1 00:00:00,040 --> 00:00:02,320 The following content is provided under a Creative 2 00:00:02,320 --> 00:00:03,590 Commons license. 3 00:00:03,590 --> 00:00:05,930 Your support will help MIT OpenCourseWare 4 00:00:05,930 --> 00:00:09,990 continue to offer high quality educational resources for free. 5 00:00:09,990 --> 00:00:12,500 To make a donation, or to view additional material 6 00:00:12,500 --> 00:00:16,120 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,120 --> 00:00:21,700 at ocw.mit.edu. 8 00:00:21,700 --> 00:00:25,000 PROFESSOR: Right now, we're finishing up 9 00:00:25,000 --> 00:00:27,740 with the first unit, and I'd like 10 00:00:27,740 --> 00:00:31,890 to continue in this lecture, lecture seven, 11 00:00:31,890 --> 00:00:45,510 with some final remarks about exponents. 12 00:00:45,510 --> 00:00:49,560 So what I'd like to do is just review something 13 00:00:49,560 --> 00:00:51,910 that I did quickly last time, and make 14 00:00:51,910 --> 00:00:53,920 a few philosophical remarks about it. 15 00:00:53,920 --> 00:00:57,660 I think that the steps involved were maybe a little tricky, 16 00:00:57,660 --> 00:01:00,680 and so I'd like to go through it one more time. 17 00:01:00,680 --> 00:01:03,900 Remember, that we were talking about this number a_k, 18 00:01:03,900 --> 00:01:05,510 which is (1 + 1/k)^k. 19 00:01:08,270 --> 00:01:11,400 And what we showed was that the limit as k 20 00:01:11,400 --> 00:01:16,920 goes to infinity of a_k was e. 21 00:01:16,920 --> 00:01:19,590 So the first thing that I'd like to do 22 00:01:19,590 --> 00:01:22,890 is just explain the proof a little bit more clearly 23 00:01:22,890 --> 00:01:28,440 than I did last time with fewer symbols, 24 00:01:28,440 --> 00:01:31,830 or at least with this abbreviation of the symbol 25 00:01:31,830 --> 00:01:35,050 here, to show you what we actually did. 26 00:01:35,050 --> 00:01:41,580 So I'll just remind you of what we did last time, 27 00:01:41,580 --> 00:01:45,726 and the first observation was to check, 28 00:01:45,726 --> 00:01:47,350 rather than the limit of this function, 29 00:01:47,350 --> 00:01:49,550 but to take the log first. 30 00:01:49,550 --> 00:01:51,640 And this is typically what's done 31 00:01:51,640 --> 00:01:54,810 when you have an exponential, when you have an exponent. 32 00:01:54,810 --> 00:01:57,750 And what we found was that the limit here 33 00:01:57,750 --> 00:02:03,500 was 1 as k goes to infinity. 34 00:02:03,500 --> 00:02:05,160 So last time, this is what we did. 35 00:02:05,160 --> 00:02:07,970 And I just wanted to be careful and show you 36 00:02:07,970 --> 00:02:09,830 exactly what the next step is. 37 00:02:09,830 --> 00:02:14,360 If you exponentiate this fact; you take e to this power, 38 00:02:14,360 --> 00:02:21,230 that's going to tend to e^1, which is just e. 39 00:02:21,230 --> 00:02:26,570 And then, we just observe that this is the same as a_k. 40 00:02:26,570 --> 00:02:32,180 So the basic ingredient here is that e^ln a = a. 41 00:02:32,180 --> 00:02:36,750 That's because the log function is the inverse 42 00:02:36,750 --> 00:02:38,030 of the exponential function. 43 00:02:38,030 --> 00:02:38,860 Yes, question? 44 00:02:38,860 --> 00:02:54,190 STUDENT: [INAUDIBLE] 45 00:02:54,190 --> 00:02:58,180 PROFESSOR: So the question was, wouldn't the log of this 46 00:02:58,180 --> 00:03:01,530 be 0 because a_k is tending to 1. 47 00:03:01,530 --> 00:03:03,830 But a_k isn't tending to 1. 48 00:03:03,830 --> 00:03:06,640 Who said it was? 49 00:03:06,640 --> 00:03:10,040 If you take the logarithm, which is what we did last time, 50 00:03:10,040 --> 00:03:12,880 logarithm of a_k is indeed k * ln(1 + 1/k). 51 00:03:17,030 --> 00:03:18,520 That does not tend to 0. 52 00:03:18,520 --> 00:03:22,610 This part of it tends to 0, and this part tends to infinity. 53 00:03:22,610 --> 00:03:26,460 And they balance each other, 0 times infinity. 54 00:03:26,460 --> 00:03:28,460 We don't really know yet from this expression, 55 00:03:28,460 --> 00:03:32,189 in fact we did some cleverness with limits and derivatives, 56 00:03:32,189 --> 00:03:33,230 to figure out this limit. 57 00:03:33,230 --> 00:03:34,354 It was a very subtle thing. 58 00:03:34,354 --> 00:03:37,560 It turned out to be 1. 59 00:03:37,560 --> 00:03:38,850 All right? 60 00:03:38,850 --> 00:03:40,645 Now, the thing that I'd like to say 61 00:03:40,645 --> 00:03:43,550 - I'm sorry I'm going to erase this aside here 62 00:03:43,550 --> 00:03:46,100 - but you need to go back to your notes 63 00:03:46,100 --> 00:03:48,824 and remember that this is what we did last time. 64 00:03:48,824 --> 00:03:50,490 Because I want to have room for the next 65 00:03:50,490 --> 00:03:54,420 comment that I want to make on this little blackboard here. 66 00:03:54,420 --> 00:03:57,090 What we just derived was this property here, 67 00:03:57,090 --> 00:04:01,810 but I made a claim yesterday, and I just 68 00:04:01,810 --> 00:04:04,120 want to emphasize it again so that we realized 69 00:04:04,120 --> 00:04:07,786 what it is that we're doing. 70 00:04:07,786 --> 00:04:08,910 I looked at this backwards. 71 00:04:08,910 --> 00:04:11,430 One way you can think of this is we're evaluating this limit 72 00:04:11,430 --> 00:04:13,000 and getting an answer. 73 00:04:13,000 --> 00:04:16,524 But all equalities can be read both directions. 74 00:04:16,524 --> 00:04:17,940 And we can write it the other way: 75 00:04:17,940 --> 00:04:25,881 e equals the limit, as k goes to infinity, of this expression 76 00:04:25,881 --> 00:04:26,380 here. 77 00:04:26,380 --> 00:04:28,640 So that's just the same thing. 78 00:04:28,640 --> 00:04:30,840 And if we read it backwards, what we're saying 79 00:04:30,840 --> 00:04:35,700 is that this limit is a formula for e. 80 00:04:35,700 --> 00:04:38,325 So this is very typical of mathematics. 81 00:04:38,325 --> 00:04:40,700 You want to always reverse your perspective all the time. 82 00:04:40,700 --> 00:04:43,710 Equations work both ways, and in this case, 83 00:04:43,710 --> 00:04:46,300 we have two different things here. 84 00:04:46,300 --> 00:04:49,430 This e was what we defined as the base, 85 00:04:49,430 --> 00:04:54,461 which when you graph e^x, you get slope 1 at 0. 86 00:04:54,461 --> 00:04:56,710 And then it turns out to be equal to this limit, which 87 00:04:56,710 --> 00:04:59,060 we can calculate numerically. 88 00:04:59,060 --> 00:05:02,100 If you do this on your calculators, you, of course, 89 00:05:02,100 --> 00:05:05,420 will have a way of programming in this number 90 00:05:05,420 --> 00:05:07,290 and evaluating it for each k. 91 00:05:07,290 --> 00:05:10,880 And you'll have another button available to evaluate this one. 92 00:05:10,880 --> 00:05:12,432 So another way of saying it is it 93 00:05:12,432 --> 00:05:14,640 that there's a relationship between these two things. 94 00:05:14,640 --> 00:05:19,210 And all of calculus is a matter of getting these relationships. 95 00:05:19,210 --> 00:05:21,860 So we can look at these things in several different ways. 96 00:05:21,860 --> 00:05:23,350 And indeed, that's what we're going 97 00:05:23,350 --> 00:05:25,832 to be doing at least at the end of today 98 00:05:25,832 --> 00:05:27,040 in talking about derivatives. 99 00:05:27,040 --> 00:05:29,480 A lot of times when we talk about derivatives, 100 00:05:29,480 --> 00:05:32,230 we're trying to look at them from several perspectives 101 00:05:32,230 --> 00:05:34,780 at once. 102 00:05:34,780 --> 00:05:37,280 Okay, so I have to keep on going with exponents, 103 00:05:37,280 --> 00:05:40,130 because I have one loose end. 104 00:05:40,130 --> 00:05:44,490 One loose end that I did not cover yet. 105 00:05:44,490 --> 00:05:48,330 There's one very important formula that's left, 106 00:05:48,330 --> 00:05:51,920 and it's the derivative of the powers. 107 00:05:51,920 --> 00:05:54,070 We actually didn't do this - well we 108 00:05:54,070 --> 00:05:57,120 did it for rational numbers r. 109 00:05:57,120 --> 00:06:00,070 So this is the formula here. 110 00:06:00,070 --> 00:06:06,200 But now we're going to check this for all real numbers, r. 111 00:06:06,200 --> 00:06:09,200 So including all the irrational ones as well. 112 00:06:09,200 --> 00:06:13,690 This is also good practice for using base e 113 00:06:13,690 --> 00:06:16,480 and using logarithmic differentiation. 114 00:06:16,480 --> 00:06:20,810 So let me do this by our two methods 115 00:06:20,810 --> 00:06:26,160 that we can use to handle exponential type problems. 116 00:06:26,160 --> 00:06:32,070 So method one was base e. 117 00:06:32,070 --> 00:06:34,040 So if I just rewrite this base e again, 118 00:06:34,040 --> 00:06:36,910 that's just this formula over here. 119 00:06:36,910 --> 00:06:50,350 x^r = (e^ln x)^r, which is e^r ln x. 120 00:06:50,350 --> 00:06:55,320 Okay, so now I can differentiate this. 121 00:06:55,320 --> 00:07:04,650 So I get that d/dx (x^r), now I'm going to use prime 122 00:07:04,650 --> 00:07:07,460 notation, because I don't want to keep on writing that d/dx 123 00:07:07,460 --> 00:07:10,580 here; (e^(r ln x))'. 124 00:07:13,890 --> 00:07:18,457 And now, what I can do is I can use the chain rule. 125 00:07:18,457 --> 00:07:20,290 The chain rule says that it's the derivative 126 00:07:20,290 --> 00:07:24,380 of this times the derivative of the function. 127 00:07:24,380 --> 00:07:29,500 So the derivative of the exponential is just itself. 128 00:07:29,500 --> 00:07:31,470 And the derivative of this guy here, 129 00:07:31,470 --> 00:07:35,400 well I'll write it out once, is (r ln x)'. 130 00:07:39,700 --> 00:07:42,250 So what's that equal to? 131 00:07:42,250 --> 00:07:45,890 Well, e^(r ln x) is is just x^r. 132 00:07:45,890 --> 00:07:53,870 And this derivative here is-- Well the derivative of r is 0. 133 00:07:53,870 --> 00:07:54,860 This is a constant. 134 00:07:54,860 --> 00:07:56,680 It just factors out. 135 00:07:56,680 --> 00:08:02,070 And ln x now has derivative-- What's the derivative of ln x? 136 00:08:02,070 --> 00:08:06,570 1/x, so this is going to be times r/x. 137 00:08:06,570 --> 00:08:10,180 And now, we rewrite it in the customary form, which is r, 138 00:08:10,180 --> 00:08:13,040 we put the r in front, x^(r-1). 139 00:08:13,040 --> 00:08:13,820 Okay? 140 00:08:13,820 --> 00:08:19,060 So I just derived the formula for you. 141 00:08:19,060 --> 00:08:23,310 And it didn't matter whether r was rational or irrational, 142 00:08:23,310 --> 00:08:25,320 it's the same proof. 143 00:08:25,320 --> 00:08:29,440 Okay so now I have to show you how method two works as well. 144 00:08:29,440 --> 00:08:34,550 So let's do method two, which we call 145 00:08:34,550 --> 00:08:39,280 logarithmic differentiation. 146 00:08:39,280 --> 00:08:43,764 And so here I'll use a symbol, say u, for x^r, 147 00:08:43,764 --> 00:08:44,930 and I'll take its logarithm. 148 00:08:44,930 --> 00:08:50,480 That's r ln x. 149 00:08:50,480 --> 00:08:51,830 And now I differentiate it. 150 00:08:51,830 --> 00:08:54,160 I'll leave that in the middle, because I 151 00:08:54,160 --> 00:08:55,979 want to remember the key property 152 00:08:55,979 --> 00:08:57,270 of logarithmic differentiation. 153 00:08:57,270 --> 00:08:58,747 But first I'll differentiate it. 154 00:08:58,747 --> 00:09:01,080 Later on, what I'm going to use is that this is the same 155 00:09:01,080 --> 00:09:02,620 as u'/u. 156 00:09:02,620 --> 00:09:06,470 This is one way of evaluating a logarithmic derivative. 157 00:09:06,470 --> 00:09:08,660 And then the other is to differentiate 158 00:09:08,660 --> 00:09:10,980 the explicit function that we have over here. 159 00:09:10,980 --> 00:09:16,790 And that is just, as we said, r/x. 160 00:09:16,790 --> 00:09:25,470 So now, I multiply through, and I get u' = ur/x which is just 161 00:09:25,470 --> 00:09:29,750 x^r r/x, which is just what we did before. 162 00:09:29,750 --> 00:09:30,520 It's r x^(r-1). 163 00:09:33,410 --> 00:09:36,790 Again, you can now see by comparing 164 00:09:36,790 --> 00:09:40,110 these two pieces of arithmetic that they're basically 165 00:09:40,110 --> 00:09:41,470 the same. 166 00:09:41,470 --> 00:09:43,510 Pretty much every time you convert to base e 167 00:09:43,510 --> 00:09:45,170 or you do logarithmic differentiation, 168 00:09:45,170 --> 00:09:46,836 it'll amount to the same thing, provided 169 00:09:46,836 --> 00:09:48,270 you don't get mixed up. 170 00:09:48,270 --> 00:09:51,720 You generally have to introduce a new symbol here. 171 00:09:51,720 --> 00:09:55,840 On the other hand, you're dealing with exponents there. 172 00:09:55,840 --> 00:10:00,990 It's worth it to know both points of view. 173 00:10:00,990 --> 00:10:07,470 All right, so now I want to make one last remark before we 174 00:10:07,470 --> 00:10:09,910 finish with exponents. 175 00:10:09,910 --> 00:10:16,120 And, I'll try to sell this to you in a lot of ways 176 00:10:16,120 --> 00:10:19,490 as the course goes on, but one thing that I 177 00:10:19,490 --> 00:10:23,370 want to try to emphasize is that the natural logarithm really 178 00:10:23,370 --> 00:10:27,210 is natural. 179 00:10:27,210 --> 00:10:39,920 So, I claim that the natural log is natural. 180 00:10:39,920 --> 00:10:45,900 And the example that we're going to use for this illustration 181 00:10:45,900 --> 00:10:53,340 is economics. 182 00:10:53,340 --> 00:10:54,090 Okay? 183 00:10:54,090 --> 00:10:58,520 So let me explain to why the natural log is the one that's 184 00:10:58,520 --> 00:11:00,820 natural for economics. 185 00:11:00,820 --> 00:11:06,040 If you are imagining the price of a stock that you own 186 00:11:06,040 --> 00:11:11,160 goes down by a dollar, that's a totally meaningless statement. 187 00:11:11,160 --> 00:11:13,439 It depends on a lot of things. 188 00:11:13,439 --> 00:11:15,730 In particular, it depends on whether the original price 189 00:11:15,730 --> 00:11:18,300 was a dollar or 100 dollars. 190 00:11:18,300 --> 00:11:22,130 So there's not much meaning to these absolute numbers. 191 00:11:22,130 --> 00:11:25,080 It's always the ratios that matter. 192 00:11:25,080 --> 00:11:29,280 So, for example, I just looked up an hour ago, 193 00:11:29,280 --> 00:11:42,050 the London Exchange closed, and it was down 27.9, 194 00:11:42,050 --> 00:11:44,480 which as I said, is pretty meaningless 195 00:11:44,480 --> 00:11:50,050 unless you know what the actual total of this index is. 196 00:11:50,050 --> 00:11:54,200 It turns out it was 6,432. 197 00:11:54,200 --> 00:11:57,070 So the change in the price, divided 198 00:11:57,070 --> 00:12:03,980 by the price, which in this case is 27.9 / 6,432, 199 00:12:03,980 --> 00:12:07,550 is what matters. 200 00:12:07,550 --> 00:12:11,791 And, in this case, it happens to be .43%. 201 00:12:11,791 --> 00:12:12,290 All right? 202 00:12:12,290 --> 00:12:14,270 That's what happened today. 203 00:12:14,270 --> 00:12:18,410 And similarly, if you take the infinitesimal of this, 204 00:12:18,410 --> 00:12:21,260 people think of days as being relatively small increments 205 00:12:21,260 --> 00:12:23,900 when you're investing in a stock, 206 00:12:23,900 --> 00:12:27,240 you would be interested in the infinitesimal sense, 207 00:12:27,240 --> 00:12:28,690 you would be interested in p'/p. 208 00:12:28,690 --> 00:12:33,080 The derivative of p divided by p. 209 00:12:33,080 --> 00:12:35,530 That's just (ln p)'. 210 00:12:38,160 --> 00:12:42,275 So this is the - let me put a little box around it 211 00:12:42,275 --> 00:12:45,460 - the formula of logarithmic differentiation. 212 00:12:45,460 --> 00:12:49,700 But let me just emphasize that it has an actual significance, 213 00:12:49,700 --> 00:12:52,430 and it's the one that's used by economists and people who 214 00:12:52,430 --> 00:12:54,450 are modeling prices of things all the time. 215 00:12:54,450 --> 00:12:58,620 They never use absolute prices when there are large swings. 216 00:12:58,620 --> 00:13:01,010 They always use the log of the price. 217 00:13:01,010 --> 00:13:07,010 And there's no point in using log base 10, or log base 2. 218 00:13:07,010 --> 00:13:08,180 Those give you junk. 219 00:13:08,180 --> 00:13:11,190 They give you an extra factor of log 2. 220 00:13:11,190 --> 00:13:14,870 It's the natural log that's the obvious one to use. 221 00:13:14,870 --> 00:13:18,000 It's completely straightforward that this 222 00:13:18,000 --> 00:13:21,010 is a simpler expression than using log base 10 223 00:13:21,010 --> 00:13:24,030 and having a factor of natural log of 10 there, 224 00:13:24,030 --> 00:13:26,800 which would just mess everything up. 225 00:13:26,800 --> 00:13:29,360 All right, so this is just one illustration. 226 00:13:29,360 --> 00:13:31,680 Anything that has to do with ratios 227 00:13:31,680 --> 00:13:36,160 is going to encounter logarithms. 228 00:13:36,160 --> 00:13:41,270 All right, so that's pretty much it. 229 00:13:41,270 --> 00:13:45,822 That's all I want to say for now anyway. 230 00:13:45,822 --> 00:13:47,280 There's lots more to say, but we'll 231 00:13:47,280 --> 00:13:50,459 be saying it when we do applications of derivatives 232 00:13:50,459 --> 00:13:51,250 in the second unit. 233 00:13:51,250 --> 00:13:54,450 So now, what I'd like to do is to start a review. 234 00:13:54,450 --> 00:13:57,790 I'm just going to run through what we did in this unit. 235 00:13:57,790 --> 00:13:59,700 I'll tell you approximately what I 236 00:13:59,700 --> 00:14:06,150 expect from you on the test that's coming up tomorrow. 237 00:14:06,150 --> 00:14:14,640 And, well, so let's get started with that. 238 00:14:14,640 --> 00:14:27,050 So this is a review of Unit One. 239 00:14:27,050 --> 00:14:32,610 And I'm just going to put on the board all of the things 240 00:14:32,610 --> 00:14:35,750 that you need to think about, anyway, keep in your head. 241 00:14:35,750 --> 00:14:41,750 And there are what are called general formulas 242 00:14:41,750 --> 00:14:45,070 for derivatives. 243 00:14:45,070 --> 00:14:51,970 And then there are the specific ones. 244 00:14:51,970 --> 00:14:55,920 And let me just remind you what the general formulas are. 245 00:14:55,920 --> 00:14:58,750 There's what you do to differentiate 246 00:14:58,750 --> 00:15:04,190 a sum, a multiple of a function, the product 247 00:15:04,190 --> 00:15:08,190 rule, the quotient rule. 248 00:15:08,190 --> 00:15:11,550 Those are several general formulas. 249 00:15:11,550 --> 00:15:13,080 And then there's one more, which is 250 00:15:13,080 --> 00:15:15,780 the chain rule, which I'm going to say just 251 00:15:15,780 --> 00:15:17,450 a little bit more about. 252 00:15:17,450 --> 00:15:21,200 So the derivative of a function of a function 253 00:15:21,200 --> 00:15:26,380 is the derivative of the function times the derivative 254 00:15:26,380 --> 00:15:27,430 of the other function. 255 00:15:27,430 --> 00:15:33,780 So here, I've abbreviated u is u(x). 256 00:15:33,780 --> 00:15:36,630 Right, so this is one of two ways of writing it. 257 00:15:36,630 --> 00:15:39,850 The other way is also one that you can keep in mind 258 00:15:39,850 --> 00:15:42,470 and you might find easier to remember. 259 00:15:42,470 --> 00:15:46,690 It's probably a good idea to remember both formulas. 260 00:15:46,690 --> 00:15:49,660 And then the last type of general formula 261 00:15:49,660 --> 00:15:56,950 that we did was implicit differentiation. 262 00:15:56,950 --> 00:15:59,200 Okay? 263 00:15:59,200 --> 00:16:03,190 So when you do implicit differentiation, 264 00:16:03,190 --> 00:16:06,530 you have an equation and you don't 265 00:16:06,530 --> 00:16:09,270 try to solve for the unknown function. 266 00:16:09,270 --> 00:16:13,110 You just put it in its simplest form and you differentiate. 267 00:16:13,110 --> 00:16:20,440 So, we actually did this, in particular, for inverses. 268 00:16:20,440 --> 00:16:23,520 That was a very, very key method for calculating 269 00:16:23,520 --> 00:16:25,180 the inverses of functions. 270 00:16:25,180 --> 00:16:28,600 And it's also true that logarithmic differentiation 271 00:16:28,600 --> 00:16:31,420 is of this type. 272 00:16:31,420 --> 00:16:33,299 This is a transformation. 273 00:16:33,299 --> 00:16:34,840 We're differentiating something else. 274 00:16:34,840 --> 00:16:37,920 We're transforming the equation by taking its logarithm 275 00:16:37,920 --> 00:16:40,980 and then differentiating. 276 00:16:40,980 --> 00:16:45,200 Okay, so there are a number of different ways this is applied. 277 00:16:45,200 --> 00:16:48,450 It can also be applied, anyway, these are two of them. 278 00:16:48,450 --> 00:16:50,320 So maybe in parenthesis. 279 00:16:50,320 --> 00:16:53,120 These are just examples. 280 00:16:53,120 --> 00:16:54,350 All right. 281 00:16:54,350 --> 00:16:59,461 I'll try to give examples of at least a few of these rules 282 00:16:59,461 --> 00:16:59,960 later. 283 00:16:59,960 --> 00:17:05,670 So now, the specific functions that you know how 284 00:17:05,670 --> 00:17:08,360 to differentiate: well you know how to differentiate now x^r 285 00:17:08,360 --> 00:17:11,410 thanks to what I just did. 286 00:17:11,410 --> 00:17:15,030 We have the sine and the cosine functions, 287 00:17:15,030 --> 00:17:17,910 which you're responsible for knowing 288 00:17:17,910 --> 00:17:19,500 what their derivatives are. 289 00:17:19,500 --> 00:17:26,490 And then other trig functions like tan and secant. 290 00:17:26,490 --> 00:17:29,810 We generally don't bother with cosecants and cotangents, 291 00:17:29,810 --> 00:17:32,710 because everything can be expressed in terms of these 292 00:17:32,710 --> 00:17:33,759 anyway. 293 00:17:33,759 --> 00:17:35,550 Actually, you can really express everything 294 00:17:35,550 --> 00:17:36,906 in terms of sines and cosines. 295 00:17:36,906 --> 00:17:38,280 But what you'll find is that it's 296 00:17:38,280 --> 00:17:41,660 much more convenient to remember the derivatives of these 297 00:17:41,660 --> 00:17:42,730 as well. 298 00:17:42,730 --> 00:17:45,870 So memorize all of these. 299 00:17:45,870 --> 00:17:49,810 All right, and then we had e^x and ln x. 300 00:17:49,810 --> 00:17:53,920 And we had the inverses of the trig functions. 301 00:17:53,920 --> 00:18:00,010 These were the two that we did: the arctangent and the arcsine. 302 00:18:00,010 --> 00:18:02,220 So those are the ones you're responsible for. 303 00:18:02,220 --> 00:18:06,970 You should have enough time, anyway, to work out anything 304 00:18:06,970 --> 00:18:09,390 else, if you know these. 305 00:18:09,390 --> 00:18:11,210 All right, so basically the idea is 306 00:18:11,210 --> 00:18:13,070 you have a bunch of special formulas. 307 00:18:13,070 --> 00:18:14,820 You have a bunch of general formulas. 308 00:18:14,820 --> 00:18:16,620 You put them together, and you can 309 00:18:16,620 --> 00:18:20,970 generate pretty much anything. 310 00:18:20,970 --> 00:18:24,810 Okay, so let's do a few examples before going on 311 00:18:24,810 --> 00:18:41,290 with the review. 312 00:18:41,290 --> 00:18:48,230 Okay, so I do want to do a few examples in sort of increasing 313 00:18:48,230 --> 00:18:50,170 level of difficulty in how you would 314 00:18:50,170 --> 00:18:51,420 combine these things together. 315 00:18:51,420 --> 00:18:55,980 So first of all, you should remember 316 00:18:55,980 --> 00:19:00,885 that if you differentiate the secant function, that's just 317 00:19:00,885 --> 00:19:03,780 - oh I just realized that I wanted to say something 318 00:19:03,780 --> 00:19:06,630 else before - so forget that. 319 00:19:06,630 --> 00:19:08,060 We'll do that in a second. 320 00:19:08,060 --> 00:19:10,980 I wanted to make some general remarks. 321 00:19:10,980 --> 00:19:17,600 So there's one rule that you discussed in my absence, which 322 00:19:17,600 --> 00:19:19,070 is the chain rule. 323 00:19:19,070 --> 00:19:21,780 And I do want to make just a couple of remarks 324 00:19:21,780 --> 00:19:26,160 about the chain rule now to remind you of what it is, 325 00:19:26,160 --> 00:19:30,160 and also to present some consequences. 326 00:19:30,160 --> 00:19:39,190 So, a little bit of extra on the chain rule. 327 00:19:39,190 --> 00:19:43,720 The first thing that I want say is that we didn't really 328 00:19:43,720 --> 00:19:46,660 fully explain why it's true. 329 00:19:46,660 --> 00:19:54,140 And I do want to just explain it by example, okay? 330 00:19:54,140 --> 00:19:59,720 So imagine that you have a function which is, say, 331 00:19:59,720 --> 00:20:02,000 10x + b. 332 00:20:02,000 --> 00:20:02,500 All right? 333 00:20:02,500 --> 00:20:04,980 So y = 10x + b. 334 00:20:04,980 --> 00:20:09,970 Then obviously, y is changing 10 times as fast as b, right? 335 00:20:09,970 --> 00:20:18,060 The issue is this number here, dy/dx, is 10. 336 00:20:18,060 --> 00:20:20,460 And now if x is a function of something, 337 00:20:20,460 --> 00:20:34,290 say t, shifted by some other constant here, then dx/dt = 5. 338 00:20:34,290 --> 00:20:38,800 Now all the chain rule is saying is that if y is going 10 times 339 00:20:38,800 --> 00:20:44,610 as fast as t, I'm sorry as x, and x is going 5 times 340 00:20:44,610 --> 00:20:50,620 as fast as t, then y is going 50 times as fast as t. 341 00:20:50,620 --> 00:20:53,530 And algebraically, all this means is if I plug 342 00:20:53,530 --> 00:20:57,100 in and substitute, which is what the composition of the two 343 00:20:57,100 --> 00:21:04,760 functions amounts to, 10(5t + a) + b and I multiply it out, 344 00:21:04,760 --> 00:21:09,200 I get 50t + 10a + b. 345 00:21:09,200 --> 00:21:11,697 Now these terms don't matter. 346 00:21:11,697 --> 00:21:13,030 The constant terms don't matter. 347 00:21:13,030 --> 00:21:14,800 The rate is 50. 348 00:21:14,800 --> 00:21:17,130 And so the consequence, if we put them together, 349 00:21:17,130 --> 00:21:30,170 is that dy/dt = 10*5, which is 50. 350 00:21:30,170 --> 00:21:31,990 All right, so this is in a nutshell 351 00:21:31,990 --> 00:21:33,630 why the chain rule works. 352 00:21:33,630 --> 00:21:39,450 And why these rates multiply. 353 00:21:39,450 --> 00:21:42,460 The second thing that I wanted to say about the chain rule 354 00:21:42,460 --> 00:21:45,050 is that it has a few consequences that 355 00:21:45,050 --> 00:21:47,380 make some of the other rules a little easier 356 00:21:47,380 --> 00:21:50,220 to remember or possibly to avoid. 357 00:21:50,220 --> 00:21:54,510 The messiest rule in my humble opinion 358 00:21:54,510 --> 00:21:59,430 is the quotient rule, which is kind of a nuisance to remember. 359 00:21:59,430 --> 00:22:01,150 So let me just remind you, if you 360 00:22:01,150 --> 00:22:03,840 take just the reciprocal of a function, 361 00:22:03,840 --> 00:22:05,990 and you differentiate it, there's 362 00:22:05,990 --> 00:22:08,170 another way of looking at this. 363 00:22:08,170 --> 00:22:09,720 And it's actually the way that I use, 364 00:22:09,720 --> 00:22:12,800 so I want to encourage you to think about it this way too. 365 00:22:12,800 --> 00:22:15,670 This is the same as (v^(-1))'. 366 00:22:15,670 --> 00:22:16,700 . 367 00:22:16,700 --> 00:22:18,840 And now instead of using the quotient rule, which 368 00:22:18,840 --> 00:22:23,840 we could've used, we can use the chain rule here 369 00:22:23,840 --> 00:22:29,720 with the power -1, which works by the power law. 370 00:22:29,720 --> 00:22:30,960 So what is this equal to? 371 00:22:30,960 --> 00:22:33,640 This is equal to -v^(-2) v'. 372 00:22:38,930 --> 00:22:42,730 So here, I've applied the chain rule rather than 373 00:22:42,730 --> 00:22:47,370 the quotient rule. 374 00:22:47,370 --> 00:22:54,060 And similarly, suppose I wanted to derive the full quotient 375 00:22:54,060 --> 00:22:54,560 rule. 376 00:22:54,560 --> 00:22:57,240 Well, now this may or may not be easier. 377 00:22:57,240 --> 00:22:59,770 But this is one way of remembering what's going on. 378 00:22:59,770 --> 00:23:05,300 If you convert it to uv^(-1) and you differentiate that, 379 00:23:05,300 --> 00:23:09,180 now I can use the product rule on this. 380 00:23:09,180 --> 00:23:11,970 Of course, I have to use the chain rule and this rule 381 00:23:11,970 --> 00:23:13,030 as well. 382 00:23:13,030 --> 00:23:15,620 So what do I get? 383 00:23:15,620 --> 00:23:21,519 I get u', the inverse, plus u, and then I have 384 00:23:21,519 --> 00:23:22,810 to differentiate the v inverse. 385 00:23:22,810 --> 00:23:24,490 That's the formula right up here. 386 00:23:24,490 --> 00:23:25,290 That's -v^(-2) v'. 387 00:23:30,300 --> 00:23:33,230 So that's one way of doing it. 388 00:23:33,230 --> 00:23:35,760 This actually explains the funny minus sign 389 00:23:35,760 --> 00:23:38,560 when you differentiate v in the formula. 390 00:23:38,560 --> 00:23:41,280 The other formula, the other way that we did it, 391 00:23:41,280 --> 00:23:44,370 was by putting this over a common denominator. 392 00:23:44,370 --> 00:23:49,330 The common denominator was v^2. 393 00:23:49,330 --> 00:23:51,580 This comes from this v v^(-2). 394 00:23:51,580 --> 00:23:54,730 And then the second term is -uv'. 395 00:23:57,250 --> 00:24:00,020 And the first term, we have to multiply by an extra factor 396 00:24:00,020 --> 00:24:02,190 of v, because we have a v^2 in the denominator. 397 00:24:02,190 --> 00:24:07,720 So it's u'v. All right, so this is the quotient rule as we 398 00:24:07,720 --> 00:24:11,247 wrote it down in lecture, and this is just another way 399 00:24:11,247 --> 00:24:13,580 of remembering it or deriving it without remembering it, 400 00:24:13,580 --> 00:24:16,700 if you just remember the chain rule and the product rule. 401 00:24:16,700 --> 00:24:19,710 Okay, so you'll find that in many contexts, 402 00:24:19,710 --> 00:24:25,910 it's easier to do one or the other. 403 00:24:25,910 --> 00:24:29,210 Okay, so now I'm ready to differentiate the secant 404 00:24:29,210 --> 00:24:30,990 and a few such functions. 405 00:24:30,990 --> 00:24:36,200 So we'll do some examples here here. 406 00:24:36,200 --> 00:24:39,030 So here's the secant function, and I 407 00:24:39,030 --> 00:24:44,820 want to use that formula up there for the reciprocal. 408 00:24:44,820 --> 00:24:48,090 This is the way I think of it. 409 00:24:48,090 --> 00:24:53,150 This is the cosine function to the power -1. 410 00:24:53,150 --> 00:24:58,750 And so, the formula here is just what? 411 00:24:58,750 --> 00:25:04,030 It's just -(cos x)^(-2) times -sin x. 412 00:25:20,280 --> 00:25:22,550 So now this is usually written in a different fashion, 413 00:25:22,550 --> 00:25:25,170 so that's why I'm doing this for a reason actually. 414 00:25:25,170 --> 00:25:27,810 Which is although there are several formulas for things, 415 00:25:27,810 --> 00:25:29,810 with trig functions, there are usually 416 00:25:29,810 --> 00:25:31,854 five ways of writing something. 417 00:25:31,854 --> 00:25:33,520 So I'm writing this one down so that you 418 00:25:33,520 --> 00:25:36,780 know what the standard way of presenting it is. 419 00:25:36,780 --> 00:25:39,760 So what happens here is that we have two minus signs 420 00:25:39,760 --> 00:25:40,300 cancelling. 421 00:25:40,300 --> 00:25:44,360 And we get sin x / cos^2 x. 422 00:25:44,360 --> 00:25:46,430 That's a perfectly acceptable answer, 423 00:25:46,430 --> 00:25:49,470 but there's a customary way in which is written. 424 00:25:49,470 --> 00:25:55,890 It's written (1 / cos x) (sin x / cos x). 425 00:25:55,890 --> 00:25:57,530 And then we get rid of the denominators 426 00:25:57,530 --> 00:26:00,710 by rewriting it in terms of secant and tangent, 427 00:26:00,710 --> 00:26:04,100 so sec x tan x. 428 00:26:04,100 --> 00:26:07,680 So this is the form that's generally 429 00:26:07,680 --> 00:26:11,790 used when you see these formulas written in textbooks. 430 00:26:11,790 --> 00:26:14,440 And so you know, you need to watch out, 431 00:26:14,440 --> 00:26:16,810 because if you ever want to use this kind of calculus, 432 00:26:16,810 --> 00:26:22,840 you'll have not be put off by all the secants and tangents. 433 00:26:22,840 --> 00:26:26,830 All right, so getting slightly more complicated, 434 00:26:26,830 --> 00:26:28,750 how about if we differentiate ln(sec x)? 435 00:26:37,400 --> 00:26:39,560 If you differentiate the natural log, 436 00:26:39,560 --> 00:26:49,450 that's just going to be (sec x)' / sec x. 437 00:26:49,450 --> 00:26:51,250 And plugging in the formula that we 438 00:26:51,250 --> 00:27:00,330 had before, that's sec x tan x / sec x, which is tan x. 439 00:27:00,330 --> 00:27:03,850 So this one also has a very nice form. 440 00:27:03,850 --> 00:27:07,940 And you might say that this is kind of an ugly function, 441 00:27:07,940 --> 00:27:14,120 but the strange thing is that the natural log was invented 442 00:27:14,120 --> 00:27:19,030 before the exponential by a guy named Napier, exactly 443 00:27:19,030 --> 00:27:21,720 in order to evaluate functions like this. 444 00:27:21,720 --> 00:27:25,930 These are the functions that people cared about a lot, 445 00:27:25,930 --> 00:27:28,890 because they were used in navigation. 446 00:27:28,890 --> 00:27:32,640 You wanted to multiply sines and cosines together 447 00:27:32,640 --> 00:27:34,030 to do navigation. 448 00:27:34,030 --> 00:27:38,794 And the multiplication he encoded using a logarithm. 449 00:27:38,794 --> 00:27:40,710 So these were invented long before people even 450 00:27:40,710 --> 00:27:42,794 knew about exponents. 451 00:27:42,794 --> 00:27:44,460 And it was a surprise, actually, that it 452 00:27:44,460 --> 00:27:46,100 was connected to exponents. 453 00:27:46,100 --> 00:27:48,650 So the natural log was invented before the log base 10 454 00:27:48,650 --> 00:27:52,650 and everything else, exactly for this kind of purpose. 455 00:27:52,650 --> 00:27:54,550 Anyway, so this is a nice function, 456 00:27:54,550 --> 00:27:58,010 which was very important, so that your ships wouldn't 457 00:27:58,010 --> 00:28:03,770 crash into the reef. 458 00:28:03,770 --> 00:28:05,570 Okay, let's continue here. 459 00:28:05,570 --> 00:28:08,870 So there's another kind of function 460 00:28:08,870 --> 00:28:10,490 that I want to discuss with you. 461 00:28:10,490 --> 00:28:12,460 And these are the kinds in which there's 462 00:28:12,460 --> 00:28:19,380 a choice as to which of these rules to apply. 463 00:28:19,380 --> 00:28:25,130 And I'll just give a couple of examples of that. 464 00:28:25,130 --> 00:28:27,620 There usually is a better and a worse way, 465 00:28:27,620 --> 00:28:38,430 so let me illustrate that. 466 00:28:38,430 --> 00:28:41,120 Okay, yet another example. 467 00:28:41,120 --> 00:28:43,830 I hope you've seen some of these before. 468 00:28:43,830 --> 00:28:46,660 Say (x^10 + 8x)^6. 469 00:28:51,010 --> 00:28:52,990 So it's a little bit more complicated than what 470 00:28:52,990 --> 00:29:00,330 we had before, because there were several more symbols here. 471 00:29:00,330 --> 00:29:03,210 So what should we do at this point? 472 00:29:03,210 --> 00:29:06,210 There's one choice which I claim is a bad idea, 473 00:29:06,210 --> 00:29:10,810 and that is to expand this out to the 6th power. 474 00:29:10,810 --> 00:29:13,530 That's a bad idea, because it's very long. 475 00:29:13,530 --> 00:29:15,990 And then your answer will also be very long. 476 00:29:15,990 --> 00:29:19,521 It will fill the entire exam paper, for instance. 477 00:29:19,521 --> 00:29:20,020 Yeah? 478 00:29:20,020 --> 00:29:21,380 STUDENT: Can you use the chain rule? 479 00:29:21,380 --> 00:29:21,970 PROFESSOR: Chain rule. 480 00:29:21,970 --> 00:29:22,580 That's it. 481 00:29:22,580 --> 00:29:23,500 We use the chain rule. 482 00:29:23,500 --> 00:29:26,620 So fortunately, this is relatively easy 483 00:29:26,620 --> 00:29:27,620 using the chain rule. 484 00:29:27,620 --> 00:29:30,790 We just think of this box as being the function. 485 00:29:30,790 --> 00:29:34,560 And we take 6 times this guy to the 5th, 486 00:29:34,560 --> 00:29:37,570 times the derivative of this guy, which is 10x^9 + 8. 487 00:29:41,500 --> 00:29:43,910 And this is, filling this in, it's x^10 + 8x. 488 00:29:43,910 --> 00:29:46,140 And that's it. 489 00:29:46,140 --> 00:29:50,270 That's all you need to do differentiate things like this. 490 00:29:50,270 --> 00:29:55,140 The chain rule is very effective. 491 00:29:55,140 --> 00:29:59,864 STUDENT: [INAUDIBLE] 492 00:29:59,864 --> 00:30:01,280 PROFESSOR: That's a good question. 493 00:30:01,280 --> 00:30:04,330 So I'm not really willing to answer too many questions 494 00:30:04,330 --> 00:30:07,500 about what's going to be on the exam. 495 00:30:07,500 --> 00:30:09,070 But the question that was just asked 496 00:30:09,070 --> 00:30:13,200 is exactly the kind of question I'm very happy to answer. 497 00:30:13,200 --> 00:30:18,350 Okay, the question was, in what form is-- what 498 00:30:18,350 --> 00:30:20,090 form is an acceptable answer? 499 00:30:20,090 --> 00:30:23,560 Now in real life, that is a really serious question. 500 00:30:23,560 --> 00:30:25,200 When you ask a computer a question 501 00:30:25,200 --> 00:30:28,930 and it gives you 500 million sheets of printout, 502 00:30:28,930 --> 00:30:31,380 it's useless. 503 00:30:31,380 --> 00:30:33,960 And you really care what form answers are in, 504 00:30:33,960 --> 00:30:35,730 and indeed, somebody might really 505 00:30:35,730 --> 00:30:39,000 care what this thing to the 6th power is, 506 00:30:39,000 --> 00:30:42,090 and then you would be forced to discuss things in terms 507 00:30:42,090 --> 00:30:46,110 of that other functional form. 508 00:30:46,110 --> 00:30:50,410 For the purposes of this exam, this is okay form. 509 00:30:50,410 --> 00:30:54,490 And, in fact, any correct form is an okay form. 510 00:30:54,490 --> 00:30:57,770 I recommend strongly that you not try to simplify things 511 00:30:57,770 --> 00:30:59,700 unless we tell you to. 512 00:30:59,700 --> 00:31:04,860 Sometimes it will be to your advantage to simplify things. 513 00:31:04,860 --> 00:31:08,010 Sometimes we'll say simplify. 514 00:31:08,010 --> 00:31:10,390 It takes a good deal of experience 515 00:31:10,390 --> 00:31:13,121 to know when it's really worth it to simplify expressions. 516 00:31:13,121 --> 00:31:13,620 Yes? 517 00:31:13,620 --> 00:31:19,530 STUDENT: [INAUDIBLE] 518 00:31:19,530 --> 00:31:23,590 PROFESSOR: Right, so turning to this example. 519 00:31:23,590 --> 00:31:25,520 The question is what is this derivative? 520 00:31:25,520 --> 00:31:27,240 And here's an answer. 521 00:31:27,240 --> 00:31:29,500 That's the end of the problem. 522 00:31:29,500 --> 00:31:31,810 This is a more customary form. 523 00:31:31,810 --> 00:31:37,160 But this is answer is okay. 524 00:31:37,160 --> 00:31:38,610 Same issue. 525 00:31:38,610 --> 00:31:40,970 That's exactly the point. 526 00:31:40,970 --> 00:31:41,660 Yes? 527 00:31:41,660 --> 00:31:51,460 STUDENT: [INAUDIBLE] 528 00:31:51,460 --> 00:31:59,032 PROFESSOR: The question is, do you have to show the work? 529 00:31:59,032 --> 00:32:00,240 Do you have to show the work? 530 00:32:00,240 --> 00:32:04,870 Well if I ask you what is d/dx of sec x, 531 00:32:04,870 --> 00:32:06,650 then if you wrote down this answer 532 00:32:06,650 --> 00:32:09,510 or you wrote down this answer showing no work, 533 00:32:09,510 --> 00:32:11,200 that would be acceptable. 534 00:32:11,200 --> 00:32:15,950 If the question was derive the formula for this 535 00:32:15,950 --> 00:32:18,650 from the formula for the derivative of the cosine 536 00:32:18,650 --> 00:32:21,160 or something like that, then it would not be acceptable. 537 00:32:21,160 --> 00:32:24,340 You'd have to carry out this arithmetic. 538 00:32:24,340 --> 00:32:28,470 So, in other words, typically this 539 00:32:28,470 --> 00:32:32,290 will come up, for instance, in various contexts. 540 00:32:32,290 --> 00:32:34,830 You just basically have to follow directions. 541 00:32:34,830 --> 00:32:35,330 Yes? 542 00:32:35,330 --> 00:32:41,424 STUDENT: [INAUDIBLE] 543 00:32:41,424 --> 00:32:43,090 PROFESSOR: The next question is, are you 544 00:32:43,090 --> 00:32:44,465 expected to be able to prove what 545 00:32:44,465 --> 00:32:46,180 the derivative of the sine function is? 546 00:32:46,180 --> 00:32:49,580 The short answer to that is yes. 547 00:32:49,580 --> 00:32:51,630 But I will be getting to that when I discuss 548 00:32:51,630 --> 00:32:54,240 the rest of the material here. 549 00:32:54,240 --> 00:32:58,430 We're almost there. 550 00:32:58,430 --> 00:33:02,640 Okay, so let me just finish these examples 551 00:33:02,640 --> 00:33:04,880 with one last one. 552 00:33:04,880 --> 00:33:06,880 And then we'll talk about this question 553 00:33:06,880 --> 00:33:10,630 of things like the derivative of the sine function, 554 00:33:10,630 --> 00:33:12,060 and deriving it. 555 00:33:12,060 --> 00:33:15,620 So the last example that I'd like to write down is the one 556 00:33:15,620 --> 00:33:18,940 that I promised you in the first lecture, 557 00:33:18,940 --> 00:33:26,172 namely to differentiate e^(x tan^(-1) x). 558 00:33:26,172 --> 00:33:28,380 Basically you're supposed to be able to differentiate 559 00:33:28,380 --> 00:33:29,350 any function. 560 00:33:29,350 --> 00:33:32,390 So this is the one that we mentioned at the beginning. 561 00:33:32,390 --> 00:33:34,130 So here it is. 562 00:33:34,130 --> 00:33:37,280 Let's do it. 563 00:33:37,280 --> 00:33:38,170 So what is it? 564 00:33:38,170 --> 00:33:45,742 Well, it's just equal to - I have to differentiate. 565 00:33:45,742 --> 00:33:47,200 I have to use the chain rule - it's 566 00:33:47,200 --> 00:33:52,930 equal to the exponential times the derivative 567 00:33:52,930 --> 00:33:58,200 of this expression here. 568 00:33:58,200 --> 00:33:59,260 That's the chain rule. 569 00:33:59,260 --> 00:34:01,700 That's the first step. 570 00:34:01,700 --> 00:34:06,440 And now I have to apply the product rule here. 571 00:34:06,440 --> 00:34:10,820 So I have e^(x tan^(-1) x). 572 00:34:10,820 --> 00:34:15,809 And I differentiate the first factor, so I get tan^(-1) x. 573 00:34:15,809 --> 00:34:17,600 Add to it what happens when I differentiate 574 00:34:17,600 --> 00:34:19,670 the second factor, leaving alone the x. 575 00:34:19,670 --> 00:34:21,450 So that's x / (1+x^2). 576 00:34:24,310 --> 00:34:26,300 And that's it. 577 00:34:26,300 --> 00:34:28,780 That's the end of the problem. 578 00:34:28,780 --> 00:34:30,590 It wasn't that hard. 579 00:34:30,590 --> 00:34:35,330 Of course, it requires you to remember all of the rules, 580 00:34:35,330 --> 00:34:37,300 and a lot of formulas underlying them. 581 00:34:37,300 --> 00:34:39,560 So that's consistent with what I just told you. 582 00:34:39,560 --> 00:34:42,060 I told you that you wanted to know this. 583 00:34:42,060 --> 00:34:44,740 I told you that you needed to know this product rule, 584 00:34:44,740 --> 00:34:50,419 and that you needed to know the chain rule. 585 00:34:50,419 --> 00:34:51,960 And I guess there was one more thing, 586 00:34:51,960 --> 00:34:55,260 the derivative of e^x came into play there. 587 00:34:55,260 --> 00:34:59,040 So of these formulas, we used four of them 588 00:34:59,040 --> 00:35:03,810 in this one calculation. 589 00:35:03,810 --> 00:35:15,880 Okay, so now what other things did we talk about in Unit One? 590 00:35:15,880 --> 00:35:23,590 So the main other thing that we talked about 591 00:35:23,590 --> 00:35:33,120 was the definition of a derivative. 592 00:35:33,120 --> 00:35:40,020 And also there was sort of a goal 593 00:35:40,020 --> 00:35:51,050 which was to get to the meaning of the derivative. 594 00:35:51,050 --> 00:35:56,520 So these are things - so we had a couple of ways of looking 595 00:35:56,520 --> 00:35:59,170 at it, or at least a couple that I'm 596 00:35:59,170 --> 00:36:01,780 going to emphasize right now. 597 00:36:01,780 --> 00:36:06,270 But first, let me remind you what the formula is. 598 00:36:06,270 --> 00:36:13,900 The derivative is the limit as delta x goes to 0 of (f(x + 599 00:36:13,900 --> 00:36:19,040 delta x) - f(x)) / delta x. 600 00:36:19,040 --> 00:36:22,430 So that's it, and this is certainly 601 00:36:22,430 --> 00:36:25,640 going to be a central focus here. 602 00:36:25,640 --> 00:36:29,600 And you want to be able to recognize this formula 603 00:36:29,600 --> 00:36:42,760 in a number of ways. 604 00:36:42,760 --> 00:36:44,520 So, how do we use this? 605 00:36:44,520 --> 00:36:48,950 Well one thing we did was we calculated a bunch 606 00:36:48,950 --> 00:36:51,450 of these rates of change. 607 00:36:51,450 --> 00:36:53,580 In fact, more or less, they're the ones which 608 00:36:53,580 --> 00:36:55,760 are written right over here. 609 00:36:55,760 --> 00:36:57,210 This list of functions here. 610 00:36:57,210 --> 00:37:01,470 Now, which ones did we start out with just straight 611 00:37:01,470 --> 00:37:03,800 from the definition here? 612 00:37:03,800 --> 00:37:04,840 Which of these things? 613 00:37:04,840 --> 00:37:06,215 There were a whole bunch of them. 614 00:37:06,215 --> 00:37:09,180 So we started out with a function 1/x. 615 00:37:09,180 --> 00:37:11,530 We did x^n. 616 00:37:11,530 --> 00:37:14,530 We did sine x. 617 00:37:14,530 --> 00:37:16,880 We did cosine x. 618 00:37:16,880 --> 00:37:19,205 Now there was a little bit of subtlety with sine 619 00:37:19,205 --> 00:37:21,110 x and cosine x. 620 00:37:21,110 --> 00:37:25,210 We got them using something else. 621 00:37:25,210 --> 00:37:26,880 We didn't quite get them all the way. 622 00:37:26,880 --> 00:37:31,790 We got them using the case x = 0. 623 00:37:31,790 --> 00:37:34,530 We got them from the derivative at x = 0, 624 00:37:34,530 --> 00:37:37,680 we got the formulas for the derivatives of sine and cosine. 625 00:37:37,680 --> 00:37:40,890 But that was an argument which involved plugging in sin 626 00:37:40,890 --> 00:37:44,460 (x + delta x), and running through. 627 00:37:44,460 --> 00:37:45,840 So that's one example. 628 00:37:45,840 --> 00:37:50,630 We also did a^x. 629 00:37:50,630 --> 00:37:53,270 And that may be it. 630 00:37:53,270 --> 00:37:58,350 Oh yeah, I think that's about it. 631 00:37:58,350 --> 00:38:00,450 That may be about it. 632 00:38:00,450 --> 00:38:00,950 No. 633 00:38:00,950 --> 00:38:01,620 It isn't. 634 00:38:01,620 --> 00:38:03,910 Okay, so let me make a connection here which you 635 00:38:03,910 --> 00:38:07,770 probably haven't yet made, which is that we did it for (u v)'. 636 00:38:10,520 --> 00:38:15,690 And we also did it for (u / v)'. 637 00:38:15,690 --> 00:38:17,460 So sorry, I shouldn't write primes, 638 00:38:17,460 --> 00:38:20,500 because that's not consistent with the claim there. 639 00:38:20,500 --> 00:38:24,900 I differentiated the product; I differentiated the quotient 640 00:38:24,900 --> 00:38:28,110 using the same delta x notation. 641 00:38:28,110 --> 00:38:32,760 I guess I forgot that because I wasn't there when it happened. 642 00:38:32,760 --> 00:38:36,550 So look, these are the ones that you do by this. 643 00:38:36,550 --> 00:38:39,310 And, of course, you might have to reduce them to other things. 644 00:38:39,310 --> 00:38:42,190 These involve using something else. 645 00:38:42,190 --> 00:38:46,610 This one involves using the slope of this function at 0, 646 00:38:46,610 --> 00:38:48,600 just the way the sine and the cosine did. 647 00:38:48,600 --> 00:38:52,410 This one involves the slopes of the individual functions, u 648 00:38:52,410 --> 00:38:54,827 and v. And this one also involves the individual-- 649 00:38:54,827 --> 00:38:56,410 So, in other words, it doesn't get you 650 00:38:56,410 --> 00:38:58,390 all the way through to the end, but it's 651 00:38:58,390 --> 00:39:03,010 expressed in terms of something simpler in each of these cases. 652 00:39:03,010 --> 00:39:05,840 And I could go on. 653 00:39:05,840 --> 00:39:09,280 We didn't do these in class, but you're certainly-- 654 00:39:09,280 --> 00:39:12,170 e^x is a perfectly okay one on one of the exams. 655 00:39:12,170 --> 00:39:14,561 We ask you for 1/x^2. 656 00:39:14,561 --> 00:39:16,310 In other words, I'm not claiming that it's 657 00:39:16,310 --> 00:39:18,380 going to be one on this list, but it certainly 658 00:39:18,380 --> 00:39:19,671 can be any one of these. 659 00:39:19,671 --> 00:39:21,170 But we're not going to ask you to go 660 00:39:21,170 --> 00:39:26,380 all the way through to the beginning in these formulas. 661 00:39:26,380 --> 00:39:28,940 There are also some fundamental limits that I certainly 662 00:39:28,940 --> 00:39:31,210 want you to know about. 663 00:39:31,210 --> 00:39:34,680 And these you can derive in reverse. 664 00:39:34,680 --> 00:39:58,880 So I will describe that now. 665 00:39:58,880 --> 00:40:06,800 So let me also emphasize the following thing: I want 666 00:40:06,800 --> 00:40:18,590 to read this backwards now. 667 00:40:18,590 --> 00:40:21,370 This is the theme from the very beginning of this lecture. 668 00:40:21,370 --> 00:40:25,210 Namely, if you're given the function f, 669 00:40:25,210 --> 00:40:27,811 you can figure out its derivative by this formula 670 00:40:27,811 --> 00:40:28,310 here. 671 00:40:28,310 --> 00:40:29,860 That is the formula for this in terms of what's 672 00:40:29,860 --> 00:40:30,920 on the right hand side. 673 00:40:30,920 --> 00:40:34,400 On the other hand, you can also use 674 00:40:34,400 --> 00:40:45,280 the formula in that direction, and if you 675 00:40:45,280 --> 00:40:48,310 know the slope of something, you can figure out 676 00:40:48,310 --> 00:40:49,170 what the limit is. 677 00:40:49,170 --> 00:40:54,570 For example, I'll use the letter x here, 678 00:40:54,570 --> 00:40:56,040 even though it's cheating. 679 00:40:56,040 --> 00:40:59,530 Maybe I'll call it delta x so it's clearer to you. 680 00:40:59,530 --> 00:41:06,900 Maybe I'll call it u. 681 00:41:06,900 --> 00:41:10,370 Suppose you look at this limit here. 682 00:41:10,370 --> 00:41:14,800 Well, I claim that you should recognize that is 683 00:41:14,800 --> 00:41:19,870 the derivative with respect to u of the function e^u at u = 0, 684 00:41:19,870 --> 00:41:22,660 which of course we know to be 1. 685 00:41:22,660 --> 00:41:25,420 So this is reading this formula in reverse. 686 00:41:25,420 --> 00:41:27,940 It's recognizing that one of these limits - 687 00:41:27,940 --> 00:41:35,160 let me rewrite this again here - one of these so-called 688 00:41:35,160 --> 00:41:39,390 difference quotient limits is a derivative. 689 00:41:39,390 --> 00:41:42,190 And since we know a formula for that derivative, 690 00:41:42,190 --> 00:41:49,940 we can evaluate it. 691 00:41:49,940 --> 00:41:54,150 And lastly, there's one other type of thing 692 00:41:54,150 --> 00:41:57,550 which I think you should know. 693 00:41:57,550 --> 00:41:59,767 These are the ones you do with difference quotients. 694 00:41:59,767 --> 00:42:01,350 There are also other formulas that you 695 00:42:01,350 --> 00:42:03,000 want to be able to derive. 696 00:42:03,000 --> 00:42:19,740 You want to be able to derive formulas 697 00:42:19,740 --> 00:42:27,670 by implicit differentiation. 698 00:42:27,670 --> 00:42:30,220 In other words, the basic idea is 699 00:42:30,220 --> 00:42:32,150 to take whatever equation you've got 700 00:42:32,150 --> 00:42:36,560 and simplify it as much as possible, 701 00:42:36,560 --> 00:42:41,260 without insisting that you solve for y. 702 00:42:41,260 --> 00:42:44,080 That's not necessarily the most appropriate way 703 00:42:44,080 --> 00:42:45,630 to get the rate of change. 704 00:42:45,630 --> 00:42:51,910 The much simpler formula is sin y = x. 705 00:42:51,910 --> 00:42:59,780 And that one is easier to differentiate implicitly. 706 00:42:59,780 --> 00:43:02,900 So I should say, do this kind of thing. 707 00:43:02,900 --> 00:43:05,550 So that's, if you like, a typical derivation 708 00:43:05,550 --> 00:43:08,390 that you might see. 709 00:43:08,390 --> 00:43:13,070 And then there's one last type of problem that you'll face, 710 00:43:13,070 --> 00:43:21,590 and it's the other thing that I claim we discussed. 711 00:43:21,590 --> 00:43:26,580 And it goes all the way back to the first lecture. 712 00:43:26,580 --> 00:43:33,700 So the last thing that we'll be talking about is tangent lines. 713 00:43:33,700 --> 00:43:34,200 All right? 714 00:43:34,200 --> 00:43:38,760 The geometric point of view of a derivative. 715 00:43:38,760 --> 00:43:41,900 And we'll be doing more of this in next the unit. 716 00:43:41,900 --> 00:43:44,850 So first of all, you'll be expected 717 00:43:44,850 --> 00:43:52,380 to be able to compute the tangent line. 718 00:43:52,380 --> 00:43:56,400 That's often fairly straightforward. 719 00:43:56,400 --> 00:44:03,100 And the second thing is to graph y' , 720 00:44:03,100 --> 00:44:07,370 the derivative of a function. 721 00:44:07,370 --> 00:44:09,350 And the third thing, which I'm going 722 00:44:09,350 --> 00:44:11,550 to throw in here, because I regard it 723 00:44:11,550 --> 00:44:14,900 in a sort of geometric vein, although it's got 724 00:44:14,900 --> 00:44:16,690 an analytical aspect to it. 725 00:44:16,690 --> 00:44:18,870 So this is a picture. 726 00:44:18,870 --> 00:44:20,710 This is a computation. 727 00:44:20,710 --> 00:44:23,080 And if you combine the two together, 728 00:44:23,080 --> 00:44:24,270 you get something else. 729 00:44:24,270 --> 00:44:37,870 And so this is to recognize differentiable functions. 730 00:44:37,870 --> 00:44:40,190 Alright, so how do you do this? 731 00:44:40,190 --> 00:44:43,600 Well, we really only have one way of doing it. 732 00:44:43,600 --> 00:44:54,580 We're going to check the left and right tangents. 733 00:44:54,580 --> 00:44:59,450 They must be equal. 734 00:44:59,450 --> 00:45:04,280 So again, this is a property that you 735 00:45:04,280 --> 00:45:06,830 should be familiar with from some of your exercises. 736 00:45:06,830 --> 00:45:09,660 And the idea is simply, that if the tangent line exists, 737 00:45:09,660 --> 00:45:14,770 it's the same from the right and from the left. 738 00:45:14,770 --> 00:45:20,200 Okay, now I'm going to just do one example here 739 00:45:20,200 --> 00:45:25,450 from this sort of qualitative sketching skill 740 00:45:25,450 --> 00:45:27,289 to give you an example here. 741 00:45:27,289 --> 00:45:28,830 And what I'm going to do is I'm going 742 00:45:28,830 --> 00:45:34,750 to draw a graph of a function like this. 743 00:45:34,750 --> 00:45:38,490 And what I want to do underneath is draw 744 00:45:38,490 --> 00:45:41,600 the graph of the derivative. 745 00:45:41,600 --> 00:45:45,900 So this is the function y = f(x), 746 00:45:45,900 --> 00:45:48,350 and here I'm going to draw the graph of the function y = 747 00:45:48,350 --> 00:45:56,490 f'(x) right underneath it. 748 00:45:56,490 --> 00:46:00,330 So now, let's think about what it's supposed to look like. 749 00:46:00,330 --> 00:46:05,840 And the one step that you need to make in order to do this, 750 00:46:05,840 --> 00:46:08,660 is to draw a few tangent lines. 751 00:46:08,660 --> 00:46:13,210 I'm just going to draw one down here. 752 00:46:13,210 --> 00:46:18,730 And I'm going to draw one up here. 753 00:46:18,730 --> 00:46:22,740 Now, the tangent lines here - notice 754 00:46:22,740 --> 00:46:27,266 that the slope of these tangent lines are all positive. 755 00:46:27,266 --> 00:46:28,640 So everything I draw down here is 756 00:46:28,640 --> 00:46:33,880 going to be above the x-axis. 757 00:46:33,880 --> 00:46:36,190 Furthermore, as I go further to the left, 758 00:46:36,190 --> 00:46:37,830 they get steeper and steeper. 759 00:46:37,830 --> 00:46:39,500 So they're getting higher and higher. 760 00:46:39,500 --> 00:46:44,020 So the function is coming down like this. 761 00:46:44,020 --> 00:46:45,350 It starts up there. 762 00:46:45,350 --> 00:46:50,570 Maybe I'll draw it in green to illustrate the graph here. 763 00:46:50,570 --> 00:46:56,910 So that's this function here. 764 00:46:56,910 --> 00:46:59,750 As we get farther out, it's getting flatter and flatter. 765 00:46:59,750 --> 00:47:06,270 So it's leveling off, but above the axis like that. 766 00:47:06,270 --> 00:47:08,280 So one of the things to emphasize 767 00:47:08,280 --> 00:47:10,830 is, you should not expect the derivative 768 00:47:10,830 --> 00:47:12,090 to look like the function. 769 00:47:12,090 --> 00:47:13,320 It's totally different. 770 00:47:13,320 --> 00:47:17,280 It's keeping track at each point of its tangent line. 771 00:47:17,280 --> 00:47:19,780 On the other hand, you should get some kind of physical feel 772 00:47:19,780 --> 00:47:23,619 for it, and we'll be practicing this more in the next unit. 773 00:47:23,619 --> 00:47:25,660 So let me give you an example of a function which 774 00:47:25,660 --> 00:47:27,800 does exactly this. 775 00:47:27,800 --> 00:47:33,240 And it's the function y = ln x. 776 00:47:33,240 --> 00:47:38,560 If you differentiate it, you get y' = 1/x. 777 00:47:38,560 --> 00:47:44,630 And this plot above is, roughly speaking, the logarithm. 778 00:47:44,630 --> 00:47:50,230 And this plot underneath is the function 1/x. 779 00:47:50,230 --> 00:47:53,230 We still have time for one question. 780 00:47:53,230 --> 00:47:58,580 And so, fire away. 781 00:47:58,580 --> 00:48:03,207 Yes? 782 00:48:03,207 --> 00:48:04,040 STUDENT: [INAUDIBLE] 783 00:48:04,040 --> 00:48:05,580 PROFESSOR: The question is, can you 784 00:48:05,580 --> 00:48:09,770 show how you derive the inverse tangent of x. 785 00:48:09,770 --> 00:48:13,350 So that's in a lecture. 786 00:48:13,350 --> 00:48:17,060 I'm happy to do it right now, but it's going 787 00:48:17,060 --> 00:48:20,420 to take me a whole two minutes. 788 00:48:20,420 --> 00:48:27,560 So, here's how you do it. y = tan^(-1) x. 789 00:48:27,560 --> 00:48:30,230 And now this is hopeless to differentiate, 790 00:48:30,230 --> 00:48:34,720 so I rewrite it as tan y = x. 791 00:48:34,720 --> 00:48:38,440 And now I have to differentiate that. 792 00:48:38,440 --> 00:48:42,360 So when I differentiate it, I get 793 00:48:42,360 --> 00:48:44,260 the derivative of tan y with respect 794 00:48:44,260 --> 00:48:46,560 to x-- with respect to y. 795 00:48:46,560 --> 00:48:48,210 That's 1 / (1 + y^2) times y'. 796 00:48:51,120 --> 00:48:52,850 So this is a hard step. 797 00:48:52,850 --> 00:48:53,930 That's the chain rule. 798 00:48:53,930 --> 00:48:55,860 And on the left side I get 1. 799 00:48:55,860 --> 00:48:57,650 So I'm doing this super fast because we 800 00:48:57,650 --> 00:49:00,720 have thirty seconds left. 801 00:49:00,720 --> 00:49:02,840 But this is the hard step right here. 802 00:49:02,840 --> 00:49:04,900 And it needs for you to know that d/dy tan 803 00:49:04,900 --> 00:49:14,432 y is equal to one over-- Oh, bad bad bad, secant squared. 804 00:49:14,432 --> 00:49:22,810 I was ahead of myself so fast. 805 00:49:22,810 --> 00:49:24,920 So here's the identity. 806 00:49:24,920 --> 00:49:28,500 So you need have known this in advance. 807 00:49:28,500 --> 00:49:30,740 And that's the input into this equation. 808 00:49:30,740 --> 00:49:44,000 So now, what we have is that y' = 1 / sec^2 y y, 809 00:49:44,000 --> 00:49:51,380 which is the same thing as cos^2 y. 810 00:49:51,380 --> 00:49:54,170 Now, the last bit of the problem is 811 00:49:54,170 --> 00:49:57,930 to rewrite this in terms of x. 812 00:49:57,930 --> 00:50:02,664 And that you have to do with a right triangle. 813 00:50:02,664 --> 00:50:05,618 If this is x and this is 1, then the angle 814 00:50:05,618 --> 00:50:09,416 is y, because the tangent of y is x. 815 00:50:09,416 --> 00:50:14,902 So this expresses the fact that the tangent of y is x. 816 00:50:14,902 --> 00:50:18,700 And then the hypotenuse is the square root of 1 + x^2. 817 00:50:21,654 --> 00:50:27,140 And so the cosine is 1 divided by that. 818 00:50:27,140 --> 00:50:30,938 So this thing is 1 divided by the square root of 1 + x^2, 819 00:50:30,938 --> 00:50:36,424 the quantity squared. 820 00:50:36,424 --> 00:50:40,644 So, and then the last little bit here, since I'm racing along, 821 00:50:40,644 --> 00:50:45,286 is that it's 1 / (1 + x^2), which I incorrectly wrote over 822 00:50:45,286 --> 00:50:46,130 here. 823 00:50:46,130 --> 00:50:48,662 Okay, so good luck on the test. 824 00:50:48,662 --> 00:50:50,756 See you tomorrow.