formula over here. x ^ r = (e^ln x)^r, which 115 00:06:44,53 --> 00:06:50,35 is e ^ r ln x. 116 00:06:50,35 --> 00:06:55,32 Okay, so now I can differentiate this. 117 00:06:55,32 --> 00:07:04,65 So I get that d / dx (x^r), now I'm going to use prime 118 00:07:04,65 --> 00:07:06,86 notation, because I don't want to keep on writing that d / 119 00:07:06,86 --> 00:07:10,58 dx here; e ^ (r ln x)'. 120 00:07:10,58 --> 00:07:13,89 121 00:07:13,89 --> 00:07:18,61 And now, what I can do is I can use the chain rule. 122 00:07:18,61 --> 00:07:21,64 The chain rule says that it's the derivative of a this times 123 00:07:21,64 --> 00:07:24,38 the derivative of the function. 124 00:07:24,38 --> 00:07:29,5 So the derivative of the exponential is just itself. 125 00:07:29,5 --> 00:07:32,04 And the derivative of this guy here, well I'll write 126 00:07:32,04 --> 00:07:35,4 it out once, is (r ln x)'. 127 00:07:35,4 --> 00:07:39,7 128 00:07:39,7 --> 00:07:42,25 So what's that equal to? 129 00:07:42,25 --> 00:07:45,89 Well, each of the r ln x's is just x ^ r. 130 00:07:45,89 --> 00:07:50,13 And this derivative here is... 131 00:07:50,13 --> 00:07:53,87 Well the derivative of r is 0. 132 00:07:53,87 --> 00:07:54,86 This is a constant. 133 00:07:54,86 --> 00:07:56,68 It just factors out. 134 00:07:56,68 --> 00:07:58,94 And ln x now the derivative... 135 00:07:58,94 --> 00:08:01,69 What's the derivative of ln x? 136 00:08:01,69 --> 00:08:06,57 1 / x, so this is going to be times r / x. 137 00:08:06,57 --> 00:08:10,21 And now, we rewrite it in the customary form, which is r, we 138 00:08:10,21 --> 00:08:13,04 put the r in front, x ^ (r - 1). 139 00:08:13,04 --> 00:08:13,82 Ok? 140 00:08:13,82 --> 00:08:19,06 So I just derived the formula for you. 141 00:08:19,06 --> 00:08:23,31 And it didn't matter whether r was rational or irrational, 142 00:08:23,31 --> 00:08:25,32 it's the same proof. 143 00:08:25,32 --> 00:08:29,44 Okay so now I have to show you how method two works as well. 144 00:08:29,44 --> 00:08:35,18 So let's do method two, which we called logarithmic 145 00:08:35,18 --> 00:08:39,28 differentiation. 146 00:08:39,28 --> 00:08:44,37 And so here I'll use a symbol, say u for x ^ r, and 147 00:08:44,37 --> 00:08:44,93 I'll take its logarithm. 148 00:08:44,93 --> 00:08:50,48 That's r ln x. 149 00:08:50,48 --> 00:08:51,83 And now I differentiate it. 150 00:08:51,83 --> 00:08:54,99 I'll leave that in the middle, because I want to remember 151 00:08:54,99 --> 00:08:57,27 the key property of logarithmic differention. 152 00:08:57,27 --> 00:08:58,95 But first I'll differentiate it. 153 00:08:58,95 --> 00:09:00,63 Later on, what I'm going to use is that this 154 00:09:00,63 --> 00:09:02,62 is the same as u'/u. 155 00:09:02,62 --> 00:09:06,47 This is one way of evaluating a logarithmic derivative. 156 00:09:06,47 --> 00:09:09,23 And then the other is to differentiate the explicit 157 00:09:09,23 --> 00:09:10,98 function that we have over here. 158 00:09:10,98 --> 00:09:16,79 And that is just, as we said, r /x. 159 00:09:16,79 --> 00:09:24,54 So now, I multiply through, and I get u' = u(r/x), which 160 00:09:24,54 --> 00:09:29,75 is just x^r(r/x), which is just what we did before. 161 00:09:29,75 --> 00:09:33,41 It's r x ^ (r - 1). 162 00:09:33,41 --> 00:09:38,11 Again, you can now see by comparing these two pieces 163 00:09:38,11 --> 00:09:41,47 of arithmetic that they're basically the same. 164 00:09:41,47 --> 00:09:43,95 Pretty much every time you convert to base c or you do 165 00:09:43,95 --> 00:09:46,09 logarithmic differentiation, it'll amount to the same 166 00:09:46,09 --> 00:09:48,27 thing, provided you don't get mixed up. 167 00:09:48,27 --> 00:09:51,72 You generally have to introduce a new symbol here. 168 00:09:51,72 --> 00:09:55,84 On the other hand, you're dealing with exponents there. 169 00:09:55,84 --> 00:10:00,99 It's worth it to know both points of view. 170 00:10:00,99 --> 00:10:05,6 All right, so now I want to make one last remark before 171 00:10:05,6 --> 00:10:09,91 we finish with exponents. 172 00:10:09,91 --> 00:10:16,34 And, I'll try to sell this to you in a lot of ways as the 173 00:10:16,34 --> 00:10:21,49 court goes on, but one thing that I want to try to 174 00:10:21,49 --> 00:10:27,21 emphasize is that the natural logarithm really is natural. 175 00:10:27,21 --> 00:10:39,92 So, I claim that the natural log is natural. 176 00:10:39,92 --> 00:10:45,72 And the example that we're going to use for this 177 00:10:45,72 --> 00:10:53,34 illustration is economics. 178 00:10:53,34 --> 00:10:54,09 Okay? 179 00:10:54,09 --> 00:10:58,52 So let me explain to why the natural log is the one that's 180 00:10:58,52 --> 00:11:00,82 natural for economics. 181 00:11:00,82 --> 00:11:06,2 If you are imagining the price of a stock that you own goes 182 00:11:06,2 --> 00:11:11,16 down by a dollar, that's a totally meaningless statement. 183 00:11:11,16 --> 00:11:13,48 It depends on a lot of things. 184 00:11:13,48 --> 00:11:15,73 In particular, it depends on whether the original price 185 00:11:15,73 --> 00:11:18,3 was a dollar or 100 dollars. 186 00:11:18,3 --> 00:11:22,13 So there's not much meaning to these absolute numbers. 187 00:11:22,13 --> 00:11:25,08 It's always the ratios that matter. 188 00:11:25,08 --> 00:11:30,83 So, for example, I just looked up an hour ago, the London 189 00:11:30,83 --> 00:11:43,64 Exchange close, and it was down 27.9, which as I said, is 190 00:11:43,64 --> 00:11:47,39 pretty meaningless unless you know what the actual 191 00:11:47,39 --> 00:11:50,05 total of this index is. 192 00:11:50,05 --> 00:11:54,2 It turns out it was 6,432. 193 00:11:54,2 --> 00:11:59,02 So the change in the price, divided by the price, which 194 00:11:59,02 --> 00:12:07,55 in this case is 27.9 / 6,432, is what matters. 195 00:12:07,55 --> 00:12:12,01 And, in this case, it happens to be 0.43%. 196 00:12:12,01 --> 00:12:12,29 All right? 197 00:12:12,29 --> 00:12:14,27 That's what happened today. 198 00:12:14,27 --> 00:12:18,7 And similarly, if you take the infinitessimal of this, people 199 00:12:18,7 --> 00:12:21,26 think of days as being relatively small increments 200 00:12:21,26 --> 00:12:25,11 when you're investing in a stock, you would be interested 201 00:12:25,11 --> 00:12:28,69 in the infinitessimal sense, p'/p. 202 00:12:28,69 --> 00:12:33,08 The derivative of p / p. 203 00:12:33,08 --> 00:12:35,53 That's just (ln p)'. 204 00:12:35,53 --> 00:12:38,16 205 00:12:38,16 --> 00:12:42,66 So this is the - let me put a little box around it - the 206 00:12:42,66 --> 00:12:45,46 formula of logarithmic differentiation. 207 00:12:45,46 --> 00:12:49,7 But let me just emphasize that it has an actual significance, 208 00:12:49,7 --> 00:12:52,43 and it's the one that's used by economists and people who 209 00:12:52,43 --> 00:12:54,45 are modeling prices of things all the time. 210 00:12:54,45 --> 00:12:58,62 They never use absolute prices when there are large swings. 211 00:12:58,62 --> 00:13:01,01 They always use the log of the price. 212 00:13:01,01 --> 00:13:07,01 And there's no point in using log base 10, or log base 2. 213 00:13:07,01 --> 00:13:08,18 Those give you junk. 214 00:13:08,18 --> 00:13:11,19 They give you an extra factor of log 2. 215 00:13:11,19 --> 00:13:14,87 It's the natural log that's the obvious one to use. 216 00:13:14,87 --> 00:13:19,38 It's completely straightforward that this is a simpler 217 00:13:19,38 --> 00:13:22,6 expression than using log base 10 and having a factor of 218 00:13:22,6 --> 00:13:26,8 natural log of 10 there, which would just mess everything up. 219 00:13:26,8 --> 00:13:29,36 All right, so this is just one illustration. 220 00:13:29,36 --> 00:13:32,2 Anything that has to do with ratios is going to 221 00:13:32,2 --> 00:13:36,16 encounter logarithms. 222 00:13:36,16 --> 00:13:41,27 All right, so that's pretty much it. 223 00:13:41,27 --> 00:13:45,99 That's all I want to say for now anyway. 224 00:13:45,99 --> 00:13:48,28 There's lots more to say, but we'll be saying it when we do 225 00:13:48,28 --> 00:13:51,25 applications of derivatives in the second unit. 226 00:13:51,25 --> 00:13:54,45 So now, what I'd like to do is to start a review. 227 00:13:54,45 --> 00:13:57,79 I'm just going to run through what we did in this unit. 228 00:13:57,79 --> 00:14:01,58 I'll tell you approximately what I expect from you on the 229 00:14:01,58 --> 00:14:06,15 test that's coming up tomorrow. 230 00:14:06,15 --> 00:14:14,64 And, well, so let's get started with that. 231 00:14:14,64 --> 00:14:27,05 So this is a review of Unit One. 232 00:14:27,05 --> 00:14:32,61 And I'm just going to put on the board all of the things 233 00:14:32,61 --> 00:14:35,75 that you need to think about, anyway, keep in your head. 234 00:14:35,75 --> 00:14:41,75 And there what are called general formulas 235 00:14:41,75 --> 00:14:45,07 for derivatives. 236 00:14:45,07 --> 00:14:51,97 And then there are the specific ones. 237 00:14:51,97 --> 00:14:55,92 And let me just remind you what the general formulas are. 238 00:14:55,92 --> 00:15:01,44 There's what you do to differentiate a sum, a multiple 239 00:15:01,44 --> 00:15:08,19 of a function, the product rule, the quotient rule. 240 00:15:08,19 --> 00:15:11,55 Those are several general formulas. 241 00:15:11,55 --> 00:15:15,02 And then there's one more, which is the chain rule, which 242 00:15:15,02 --> 00:15:17,45 I'm going to say just a little bit more about. 243 00:15:17,45 --> 00:15:22,35 So the derivative of a function of a function is the derivative 244 00:15:22,35 --> 00:15:26,65 of the function times the derivative of the 245 00:15:26,65 --> 00:15:27,43 other function. 246 00:15:27,43 --> 00:15:33,78 So here, I've abbreviated u = u(x). 247 00:15:33,78 --> 00:15:36,63 Right, so this is one of two ways of writing it. 248 00:15:36,63 --> 00:15:40,03 The other way is also one that you can keep in mind and you 249 00:15:40,03 --> 00:15:42,47 might find easier to remember. 250 00:15:42,47 --> 00:15:46,69 It's probably a good idea to remember both formulas. 251 00:15:46,69 --> 00:15:53,27 And then the last type of general formula that we did 252 00:15:53,27 --> 00:15:56,95 was implicit differentiation. 253 00:15:56,95 --> 00:15:59,2 Okay? 254 00:15:59,2 --> 00:16:04,7 So when you do implicit differentiation, you have an 255 00:16:04,7 --> 00:16:09,27 equation and you don't try to solve for the unknown function. 256 00:16:09,27 --> 00:16:13,11 You just put it in its simplest form and you differentiate. 257 00:16:13,11 --> 00:16:20,44 So, we actually did this, in particular, for inverses. 258 00:16:20,44 --> 00:16:23,805 That was a very, very key method for calculating the 259 00:16:23,805 --> 00:16:25,18 inverses of functions. 260 00:16:25,18 --> 00:16:28,6 And it's also true that logarithmic differentiation 261 00:16:28,6 --> 00:16:31,42 is of this type. 262 00:16:31,42 --> 00:16:33,31 This is a transformation. 263 00:16:33,31 --> 00:16:34,84 We're differentiating something else. 264 00:16:34,84 --> 00:16:37,92 We're transforming the equation by taking its logarithm 265 00:16:37,92 --> 00:16:40,98 and then differentiating. 266 00:16:40,98 --> 00:16:45,2 Ok, so there are a number of different ways this is applied. 267 00:16:45,2 --> 00:16:48,45 It can also be applied, anyway, these are two of them. 268 00:16:48,45 --> 00:16:50,32 So maybe in parenthesis. 269 00:16:50,32 --> 00:16:53,12 These are just examples. 270 00:16:53,12 --> 00:16:54,35 All right. 271 00:16:54,35 --> 00:16:58,55 I'll try to give examples of at least a few of 272 00:16:58,55 --> 00:16:59,96 these rules later. 273 00:16:59,96 --> 00:17:05,76 So now, the specific functions that you know how to 274 00:17:05,76 --> 00:17:08,36 differentiate: well you know how to differentiate now x ^ r 275 00:17:08,36 --> 00:17:11,41 thanks to what I just did. 276 00:17:11,41 --> 00:17:15,77 We have the sine and the cosine functions, which you're 277 00:17:15,77 --> 00:17:19,5 responsible for knowing what their derivatives are. 278 00:17:19,5 --> 00:17:26,49 And then other trig functions like tan and secant. 279 00:17:26,49 --> 00:17:29,81 We generally don't bother with cosecants and cotangents, 280 00:17:29,81 --> 00:17:32,19 because everything can be expressed in terms 281 00:17:32,19 --> 00:17:34,03 of these anyway. 282 00:17:34,03 --> 00:17:35,87 Actually, you can really express everything in terms 283 00:17:35,87 --> 00:17:36,95 of sines and cosines. 284 00:17:36,95 --> 00:17:39,76 But what you'll find is that it's much more convenient to 285 00:17:39,76 --> 00:17:42,73 remember the derivatives of these as well. 286 00:17:42,73 --> 00:17:45,87 So memorize all of these. 287 00:17:45,87 --> 00:17:49,81 All right, and then we had e^x and ln x. 288 00:17:49,81 --> 00:17:53,92 And we had the inverses of the trig functions. 289 00:17:53,92 --> 00:18:00,01 These were the two that we did: the arctangent and the arcsin. 290 00:18:00,01 --> 00:18:02,22 So those are the ones you're responsible for. 291 00:18:02,22 --> 00:18:06,97 You should have enough time, anyway, to work out anything 292 00:18:06,97 --> 00:18:09,39 else, if you know these. 293 00:18:09,39 --> 00:18:11,8 All right, so basically the idea is you have a bunch 294 00:18:11,8 --> 00:18:13,07 of special formulas. 295 00:18:13,07 --> 00:18:14,82 You have a bunch of general formulas. 296 00:18:14,82 --> 00:18:17,87 You put them together, and you can generate 297 00:18:17,87 --> 00:18:20,97 pretty much anything. 298 00:18:20,97 --> 00:18:24,6 Okay, so let's do a few examples before going 299 00:18:24,6 --> 00:18:41,29 on with the review. 300 00:18:41,29 --> 00:18:48,23 Okay, so I do want to do a few examples in sort of increasing 301 00:18:48,23 --> 00:18:50,84 level of difficulty and how you would combine these 302 00:18:50,84 --> 00:18:51,42 things together. 303 00:18:51,42 --> 00:18:56,38 So first of all, you should remember that if you 304 00:18:56,38 --> 00:19:02,35 differentiate the secant function, that's just - oh I 305 00:19:02,35 --> 00:19:04,01 just realized that I wanted to say something else 306 00:19:04,01 --> 00:19:06,63 before - so forget that. 307 00:19:06,63 --> 00:19:08,06 We'll do that in a second. 308 00:19:08,06 --> 00:19:10,98 I wanted to make some general remarks. 309 00:19:10,98 --> 00:19:17,29 So there's one rule that you discussed in my absence, 310 00:19:17,29 --> 00:19:19,07 which is the chain rule. 311 00:19:19,07 --> 00:19:22,1 And I do want to make just a couple of remarks about the 312 00:19:22,1 --> 00:19:28,1 chain rule now to remind you of what it is, and also to 313 00:19:28,1 --> 00:19:30,16 present some consequences. 314 00:19:30,16 --> 00:19:39,19 So, a little bit of extra on the chain rule. 315 00:19:39,19 --> 00:19:43,72 The first thing that I want say is that we didn't really 316 00:19:43,72 --> 00:19:46,66 fully explain why it's true. 317 00:19:46,66 --> 00:19:54,14 And I do want to just explain it by example, okay? 318 00:19:54,14 --> 00:20:01,18 So imagine that you have a function which is, say, 10x 319 00:20:01,18 --> 00:20:02,01 b. 320 00:20:02,01 --> 00:20:02,5 All right? 321 00:20:02,5 --> 00:20:03,69 So y = 10x 322 00:20:03,69 --> 00:20:04,98 b. 323 00:20:04,98 --> 00:20:09,97 Then obviously, y is changing 10 times as fast as b, right? 324 00:20:09,97 --> 00:20:18,06 The issue is this number here, dy / dx = 10. 325 00:20:18,06 --> 00:20:23,53 And now if x is a function of something, say t, shifted 326 00:20:23,53 --> 00:20:34,29 by some other constant here, then dx/dt = 5. 327 00:20:34,29 --> 00:20:38,51 Now all the chain rule is saying is that if y is going 10 328 00:20:38,51 --> 00:20:44,74 times as fast as t, I'm sorry as x, and x is going 5 times as 329 00:20:44,74 --> 00:20:50,62 fast as t, then y is going 50 times as fast as t. 330 00:20:50,62 --> 00:20:54,85 And algebraically, all this means is if I plug in and 331 00:20:54,85 --> 00:20:57,1 substitute, which is what the composition of the two 332 00:20:57,1 --> 00:21:04,57 functions amounts to, 10(5t + a) + b and I multiply 333 00:21:04,57 --> 00:21:07,91 it out, I get 50t 334 00:21:07,91 --> 00:21:09,2 10a b. 335 00:21:09,2 --> 00:21:12 Now these terms don't matter. 336 00:21:12 --> 00:21:13,03 The constant terms don't matter. 337 00:21:13,03 --> 00:21:14,8 The rate is 50. 338 00:21:14,8 --> 00:21:17,6 And so the consequence, if we put them together, is that 339 00:21:17,6 --> 00:21:30,17 dy/dt = 10*5, which is 50. 340 00:21:30,17 --> 00:21:32,4 All right, so this is in a nutshell why the 341 00:21:32,4 --> 00:21:33,63 chain rule works. 342 00:21:33,63 --> 00:21:39,45 And why these rates multiply. 343 00:21:39,45 --> 00:21:43,05 The second thing that I wanted to say about the chain rule is 344 00:21:43,05 --> 00:21:45,89 that it has a few consequences that make some of the other 345 00:21:45,89 --> 00:21:50,22 rules a little easier to remember or possibly to avoid. 346 00:21:50,22 --> 00:21:55,46 The messiest rule in my humble opinion is the quotient 347 00:21:55,46 --> 00:21:59,43 rule, which is kind of a nuisance to remember. 348 00:21:59,43 --> 00:22:02,49 So let me just remind you, if you take just the reciprocal 349 00:22:02,49 --> 00:22:05,99 of a function, and you differentiate it, there's 350 00:22:05,99 --> 00:22:08,17 another way of looking at this. 351 00:22:08,17 --> 00:22:11,03 And it's actually the way that I use, so I want to encourage 352 00:22:11,03 --> 00:22:12,8 you to think about it this way too. 353 00:22:12,8 --> 00:22:16,7 This is the same as (v^-1)' power . 354 00:22:16,7 --> 00:22:18,97 And now instead of using the quotient rule, which we 355 00:22:18,97 --> 00:22:25,69 could've used, we can use the chain rule here with the 356 00:22:25,69 --> 00:22:29,72 power -1, which works by the power law. 357 00:22:29,72 --> 00:22:30,96 So what is this equal to? 358 00:22:30,96 --> 00:22:33,64 This is equal to (-v^-2)(v'). 359 00:22:33,64 --> 00:22:38,93 360 00:22:38,93 --> 00:22:42,73 So here, I've applied the chain rule rather than 361 00:22:42,73 --> 00:22:47,37 the quotient rule. 362 00:22:47,37 --> 00:22:53,74 And similarly, suppose I wanted to derive the 363 00:22:53,74 --> 00:22:54,56 full quotient rule. 364 00:22:54,56 --> 00:22:57,24 Well, now this may or may not be easier. 365 00:22:57,24 --> 00:22:59,77 But this is one way of remembering what's going on. 366 00:22:59,77 --> 00:23:04,67 If you convert it to (uv^-1) and you differentiate 367 00:23:04,67 --> 00:23:09,18 that, now I can use the product rule on this. 368 00:23:09,18 --> 00:23:11,43 Of course, I have to use the chain rule and 369 00:23:11,43 --> 00:23:13,03 this rule as well. 370 00:23:13,03 --> 00:23:15,62 So what do I get? 371 00:23:15,62 --> 00:23:20,69 I get u' , the inverse, 372 00:23:20,69 --> 00:23:22,81 u, and then I have to differentiate the v inverse. 373 00:23:22,81 --> 00:23:24,49 That's the formula right up here. 374 00:23:24,49 --> 00:23:25,29 That's (-v^-2)(v'). 375 00:23:25,29 --> 00:23:30,3 376 00:23:30,3 --> 00:23:33,23 So that's one way of doing it. 377 00:23:33,23 --> 00:23:35,93 This actually explains the funny minus sign when you 378 00:23:35,93 --> 00:23:38,56 differentiate v in the formula. 379 00:23:38,56 --> 00:23:41,73 The other formula, the other way that we did it, was by 380 00:23:41,73 --> 00:23:44,37 putting this over a common denominator. 381 00:23:44,37 --> 00:23:49,33 The common denominator was v^2. 382 00:23:49,33 --> 00:23:51,58 This comes from this v ^ -2. 383 00:23:51,58 --> 00:23:54,73 And then the second term is - u v'. 384 00:23:54,73 --> 00:23:57,25 385 00:23:57,25 --> 00:24:00,15 And the first term, we have to multiply by an extra factor of 386 00:24:00,15 --> 00:24:02,19 v, because we have a v^2 in the denominator. 387 00:24:02,19 --> 00:24:04,32 So it's u'v. 388 00:24:04,32 --> 00:24:08,11 All right, so this is the quotient rule as we wrote it 389 00:24:08,11 --> 00:24:11,515 down in lecture, and this is just another way of remembering 390 00:24:11,515 --> 00:24:13,85 it or deriving it without remembering it, if you just 391 00:24:13,85 --> 00:24:16,7 remember the chain rule and the product rule. 392 00:24:16,7 --> 00:24:21,46 Okay, so you'll find that in many contexts, it's easier 393 00:24:21,46 --> 00:24:25,91 to do one or the other. 394 00:24:25,91 --> 00:24:29,21 Okay, so now i'm ready to differentiate the secant 395 00:24:29,21 --> 00:24:30,99 and a few such functions. 396 00:24:30,99 --> 00:24:36,2 So we'll do some examples here here. 397 00:24:36,2 --> 00:24:40,42 So here's the secant function, and I want to use that formula 398 00:24:40,42 --> 00:24:44,82 up there for the reciprocal. 399 00:24:44,82 --> 00:24:48,09 This is the way I think of it. 400 00:24:48,09 --> 00:24:53,15 This is the cos x ^ -1. 401 00:24:53,15 --> 00:24:58,75 And so, the formula here is just what? 402 00:24:58,75 --> 00:25:04,03 It's just - (cos x) ^ -2 (-sin x). 403 00:25:04,03 --> 00:25:20,28 404 00:25:20,28 --> 00:25:22,69 So now this is usually written in a different fashion, so 405 00:25:22,69 --> 00:25:25,17 that's why I'm doing this for a reason actually. 406 00:25:25,17 --> 00:25:27,81 Which is although there are several formulas for things, 407 00:25:27,81 --> 00:25:30,51 with trig functions, there are usually five ways 408 00:25:30,51 --> 00:25:31,87 of writing something. 409 00:25:31,87 --> 00:25:34,43 So I'm writing this one down so that you know what the standard 410 00:25:34,43 --> 00:25:36,78 way of presenting it is. 411 00:25:36,78 --> 00:25:39,43 So what happens here is that we have two minus 412 00:25:39,43 --> 00:25:40,3 signs cancelling. 413 00:25:40,3 --> 00:25:44,36 And we get sin x / cos^2 x. 414 00:25:44,36 --> 00:25:48,1 That's a perfectly acceptable answer, but there's a customary 415 00:25:48,1 --> 00:25:49,47 way in which is written. 416 00:25:49,47 --> 00:25:55,89 It's written (1 / cos x) (sin x / cos x). 417 00:25:55,89 --> 00:25:58,32 And then we get rid of the denominators by rewriting 418 00:25:58,32 --> 00:26:04,1 it in terms of secant and tangent, so sec x tan x. 419 00:26:04,1 --> 00:26:09,63 So this is the form that's generally used when 420 00:26:09,63 --> 00:26:11,79 you see these formulas written in textbooks. 421 00:26:11,79 --> 00:26:15,04 And so you know, you need to watch out, because if you ever 422 00:26:15,04 --> 00:26:18,64 want to use this kind of Calculus, you'll have not 423 00:26:18,64 --> 00:26:22,84 be put off by all the secants and tangents. 424 00:26:22,84 --> 00:26:27,27 All right, so getting slightly more complicated, how about if 425 00:26:27,27 --> 00:26:28,75 we differentiate ln(sec x)? 426 00:26:28,75 --> 00:26:37,4 427 00:26:37,4 --> 00:26:40,48 If you differentiate the natural log, that's just going 428 00:26:40,48 --> 00:26:49,45 to be (sec x)' / sec x. 429 00:26:49,45 --> 00:26:52,46 And plugging in the formula that we had before, that's 430 00:26:52,46 --> 00:27:00,33 (sec x) (tan x) / sec x, which is tan x. 431 00:27:00,33 --> 00:27:03,85 So this one also has a very nice form. 432 00:27:03,85 --> 00:27:08,23 And you might say that this is kind of an ugly function, but 433 00:27:08,23 --> 00:27:14,7 the strange thing is that the natural log was invented before 434 00:27:14,7 --> 00:27:19,56 the exponential by a guy named Napier, exactly in order to 435 00:27:19,56 --> 00:27:21,72 evaluate functions like this. 436 00:27:21,72 --> 00:27:24,3 These are the functions that people cared about 437 00:27:24,3 --> 00:27:28,89 a lot, because they were used in navigation. 438 00:27:28,89 --> 00:27:32,64 You wanted to multiply sines and cosines together 439 00:27:32,64 --> 00:27:34,03 to do navigation. 440 00:27:34,03 --> 00:27:38,94 And the multiplication he encoded using a logarithm. 441 00:27:38,94 --> 00:27:40,71 So these were invented long before people even 442 00:27:40,71 --> 00:27:42,92 knew about exponents. 443 00:27:42,92 --> 00:27:44,6 And it was a surprise, actually, that it was 444 00:27:44,6 --> 00:27:46,1 connected to exponents. 445 00:27:46,1 --> 00:27:48,74 So the natural log was invented before the log base 10 and 446 00:27:48,74 --> 00:27:52,65 everything else, exactly for this kind of purpose. 447 00:27:52,65 --> 00:27:56,51 Anyway, so this is a nice function, which was very 448 00:27:56,51 --> 00:28:03,77 important, so that your ships wouldn't crash into the reef. 449 00:28:03,77 --> 00:28:05,57 Okay, let's continue here. 450 00:28:05,57 --> 00:28:09,175 So there's another kind of function that I want 451 00:28:09,175 --> 00:28:10,49 to discuss with you. 452 00:28:10,49 --> 00:28:16,59 And these are the kinds in which there's a choice as to 453 00:28:16,59 --> 00:28:19,38 which of these rules to apply. 454 00:28:19,38 --> 00:28:25,13 And I'll just give a couple of examples of that. 455 00:28:25,13 --> 00:28:28,786 There usually is a better and a worse way, so let 456 00:28:28,786 --> 00:28:38,43 me illustrate that. 457 00:28:38,43 --> 00:28:41,12 Okay, yet another example. 458 00:28:41,12 --> 00:28:43,83 I hope you've seen some of these before. 459 00:28:43,83 --> 00:28:45,58 Say (x ^ 10 460 00:28:45,58 --> 00:28:47,28 + 8x) ^ 6. 461 00:28:47,28 --> 00:28:51,01 462 00:28:51,01 --> 00:28:53,28 So it's a little bit more complicated than what we had 463 00:28:53,28 --> 00:29:00,33 before, because there were several more symbols here. 464 00:29:00,33 --> 00:29:03,21 So what should we do at this point? 465 00:29:03,21 --> 00:29:06,4 There's one choice which I claim is a bad idea, and 466 00:29:06,4 --> 00:29:10,81 that is to expand this out to the 6th power. 467 00:29:10,81 --> 00:29:13,53 That's a bad idea, because it's very long. 468 00:29:13,53 --> 00:29:15,99 And then your answer will also be very long. 469 00:29:15,99 --> 00:29:19,62 It will fill the entire exam paper, for instance. 470 00:29:19,62 --> 00:29:20,02 Yeah? 471 00:29:20,02 --> 00:29:21,38 STUDENT: Can you use the chain rule? 472 00:29:21,38 --> 00:29:21,97 PROFESSOR: Chain rule. 473 00:29:21,97 --> 00:29:22,58 That's it. 474 00:29:22,58 --> 00:29:23,5 We use the chain rule. 475 00:29:23,5 --> 00:29:26,86 So fortunately, this is relatively easy using 476 00:29:26,86 --> 00:29:27,62 the chain rule. 477 00:29:27,62 --> 00:29:30,79 We just think of this box as being the function. 478 00:29:30,79 --> 00:29:35,34 And we take 6 times this guy to the 5th, times the derivative 479 00:29:35,34 --> 00:29:39,38 of this guy, which is 10x ^ 9 480 00:29:39,38 --> 00:29:41,5 8. 481 00:29:41,5 --> 00:29:43,91 And this is, feeling this in, it's x^10 + 8x. 482 00:29:43,91 --> 00:29:46,14 And that's it. 483 00:29:46,14 --> 00:29:50,27 That's all you need to do differentiate things like this. 484 00:29:50,27 --> 00:29:55,14 The chain rule is very effective. 485 00:29:55,14 --> 00:30:00,34 STUDENT: [INAUDIBLE] 486 00:30:00,34 --> 00:30:01,28 PROFESSOR: That's a good question. 487 00:30:01,28 --> 00:30:05,83 So I'm not really willing to answer too many questions about 488 00:30:05,83 --> 00:30:07,5 what's going to be on the exam. 489 00:30:07,5 --> 00:30:10,59 But the question that was just asked is exactly the kind of 490 00:30:10,59 --> 00:30:13,2 question I'm very happy to answer. 491 00:30:13,2 --> 00:30:20,09 Ok the question was, in what form is an acceptable answer? 492 00:30:20,09 --> 00:30:23,56 Now in real life, that is a really serious question. 493 00:30:23,56 --> 00:30:26,38 When you ask a computer a question and it gives you 494 00:30:26,38 --> 00:30:31,38 500 million sheets of printout, it's useless. 495 00:30:31,38 --> 00:30:34,85 And you really care what form answers are in, and indeed, 496 00:30:34,85 --> 00:30:39 somebody might really care what this thing to the 6th power is, 497 00:30:39 --> 00:30:42,2 and then you would be forced to discuss things in terms of 498 00:30:42,2 --> 00:30:46,11 that other functional form. 499 00:30:46,11 --> 00:30:50,41 For the purposes of this exam, this is okay form. 500 00:30:50,41 --> 00:30:54,49 And, in fact, any correct form is an okay form. 501 00:30:54,49 --> 00:30:57,77 I recommend strongly that you not try to simplify things 502 00:30:57,77 --> 00:30:59,7 unless we tell you to. 503 00:30:59,7 --> 00:31:04,86 Sometimes it will be to your advantage to simplify things. 504 00:31:04,86 --> 00:31:08,01 Sometimes we'll say simplify. 505 00:31:08,01 --> 00:31:10,92 It takes a good deal of experience to know when 506 00:31:10,92 --> 00:31:13,34 it's really worth it to simplify expressions. 507 00:31:13,34 --> 00:31:13,62 Yes? 508 00:31:13,62 --> 00:31:19,53 STUDENT: [INAUDIBLE] 509 00:31:19,53 --> 00:31:23,59 PROFESSOR: Right, so turning to this example. 510 00:31:23,59 --> 00:31:25,52 The question is what is this derivative? 511 00:31:25,52 --> 00:31:27,24 And here's an answer. 512 00:31:27,24 --> 00:31:29,5 That's the end of the problem. 513 00:31:29,5 --> 00:31:31,81 This is a more customary form. 514 00:31:31,81 --> 00:31:37,16 But this is answer is ok. 515 00:31:37,16 --> 00:31:38,61 Same issue. 516 00:31:38,61 --> 00:31:40,97 That's exactly the point. 517 00:31:40,97 --> 00:31:41,66 Yes? 518 00:31:41,66 --> 00:31:51,46 STUDENT: [INAUDIBLE] 519 00:31:51,46 --> 00:31:59,41 PROFESSOR: The question is, do you have to show the work? 520 00:31:59,41 --> 00:32:00,24 Do you have to show the work? 521 00:32:00,24 --> 00:32:05,87 Well if I ask you what is d/dx of sec x, then if you wrote 522 00:32:05,87 --> 00:32:08,39 down this answer or you wrote down this answer showing no 523 00:32:08,39 --> 00:32:11,2 work, that would be acceptable. 524 00:32:11,2 --> 00:32:17,05 If the question was derive the formula for this from the 525 00:32:17,05 --> 00:32:19,25 formula for the derivative of the cosine or something like 526 00:32:19,25 --> 00:32:21,16 that, then it would not be acceptable. 527 00:32:21,16 --> 00:32:24,34 You'd have to carry out this arithmetic. 528 00:32:24,34 --> 00:32:29,28 So, in other words, typically this will come up, for 529 00:32:29,28 --> 00:32:32,29 instance, in various contexts. 530 00:32:32,29 --> 00:32:35,04 You just basically have to follow directions. 531 00:32:35,04 --> 00:32:35,33 Yes? 532 00:32:35,33 --> 00:32:41,52 STUDENT: [INAUDIBLE] 533 00:32:41,52 --> 00:32:43,68 PROFESSOR: The next question is, are you expected to be able 534 00:32:43,68 --> 00:32:46,18 to prove what the derivative of the sine function is? 535 00:32:46,18 --> 00:32:49,58 The short answer to that is yes. 536 00:32:49,58 --> 00:32:52,51 But I will be getting to that when I discuss the rest 537 00:32:52,51 --> 00:32:54,24 of the material here. 538 00:32:54,24 --> 00:32:58,43 We're almost there. 539 00:32:58,43 --> 00:33:02,64 Okay, so let me just finish these examples 540 00:33:02,64 --> 00:33:04,88 with one last one. 541 00:33:04,88 --> 00:33:09,04 And then we'll talk about this question of things like the 542 00:33:09,04 --> 00:33:12,06 derivative of the sine function, and deriving it. 543 00:33:12,06 --> 00:33:15,62 So the last example that I'd like to write down is the one 544 00:33:15,62 --> 00:33:20,75 that I promised you in the first lecture, namely to 545 00:33:20,75 --> 00:33:26,57 differentiate e ^ x arctan x. 546 00:33:26,57 --> 00:33:28,38 Basically you're supposed to be able to differentiate 547 00:33:28,38 --> 00:33:29,35 any function. 548 00:33:29,35 --> 00:33:32,39 So this is the one that we mentioned at the beginning. 549 00:33:32,39 --> 00:33:34,13 So here it is. 550 00:33:34,13 --> 00:33:37,28 Let's do it. 551 00:33:37,28 --> 00:33:38,17 So what is it? 552 00:33:38,17 --> 00:33:46,15 Well, it's just equal to - I have to differentiate. 553 00:33:46,15 --> 00:33:51,28 I have to use the chain rule - it's equal to the exponential 554 00:33:51,28 --> 00:33:58,2 times the derivative of this expression here. 555 00:33:58,2 --> 00:33:59,26 That's the chain rule. 556 00:33:59,26 --> 00:34:01,7 That's the first step. 557 00:34:01,7 --> 00:34:06,44 And now I have to apply the product rule here. 558 00:34:06,44 --> 00:34:10,82 So I have e ^ x arctan x. 559 00:34:10,82 --> 00:34:16,12 And I differentiate the first factor, so I get arctan x. 560 00:34:16,12 --> 00:34:18,01 Add to it what happens when I differentiate the second 561 00:34:18,01 --> 00:34:19,67 factor, leaving alone the x. 562 00:34:19,67 --> 00:34:21,64 So that's x / 1 563 00:34:21,64 --> 00:34:24,31 x^2. 564 00:34:24,31 --> 00:34:26,3 And that's it. 565 00:34:26,3 --> 00:34:28,78 That's the end of the problem. 566 00:34:28,78 --> 00:34:30,59 It wasn't that hard. 567 00:34:30,59 --> 00:34:35,56 Of course, it requires you to remember all of the rules, and 568 00:34:35,56 --> 00:34:37,3 a lot of formulas underlying them. 569 00:34:37,3 --> 00:34:39,56 So that's consistent with what I just told you. 570 00:34:39,56 --> 00:34:42,06 I told you that you wanted to know this. 571 00:34:42,06 --> 00:34:44,74 I told you that you needed to know this product rule, 572 00:34:44,74 --> 00:34:50,61 and that you needed to know the chain rule. 573 00:34:50,61 --> 00:34:53,086 And I guess there was one more thing, the derivative of e to 574 00:34:53,086 --> 00:34:55,26 the x came into play there. 575 00:34:55,26 --> 00:34:59,17 So of these formulas, we used four of them in 576 00:34:59,17 --> 00:35:03,81 this one calculation. 577 00:35:03,81 --> 00:35:15,88 Okay, so now what other things did we talk about in Unit One? 578 00:35:15,88 --> 00:35:25,18 So the main other thing that we talked about was the 579 00:35:25,18 --> 00:35:33,12 definition of a derivative. 580 00:35:33,12 --> 00:35:42,46 And also there was sort of a goal which was to get to the 581 00:35:42,46 --> 00:35:51,05 meaning of the derivative. 582 00:35:51,05 --> 00:35:56,75 So these are things - so we had a couple of ways of looking at 583 00:35:56,75 --> 00:35:59,4 it, or at least a couple that I'm going to 584 00:35:59,4 --> 00:36:01,78 emphasize right now. 585 00:36:01,78 --> 00:36:06,27 But first, let me remind you what the formula is. 586 00:36:06,27 --> 00:36:13,9 The derivative is the limit as delta x goes to 0 of (f(x 587 00:36:13,9 --> 00:36:19,04 delta x) - f(x)) / delta x. 588 00:36:19,04 --> 00:36:22,76 So that's it, and this is certainly going to be 589 00:36:22,76 --> 00:36:25,64 a central focus here. 590 00:36:25,64 --> 00:36:29,6 And you want to be able to recognize this formula 591 00:36:29,6 --> 00:36:42,76 in a number of ways. 592 00:36:42,76 --> 00:36:44,52 So, how do we use this? 593 00:36:44,52 --> 00:36:50,56 Well one thing we did was we calculated a bunch of 594 00:36:50,56 --> 00:36:51,45 these rates of change. 595 00:36:51,45 --> 00:36:53,66 In fact, more or less, they're the ones which are 596 00:36:53,66 --> 00:36:55,76 written right over here. 597 00:36:55,76 --> 00:36:57,21 This list of functions here. 598 00:36:57,21 --> 00:37:01,47 Now, which ones did we start out with just straight 599 00:37:01,47 --> 00:37:03,8 from the definition here? 600 00:37:03,8 --> 00:37:04,84 Which of these things? 601 00:37:04,84 --> 00:37:05,83 There were a whole bunch of them. 602 00:37:05,83 --> 00:37:09,18 So we started out with a function 1 / x. 603 00:37:09,18 --> 00:37:11,53 We did x^n. 604 00:37:11,53 --> 00:37:14,53 We did sine x. 605 00:37:14,53 --> 00:37:16,88 We did cosine x. 606 00:37:16,88 --> 00:37:18,89 Now there was a little bit of subtlety with 607 00:37:18,89 --> 00:37:21,11 sine x and cosine x. 608 00:37:21,11 --> 00:37:25,21 We got them using something else. 609 00:37:25,21 --> 00:37:26,88 We didn't quite get them all the way. 610 00:37:26,88 --> 00:37:31,79 We got them using the case x = 0. 611 00:37:31,79 --> 00:37:35,59 We got them from the derivative at x = 0, we got the formulas 612 00:37:35,59 --> 00:37:37,68 for the derivatives of sine and cosine. 613 00:37:37,68 --> 00:37:41,48 But that was an argument which involved plug in sine x 614 00:37:41,48 --> 00:37:44,46 delta x, and running through. 615 00:37:44,46 --> 00:37:45,84 So that's one example. 616 00:37:45,84 --> 00:37:50,63 We also did a ^ x. 617 00:37:50,63 --> 00:37:53,27 And that may be it. 618 00:37:53,27 --> 00:37:58,35 Oh yeah, I think that's about it. 619 00:37:58,35 --> 00:38:00,75 That may be about it. 620 00:38:00,75 --> 00:38:00,95 No. 621 00:38:00,95 --> 00:38:01,62 It isn't. 622 00:38:01,62 --> 00:38:04,62 Ok, so let me make a connection here which you probably haven't 623 00:38:04,62 --> 00:38:07,77 yet made, which is that we did it for (u v)'. 624 00:38:07,77 --> 00:38:10,52 625 00:38:10,52 --> 00:38:15,69 And we also did it for (u / v)'. 626 00:38:15,69 --> 00:38:18,06 So sorry, I shouldn't write primes, because that's 627 00:38:18,06 --> 00:38:20,5 not consistent. 628 00:38:20,5 --> 00:38:22,74 I differentiated the product; I differentiated the 629 00:38:22,74 --> 00:38:28,11 quotient using the same delta x notation. 630 00:38:28,11 --> 00:38:32,76 I guess I forgot that because I wasn't there when it happened. 631 00:38:32,76 --> 00:38:36,55 So look, these are the ones that you do by this. 632 00:38:36,55 --> 00:38:39,31 And, of course, you might have to reduce them to other things. 633 00:38:39,31 --> 00:38:42,19 These involve using something else. 634 00:38:42,19 --> 00:38:46,61 This one involves using the slope of this function at 0, 635 00:38:46,61 --> 00:38:48,6 just the way the sine and the cosine did. 636 00:38:48,6 --> 00:38:51,25 This one involves the slopes of the individual 637 00:38:51,25 --> 00:38:53,5 functions, u and v. 638 00:38:53,5 --> 00:38:55,33 And this one also involves the individual. 639 00:38:55,33 --> 00:38:57,13 So, in other words, it doesn't get you all the way through to 640 00:38:57,13 --> 00:39:01,17 the end, but it's expressed in terms of something simpler 641 00:39:01,17 --> 00:39:03,01 in each of these cases. 642 00:39:03,01 --> 00:39:05,84 And I could go on. 643 00:39:05,84 --> 00:39:09,69 We didn't do these in class, but you're certainly... e ^ x 644 00:39:09,69 --> 00:39:12,17 is a perfectly okay one on one of the exams. 645 00:39:12,17 --> 00:39:14,97 We ask you for 1 / x^2. 646 00:39:14,97 --> 00:39:16,58 In other words, I'm not claiming that it's going to 647 00:39:16,58 --> 00:39:18,8 be one on this list, but it certainly can be 648 00:39:18,8 --> 00:39:19,86 any one of these. 649 00:39:19,86 --> 00:39:21,86 But we're not going to ask you to go all the way through to 650 00:39:21,86 --> 00:39:26,38 the beginning in these formulas. 651 00:39:26,38 --> 00:39:28,94 There are also some fundamental limits that I certainly 652 00:39:28,94 --> 00:39:31,21 want you to know about. 653 00:39:31,21 --> 00:39:34,68 And these you can derive in reverse. 654 00:39:34,68 --> 00:39:58,88 So I will describe that now. 655 00:39:58,88 --> 00:40:06,91 So let me also emphasize the following thing: I want to 656 00:40:06,91 --> 00:40:18,59 read this backwards now. 657 00:40:18,59 --> 00:40:21,37 This is the theme from the very beginning of this lecture. 658 00:40:21,37 --> 00:40:25,6 Namely, if you're given the function f, you can 659 00:40:25,6 --> 00:40:28,31 figure out its derivative by its formula here. 660 00:40:28,31 --> 00:40:29,86 That is the formula for this in terms of what's 661 00:40:29,86 --> 00:40:30,92 on the right hand side. 662 00:40:30,92 --> 00:40:35,41 On the other hand, you can also use the formula in that 663 00:40:35,41 --> 00:40:47,91 direction, and if you know the slope of something, you can 664 00:40:47,91 --> 00:40:49,17 figure out what the limit is. 665 00:40:49,17 --> 00:40:54,79 For example, I'll use the letter x here, even 666 00:40:54,79 --> 00:40:56,04 though it's cheating. 667 00:40:56,04 --> 00:40:59,53 Maybe I'll call it delta x so it's clearer to you. 668 00:40:59,53 --> 00:41:06,9 Maybe I'll call it u. 669 00:41:06,9 --> 00:41:10,37 Suppose you look at this limit here. 670 00:41:10,37 --> 00:41:15,25 Well, I claim that you should recognize that is the 671 00:41:15,25 --> 00:41:19,06 derivative with respect to u of the function e^u at u = 672 00:41:19,06 --> 00:41:22,66 0, which of course we know to be 1. 673 00:41:22,66 --> 00:41:25,42 So this is reading this formula in reverse. 674 00:41:25,42 --> 00:41:28,55 It's recognizing that one of these limits - let me rewrite 675 00:41:28,55 --> 00:41:35,99 this again here - one of these so-called difference quotient 676 00:41:35,99 --> 00:41:39,39 limits is a derivative. 677 00:41:39,39 --> 00:41:42,19 And since we know a formula for that derivative, 678 00:41:42,19 --> 00:41:49,94 we can evaluate it. 679 00:41:49,94 --> 00:41:54,77 And lastly, there's one other type of thing which 680 00:41:54,77 --> 00:41:57,55 I think you should know. 681 00:41:57,55 --> 00:41:59,78 These are the ones you do with difference quotients. 682 00:41:59,78 --> 00:42:01,59 There are also other formulas that you want 683 00:42:01,59 --> 00:42:03 to be able to drive. 684 00:42:03 --> 00:42:20,89 You want to be able to derive formulas by 685 00:42:20,89 --> 00:42:27,67 implicit differentiation. 686 00:42:27,67 --> 00:42:31,73 In other words, the basic idea is to take whatever equation 687 00:42:31,73 --> 00:42:37,2 you've got and simplify it as much as possible, without 688 00:42:37,2 --> 00:42:41,26 insisting that you solve for y. 689 00:42:41,26 --> 00:42:44,18 That's not necessarily the most appropriate way to 690 00:42:44,18 --> 00:42:45,63 get the rate of change. 691 00:42:45,63 --> 00:42:51,91 The much simpler formula is sin y = x. 692 00:42:51,91 --> 00:42:59,78 And that one is easier to differentiate implicitly. 693 00:42:59,78 --> 00:43:02,9 So I should say, do this kind of thing. 694 00:43:02,9 --> 00:43:05,55 So that's, if you like, a typical derivation 695 00:43:05,55 --> 00:43:08,39 that you might see. 696 00:43:08,39 --> 00:43:13,07 And then there's one last type of problem that you'll face, 697 00:43:13,07 --> 00:43:21,59 and it's the other thing that I claim we discussed. 698 00:43:21,59 --> 00:43:26,58 And it goes all the way back to the first lecture. 699 00:43:26,58 --> 00:43:34,05 So the last thing that we'll be talking about is tangent lines. 700 00:43:34,05 --> 00:43:34,2 All right? 701 00:43:34,2 --> 00:43:38,76 The geometric point of view of a derivative. 702 00:43:38,76 --> 00:43:41,9 And we'll be doing more of this in next the unit. 703 00:43:41,9 --> 00:43:45,41 So first of all, you'll be expected to be able to 704 00:43:45,41 --> 00:43:52,38 compute the tangent line. 705 00:43:52,38 --> 00:43:56,4 That's often fairly straightforward. 706 00:43:56,4 --> 00:44:03,59 And the second thing is to graph y' , the derivative 707 00:44:03,59 --> 00:44:07,37 of a function. 708 00:44:07,37 --> 00:44:10,44 And the third thing, which I'm going to throw in here, because 709 00:44:10,44 --> 00:44:14,72 I regard it in a sort of geometric vein, although it's 710 00:44:14,72 --> 00:44:16,69 got an analytical aspect to it. 711 00:44:16,69 --> 00:44:18,87 So this is a picture. 712 00:44:18,87 --> 00:44:20,71 This is a computation. 713 00:44:20,71 --> 00:44:23,33 And if you combine the two together, you 714 00:44:23,33 --> 00:44:24,27 get something else. 715 00:44:24,27 --> 00:44:37,87 And so this is to recognize differentiable functions. 716 00:44:37,87 --> 00:44:40,19 Alright, so how do you do this? 717 00:44:40,19 --> 00:44:43,6 Well, we really only have one way of doing it. 718 00:44:43,6 --> 00:44:54,58 We're going to check the left and right tangents. 719 00:44:54,58 --> 00:44:59,45 They must be equal. 720 00:44:59,45 --> 00:45:05 So again, this is a property that you should be familiar 721 00:45:05 --> 00:45:06,83 with from some of your exercises. 722 00:45:06,83 --> 00:45:09,9 And the idea is simply, that if the tangent line exists, it's 723 00:45:09,9 --> 00:45:14,77 the same from the right and from the left. 724 00:45:14,77 --> 00:45:21,94 Ok, now I'm going to just do one example here from this sort 725 00:45:21,94 --> 00:45:27,76 of qualitative sketching skill to give you an example here. 726 00:45:27,76 --> 00:45:30,58 And what I'm going to do is I'm going to draw a graph 727 00:45:30,58 --> 00:45:34,75 of a function like this. 728 00:45:34,75 --> 00:45:38,58 And what I want to do underneath is draw the 729 00:45:38,58 --> 00:45:41,6 graph of the derivative. 730 00:45:41,6 --> 00:45:46,6 So this is the function y = f(x), and here I'm going to 731 00:45:46,6 --> 00:45:56,49 draw the graph of the function y = f'(x) right underneath it. 732 00:45:56,49 --> 00:46:00,33 So now, let's think about what it's supposed to look like. 733 00:46:00,33 --> 00:46:06,09 And the one step that you need to make in order to do this, is 734 00:46:06,09 --> 00:46:08,66 to draw a few tangent lines. 735 00:46:08,66 --> 00:46:13,21 I'm just going to draw one down here. 736 00:46:13,21 --> 00:46:18,73 And I'm going to draw one up here. 737 00:46:18,73 --> 00:46:23,78 Now, the tangent lines here - noticed that the slope of these 738 00:46:23,78 --> 00:46:27,3 tangent lines are all positive. 739 00:46:27,3 --> 00:46:31,8 So everything I draw down here is going to be 740 00:46:31,8 --> 00:46:33,88 above the x-axis. 741 00:46:33,88 --> 00:46:36,58 Furthermore,, as I go further to the left, they get 742 00:46:36,58 --> 00:46:37,83 steeper and steeper. 743 00:46:37,83 --> 00:46:39,5 So they're getting higher and higher. 744 00:46:39,5 --> 00:46:44,02 So the function is coming down like this. 745 00:46:44,02 --> 00:46:45,35 It starts up there. 746 00:46:45,35 --> 00:46:50,57 Maybe I'll draw it in green to illustrate the graph here. 747 00:46:50,57 --> 00:46:56,91 So that's this function here. 748 00:46:56,91 --> 00:46:59,75 As we get farther out, it's getting flatter and flatter. 749 00:46:59,75 --> 00:47:06,27 So it's leveling off, but above the axis like that. 750 00:47:06,27 --> 00:47:09,52 So one of the things to emphasize is, you should not 751 00:47:09,52 --> 00:47:12,09 expect the derivative to look like the function. 752 00:47:12,09 --> 00:47:13,32 It's totally different. 753 00:47:13,32 --> 00:47:17,42 It's keeping track at each point of its tangent line. 754 00:47:17,42 --> 00:47:19,78 On the other hand, you should get some kind of physical feel 755 00:47:19,78 --> 00:47:23,67 for it, and we'll be practicing this more in the next unit. 756 00:47:23,67 --> 00:47:25,5 So let me give you an example of a function 757 00:47:25,5 --> 00:47:27,8 which does exactly this. 758 00:47:27,8 --> 00:47:33,24 And it's the function y = ln x. 759 00:47:33,24 --> 00:47:38,56 If you differentiate it, you get y' = 1 / x. 760 00:47:38,56 --> 00:47:44,63 And this plot above is, roughly speaking, the logarithm. 761 00:47:44,63 --> 00:47:50,23 And this plot underneath is the function 1 / x. 762 00:47:50,23 --> 00:47:53,23 We still have time for one question. 763 00:47:53,23 --> 00:47:58,58 And so, fire away. 764 00:47:58,58 --> 00:48:03,67 Yes? 765 00:48:03,67 --> 00:48:04,04 STUDENT: [INAUDIBLE] 766 00:48:04,04 --> 00:48:06,75 PROFESSOR: The question is, can you show how you derive 767 00:48:06,75 --> 00:48:09,77 the inverse tangent of x. 768 00:48:09,77 --> 00:48:13,35 So that's in a lecture. 769 00:48:13,35 --> 00:48:17,78 I'm happy to do it right now, but it's going to make 770 00:48:17,78 --> 00:48:20,42 me a whole two minutes. 771 00:48:20,42 --> 00:48:27,56 So, here's how you do it. y = arctan x. 772 00:48:27,56 --> 00:48:30,77 And now this is hopeless to differentiate, so I 773 00:48:30,77 --> 00:48:34,72 rewrite it as tan y = x. 774 00:48:34,72 --> 00:48:38,44 And now I have to differentiate that. 775 00:48:38,44 --> 00:48:43,67 So when I differentiate it, I get the derivative of tan y 776 00:48:43,67 --> 00:48:46,56 with respect to x, so with respect to y. 777 00:48:46,56 --> 00:48:47,76 That's (1 / 1 778 00:48:47,76 --> 00:48:48,21 y^2 )y'. 779 00:48:48,21 --> 00:48:51,12 780 00:48:51,12 --> 00:48:52,85 So this is a hard step. 781 00:48:52,85 --> 00:48:53,93 That's the chain rule. 782 00:48:53,93 --> 00:48:55,86 And on the left side I get 1. 783 00:48:55,86 --> 00:48:58,73 So I'm doing this super fast because we have 784 00:48:58,73 --> 00:49:00,72 thirty seconds left. 785 00:49:00,72 --> 00:49:02,84 But this is the hard step right here. 786 00:49:02,84 --> 00:49:22,81 And it needs for you to know that d/dy (tan y) = sec^2 y. 787 00:49:22,81 --> 00:49:24,92 So here's the identity. 788 00:49:24,92 --> 00:49:28,5 So you need have known this in advance. 789 00:49:28,5 --> 00:49:30,74 And that's the input into this equation. 790 00:49:30,74 --> 00:49:44,44 So now, what we have is that y' = 1 / sec^2 y, which is 791 00:49:44,44 --> 00:49:51,38 the same thing as cos^2 y. 792 00:49:51,38 --> 00:49:54,98 Now, the last bit of the problem is to rewrite 793 00:49:54,98 --> 00:49:57,93 this in terms of x. 794 00:49:57,93 --> 00:50:02,664 And that you have to do with a right triangle. 795 00:50:02,664 --> 00:50:06,884 If this is x and this is 1, then the angle is y, because 796 00:50:06,884 --> 00:50:09,416 the tangent of y is x. 797 00:50:09,416 --> 00:50:14,902 So this expresses the fact that the tangent of y is x. 798 00:50:14,902 --> 00:50:18,489 And then the hypoteneuse is the square root of 1 799 00:50:18,489 --> 00:50:18,7 x^2. 800 00:50:18,7 --> 00:50:21,654 801 00:50:21,654 --> 00:50:27,14 And so the cosine is 1 divided by that. 802 00:50:27,14 --> 00:50:30,305 So this thing is 1 divided by the square root of 1 803 00:50:30,305 --> 00:50:36,424 x^2, the quantity squared. 804 00:50:36,424 --> 00:50:40,222 So, and then the last little bit here, since I'm racing 805 00:50:40,222 --> 00:50:44,02 along, is that it's 1 / 1 + x^2, squared, which I 806 00:50:44,02 --> 00:50:46,13 incorrectly wrote over here. 807 00:50:46,13 --> 00:50:48,662 OK, so good luck on the text. 808 00:50:48,662 --> 00:50:50,806 See you tomorrow. 809 00:50:50,806 --> 00:50:52,17