1 00:00:07,760 --> 00:00:09,250 PROFESSOR: OK. 2 00:00:09,250 --> 00:00:15,190 This video is about derivatives. 3 00:00:15,190 --> 00:00:18,630 Two rules for finding new derivatives. 4 00:00:18,630 --> 00:00:21,850 If we know the derivative of a function f-- 5 00:00:21,850 --> 00:00:23,630 say we've found that-- 6 00:00:23,630 --> 00:00:26,340 and we know the derivative of g-- 7 00:00:26,340 --> 00:00:27,710 we've found that-- 8 00:00:27,710 --> 00:00:31,250 then there are functions that we can build out of those. 9 00:00:31,250 --> 00:00:34,950 And two important and straightforward ones are the 10 00:00:34,950 --> 00:00:40,300 product, f of x times g of x, and the quotient, the ratio f 11 00:00:40,300 --> 00:00:42,440 of x over g of x. 12 00:00:42,440 --> 00:00:44,700 So those are the two rules we need. 13 00:00:44,700 --> 00:00:49,900 If we know df dx and we know dg dx, what's the derivative 14 00:00:49,900 --> 00:00:50,690 of the product? 15 00:00:50,690 --> 00:00:54,510 Well, it is not df dx times dg dx. 16 00:00:54,510 --> 00:01:01,060 And let me reduce the suspense by writing down what it is. 17 00:01:01,060 --> 00:01:07,000 It's the first one times the derivative of the second, we 18 00:01:07,000 --> 00:01:12,380 know that, plus another term, the second one times the 19 00:01:12,380 --> 00:01:15,690 derivative of the first. OK. 20 00:01:15,690 --> 00:01:19,170 So that's the rule to learn. 21 00:01:19,170 --> 00:01:22,200 Two terms, you see the pattern. 22 00:01:22,200 --> 00:01:29,680 And maybe I ought to use it, give you some examples, see 23 00:01:29,680 --> 00:01:33,250 what it's good for, and also some idea of 24 00:01:33,250 --> 00:01:35,240 where it comes from. 25 00:01:35,240 --> 00:01:37,870 And then go on to the quotient rule, 26 00:01:37,870 --> 00:01:39,510 which is a little messier. 27 00:01:39,510 --> 00:01:40,110 OK. 28 00:01:40,110 --> 00:01:45,400 So let me just start by using this in some examples. 29 00:01:45,400 --> 00:01:46,970 Right underneath, here. 30 00:01:46,970 --> 00:01:47,700 OK. 31 00:01:47,700 --> 00:01:52,500 So let me take, as a first example, f of x equals x 32 00:01:52,500 --> 00:01:56,470 squared and g of x equals x. 33 00:01:56,470 --> 00:02:00,210 So then what is p of x? 34 00:02:00,210 --> 00:02:01,950 It's x squared times x. 35 00:02:01,950 --> 00:02:04,420 I'm multiplying the functions. 36 00:02:04,420 --> 00:02:09,100 So I've got x cubed, and I want to know its derivative. 37 00:02:09,100 --> 00:02:11,970 And I know the derivatives of these guys. 38 00:02:11,970 --> 00:02:14,390 OK, so what does the rule tell me? 39 00:02:14,390 --> 00:02:20,590 It tells me that the derivative of p, dp dx-- 40 00:02:20,590 --> 00:02:22,200 so p is x cubed. 41 00:02:22,200 --> 00:02:24,640 So I'm looking for the derivative of x cubed. 42 00:02:24,640 --> 00:02:27,210 And if you know that, it's OK. 43 00:02:27,210 --> 00:02:29,450 Let's just see it come out here. 44 00:02:29,450 --> 00:02:33,860 So the derivative of x cubed, by my formula there, is the 45 00:02:33,860 --> 00:02:38,080 first one, x squared, times the derivative of the second, 46 00:02:38,080 --> 00:02:43,890 which is 1, plus the second one, x, times the derivative 47 00:02:43,890 --> 00:02:46,170 of the first, which is 2x. 48 00:02:46,170 --> 00:02:47,310 So what do we get? 49 00:02:47,310 --> 00:02:50,495 x squared, two more x squared, 3x squared. 50 00:02:54,030 --> 00:02:57,810 The derivative of x cubed is 3x squared. 51 00:02:57,810 --> 00:03:03,610 x cubed goes up faster than x squared, and this 52 00:03:03,610 --> 00:03:04,910 is a steeper slope. 53 00:03:08,010 --> 00:03:10,740 Oh, let's do x to the fourth. 54 00:03:10,740 --> 00:03:18,650 So x to the fourth-- now I'll take f to be x cubed, times x. 55 00:03:18,650 --> 00:03:21,310 Because x cubed, I just found. 56 00:03:21,310 --> 00:03:25,700 x, its derivative is 1, so I can do the derivative of x 57 00:03:25,700 --> 00:03:31,510 fourth the same way. 58 00:03:31,510 --> 00:03:32,780 It'll be f. 59 00:03:32,780 --> 00:03:36,700 So practicing that formula again with x cubed and x, it's 60 00:03:36,700 --> 00:03:44,360 x cubed times 1 plus this guy times the derivative of f. 61 00:03:44,360 --> 00:03:45,450 Right? 62 00:03:45,450 --> 00:03:48,790 I'm always going back to that formula. 63 00:03:48,790 --> 00:03:51,830 So the derivative of f, x cubed, we just found-- 64 00:03:51,830 --> 00:03:54,150 3x squared-- so I'll put it in. 65 00:03:54,150 --> 00:03:55,670 And what do we have? 66 00:03:55,670 --> 00:03:59,020 x cubed here, three more x cubeds here. 67 00:03:59,020 --> 00:04:01,200 That's a total of 4x cubed. 68 00:04:04,480 --> 00:04:04,830 OK. 69 00:04:04,830 --> 00:04:06,170 We got another one. 70 00:04:06,170 --> 00:04:07,950 Big deal. 71 00:04:07,950 --> 00:04:11,450 What is important is-- 72 00:04:11,450 --> 00:04:13,930 and it's really what math is about-- 73 00:04:13,930 --> 00:04:21,709 is the pattern, which we can probably guess from those two 74 00:04:21,709 --> 00:04:24,240 examples and the one we already knew, that the 75 00:04:24,240 --> 00:04:28,540 derivative of x squared was 2x. 76 00:04:28,540 --> 00:04:33,170 So everybody sees a 2 here and a 3 here and a 4 here, coming 77 00:04:33,170 --> 00:04:36,210 from 2, 3, and 4 there. 78 00:04:36,210 --> 00:04:41,200 And everybody also sees that the power dropped by one. 79 00:04:41,200 --> 00:04:43,660 The derivative of x squared was an x. 80 00:04:43,660 --> 00:04:45,990 The derivative of x cubed involved an x squared. 81 00:04:45,990 --> 00:04:51,520 Well, let's express this pattern in algebra. 82 00:04:51,520 --> 00:04:59,920 It's looking like the derivative of x to the n-- 83 00:04:59,920 --> 00:05:02,710 we hope for any n. 84 00:05:02,710 --> 00:05:08,590 We've got it for n equals 2, 3, 4, probably 0 and 1. 85 00:05:08,590 --> 00:05:12,540 And if the pattern continues, what do we think? 86 00:05:12,540 --> 00:05:20,160 This 4, this n shows up there, and the power drops by 1. 87 00:05:20,160 --> 00:05:29,150 So that'll be x to the n minus 1, the same power minus 1, one 88 00:05:29,150 --> 00:05:30,130 power below. 89 00:05:30,130 --> 00:05:35,160 So that's a highly important formula. 90 00:05:35,160 --> 00:05:39,820 And actually it's important to know it, not-- 91 00:05:39,820 --> 00:05:44,100 right now, well, we've done two or three examples. 92 00:05:44,100 --> 00:05:51,280 I guess the right way for me to get this for n equals-- 93 00:05:51,280 --> 00:05:56,640 so we really could check 1, 2, 3, and so on. 94 00:05:56,640 --> 00:06:02,360 All the positive integers. 95 00:06:06,360 --> 00:06:08,800 We could complete the proof. 96 00:06:08,800 --> 00:06:10,650 We could establish the pattern. 97 00:06:10,650 --> 00:06:14,880 Actually, induction would be one way to do it. 98 00:06:14,880 --> 00:06:20,780 If we know it for, as we did here, for n equals 3, then 99 00:06:20,780 --> 00:06:21,920 we've got it for 4. 100 00:06:21,920 --> 00:06:26,170 If we know it for 4, the same product formula would get it 101 00:06:26,170 --> 00:06:30,060 for 5 and onwards, and would give us that answer. 102 00:06:30,060 --> 00:06:31,310 Good. 103 00:06:32,720 --> 00:06:40,610 Even better is the fact that this formula is also true if n 104 00:06:40,610 --> 00:06:43,000 is a fraction. 105 00:06:43,000 --> 00:06:48,760 If we're doing the square root of x, you recognize the square 106 00:06:48,760 --> 00:06:50,940 root of x is x to the-- 107 00:06:50,940 --> 00:06:53,830 what's the exponent there for square root? 108 00:06:53,830 --> 00:06:55,250 1/2. 109 00:06:55,250 --> 00:06:57,540 So I would like to know for 1/2. 110 00:06:57,540 --> 00:07:02,390 OK, let me take a couple of steps to get to that one. 111 00:07:02,390 --> 00:07:03,440 All right. 112 00:07:03,440 --> 00:07:06,280 The steps I'm going to take are going to look just like 113 00:07:06,280 --> 00:07:16,350 this, but this was powers of x, and it'll be very handy if 114 00:07:16,350 --> 00:07:21,010 I can do powers of f of x. 115 00:07:21,010 --> 00:07:21,130 I'd like to know-- 116 00:07:21,130 --> 00:07:22,380 I want to find-- 117 00:07:26,010 --> 00:07:27,620 So here's what I'm headed for. 118 00:07:27,620 --> 00:07:34,050 I'd like to know the derivative of f of x to the 119 00:07:34,050 --> 00:07:39,730 n-th power equals what? 120 00:07:39,730 --> 00:07:41,290 That's what I'd like to know. 121 00:07:41,290 --> 00:07:42,690 So let me do f of x. 122 00:07:42,690 --> 00:07:44,980 Let me do it just as I did before. 123 00:07:44,980 --> 00:07:48,540 Take n equals 2, f of x squared. 124 00:07:48,540 --> 00:07:53,210 So what's the derivative of f of x squared, like sine 125 00:07:53,210 --> 00:07:55,460 squared or whatever we're squaring. 126 00:07:55,460 --> 00:07:57,030 Cosine squared. 127 00:07:57,030 --> 00:08:04,870 Well, for f of x squared, all I'm doing is I'm taking f to 128 00:08:04,870 --> 00:08:07,390 be the same as g. 129 00:08:07,390 --> 00:08:08,920 I'll use the product rule. 130 00:08:08,920 --> 00:08:14,790 If g and f are the same, then I've got something squared. 131 00:08:14,790 --> 00:08:19,480 And my product rule says that the derivative-- and I just 132 00:08:19,480 --> 00:08:21,500 copy this rule. 133 00:08:21,500 --> 00:08:26,010 Now I'm taking p is going to be f squared, right? 134 00:08:26,010 --> 00:08:30,330 Can I just write f squared equals-- 135 00:08:30,330 --> 00:08:32,950 so it's f times-- 136 00:08:32,950 --> 00:08:34,150 f is the same as g. 137 00:08:34,150 --> 00:08:35,890 Are you with me? 138 00:08:35,890 --> 00:08:38,950 I'm just using the rule in a very special case when the two 139 00:08:38,950 --> 00:08:40,350 functions are the same. 140 00:08:40,350 --> 00:08:44,320 The derivative of f squared is f. 141 00:08:44,320 --> 00:08:48,320 What do I have? f times the derivative of f, df dx. 142 00:08:48,320 --> 00:08:50,320 That's the first term. 143 00:08:50,320 --> 00:08:52,520 And then what's the second term? 144 00:08:52,520 --> 00:08:56,600 Notice I wrote f instead of g, because they're the same. 145 00:08:56,600 --> 00:08:59,440 And the second term is, again, a copy of that. 146 00:08:59,440 --> 00:09:01,890 So I have 2 of these. 147 00:09:01,890 --> 00:09:07,210 Times 2, just the way I had a 2 up there. 148 00:09:07,210 --> 00:09:10,960 This was the case of x squared. 149 00:09:10,960 --> 00:09:13,710 This is the case of f of x squared. 150 00:09:16,510 --> 00:09:19,400 Let me go one more step to f cubed. 151 00:09:19,400 --> 00:09:22,020 What am I going to do for f cubed? 152 00:09:22,020 --> 00:09:25,160 The derivative of-- 153 00:09:25,160 --> 00:09:26,390 hold on. 154 00:09:26,390 --> 00:09:30,470 I have to show you what to pay attention to here. 155 00:09:30,470 --> 00:09:32,470 To pay attention to is-- 156 00:09:32,470 --> 00:09:34,750 the 2 we're familiar with. 157 00:09:34,750 --> 00:09:37,360 This would have been the x, that's not a big deal. 158 00:09:37,360 --> 00:09:38,680 But there's something new. 159 00:09:38,680 --> 00:09:41,660 A df dx factor is coming in. 160 00:09:41,660 --> 00:09:43,150 It's going to stay with us. 161 00:09:43,150 --> 00:09:44,740 Let me see it here. 162 00:09:44,740 --> 00:09:48,740 The derivative of f of x cubed. 163 00:09:48,740 --> 00:09:51,350 Now let's practice with this one. 164 00:09:51,350 --> 00:09:51,940 OK. 165 00:09:51,940 --> 00:09:54,850 So now what am I going to take? 166 00:09:54,850 --> 00:09:58,150 How do I get f of x cubed? 167 00:09:58,150 --> 00:10:02,970 Well, I've got f, so I'd better take g to be f squared. 168 00:10:02,970 --> 00:10:05,400 Then when I multiply, I've got cubed. 169 00:10:05,400 --> 00:10:11,490 So g is now going to be f squared for this case. 170 00:10:11,490 --> 00:10:17,590 Can I take my product rule with f times f squared? 171 00:10:17,590 --> 00:10:20,190 My product rule of f times f squared is-- 172 00:10:24,420 --> 00:10:29,380 I'm doing this now with g equals f squared, just the way 173 00:10:29,380 --> 00:10:33,090 I did it over there at some point with one 174 00:10:33,090 --> 00:10:34,830 of them as a square. 175 00:10:34,830 --> 00:10:36,080 OK. 176 00:10:38,000 --> 00:10:41,150 I'm near the end of this calculation. 177 00:10:41,150 --> 00:10:41,790 OK. 178 00:10:41,790 --> 00:10:45,220 So what do I have. If this thing is cubed, I 179 00:10:45,220 --> 00:10:47,110 have f times f squared. 180 00:10:47,110 --> 00:10:48,860 That's f cubed. 181 00:10:48,860 --> 00:10:51,340 And I take its derivative by the rule. 182 00:10:51,340 --> 00:10:55,510 So I take f times the derivative of f squared, which 183 00:10:55,510 --> 00:11:01,860 I just figured out as 2f df dx. 184 00:11:01,860 --> 00:11:04,570 That's the f dg dx. 185 00:11:04,570 --> 00:11:08,950 And now I have g, which is f squared, times df dx. 186 00:11:14,860 --> 00:11:16,950 What are you seeing there? 187 00:11:16,950 --> 00:11:18,120 You're seeing-- 188 00:11:18,120 --> 00:11:21,480 well, again, these combine. 189 00:11:21,480 --> 00:11:24,760 That's what's nice about this example. 190 00:11:24,760 --> 00:11:28,460 Here I have one f squared df dx, and here I have two more. 191 00:11:28,460 --> 00:11:30,080 That's, all together, three. 192 00:11:30,080 --> 00:11:38,470 So the total was 3 times f squared times df dx. 193 00:11:38,470 --> 00:11:42,530 And let me write down what that pattern is saying. 194 00:11:42,530 --> 00:11:45,770 Here it will be n. 195 00:11:45,770 --> 00:11:48,090 Because here it was a 2. 196 00:11:48,090 --> 00:11:50,940 Here it's going to be 2 plus 1-- that's 3. 197 00:11:50,940 --> 00:11:54,450 And now if I have the n-th power, I'm expecting an n 198 00:11:54,450 --> 00:12:00,280 times the next lower power of f, f to the n 199 00:12:00,280 --> 00:12:03,760 minus 1, times what? 200 00:12:03,760 --> 00:12:06,750 Times this guy that's hanging around, df dx. 201 00:12:09,790 --> 00:12:12,540 That's my-- 202 00:12:12,540 --> 00:12:15,030 you could call that the power rule. 203 00:12:17,670 --> 00:12:20,270 The derivative of a power. 204 00:12:20,270 --> 00:12:24,040 This would be the power rule for just x to the n-th, and 205 00:12:24,040 --> 00:12:28,490 this is the derivative of a function of x to the n-th. 206 00:12:28,490 --> 00:12:30,100 There's something special here that we're 207 00:12:30,100 --> 00:12:31,890 going to see more of. 208 00:12:35,320 --> 00:12:42,170 This will be, also, an example of what's coming as maybe the 209 00:12:42,170 --> 00:12:44,860 most important rule, the chain rule. 210 00:12:44,860 --> 00:12:50,120 And typical of it is that when I take this derivative, I 211 00:12:50,120 --> 00:12:52,160 follow that same pattern-- 212 00:12:52,160 --> 00:12:58,990 n, this thing, to one lower power, but then the derivative 213 00:12:58,990 --> 00:13:00,400 of what's inside. 214 00:13:00,400 --> 00:13:01,540 Can I use those words? 215 00:13:01,540 --> 00:13:04,130 Because I'll use it again for the chain rule. 216 00:13:04,130 --> 00:13:07,640 n times one lower power, times the 217 00:13:07,640 --> 00:13:09,260 derivative of what's inside. 218 00:13:09,260 --> 00:13:12,060 And why do I want to do such a thing? 219 00:13:12,060 --> 00:13:15,450 Because I'd like to find out the derivative of the 220 00:13:15,450 --> 00:13:17,780 square root of x. 221 00:13:17,780 --> 00:13:18,730 OK. 222 00:13:18,730 --> 00:13:19,940 Can we do that? 223 00:13:19,940 --> 00:13:22,050 I want to use this, now. 224 00:13:22,050 --> 00:13:26,370 So I want to use this to find the derivative of the 225 00:13:26,370 --> 00:13:28,700 square root of x. 226 00:13:28,700 --> 00:13:29,630 OK. 227 00:13:29,630 --> 00:13:31,260 So that will be my function. 228 00:13:31,260 --> 00:13:34,390 f of x will be the square root of x. 229 00:13:34,390 --> 00:13:35,580 So this is a good example. 230 00:13:35,580 --> 00:13:38,980 That's x to the 1/2 power. 231 00:13:38,980 --> 00:13:41,100 What would I love to have happen? 232 00:13:41,100 --> 00:13:46,970 I would like this formula to continue with n equals 1/2, 233 00:13:46,970 --> 00:13:48,590 but no change in the formula. 234 00:13:48,590 --> 00:13:50,430 And that does happen. 235 00:13:50,430 --> 00:13:52,100 How can I do that? 236 00:13:52,100 --> 00:13:57,360 OK, well, square root of x is what I'm tackling. 237 00:13:59,960 --> 00:14:02,170 The easy thing would be, if I square 238 00:14:02,170 --> 00:14:05,690 that, I'll get x, right? 239 00:14:05,690 --> 00:14:07,780 The square of the square root. 240 00:14:10,510 --> 00:14:13,550 Well, square root of x squared-- 241 00:14:13,550 --> 00:14:15,130 so there's f of x. 242 00:14:15,130 --> 00:14:18,720 I'm just going to use the fact that the square root of x 243 00:14:18,720 --> 00:14:20,130 squared is x. 244 00:14:23,260 --> 00:14:24,470 Such is mathematics. 245 00:14:24,470 --> 00:14:30,830 You can write down really straightforward ideas, but it 246 00:14:30,830 --> 00:14:32,310 had to come from somewhere. 247 00:14:32,310 --> 00:14:33,670 And now what am I going to do? 248 00:14:33,670 --> 00:14:35,650 I'm going to take the derivative. 249 00:14:35,650 --> 00:14:38,840 Well, the derivative on the right side is a 1. 250 00:14:38,840 --> 00:14:41,110 The derivative of x is 1. 251 00:14:41,110 --> 00:14:44,020 What is the derivative of that left-hand side? 252 00:14:44,020 --> 00:14:45,750 Well, that fits my pattern. 253 00:14:45,750 --> 00:14:48,840 You see, here is my f of x, squared. 254 00:14:48,840 --> 00:14:50,940 And I had a little formula for the 255 00:14:50,940 --> 00:14:52,990 derivative of f of x squared. 256 00:14:52,990 --> 00:14:57,620 So the derivative of this is 2 times the thing 257 00:14:57,620 --> 00:14:59,280 to one lower power-- 258 00:14:59,280 --> 00:15:02,820 square root of x just to the first power-- 259 00:15:02,820 --> 00:15:07,260 times the derivative of what's inside, if you allow me to use 260 00:15:07,260 --> 00:15:08,210 those words. 261 00:15:08,210 --> 00:15:10,360 It's this, df dx. 262 00:15:10,360 --> 00:15:13,630 And that's of course what I actually wanted, the square 263 00:15:13,630 --> 00:15:14,880 root of x, dx. 264 00:15:22,020 --> 00:15:24,310 This lecture is not going to have too many more 265 00:15:24,310 --> 00:15:29,230 calculations, but this is a good one to see. 266 00:15:29,230 --> 00:15:31,120 That's clear. 267 00:15:31,120 --> 00:15:32,990 I take the derivative of both sides. 268 00:15:32,990 --> 00:15:34,040 That's clear. 269 00:15:34,040 --> 00:15:35,860 This is the 2 square root of x. 270 00:15:35,860 --> 00:15:40,780 And now I've got what I want, as soon as I move these over 271 00:15:40,780 --> 00:15:41,720 to the other side. 272 00:15:41,720 --> 00:15:43,710 So I divide by that. 273 00:15:43,710 --> 00:15:48,270 Can I now just do that with an eraser, or maybe just X it 274 00:15:48,270 --> 00:15:50,130 out, and put it here. 275 00:15:50,130 --> 00:15:53,920 1 over 2 square root of x. 276 00:15:53,920 --> 00:15:56,410 Am I seeing what I want for the derivative of 277 00:15:56,410 --> 00:15:57,560 square root of x? 278 00:15:57,560 --> 00:15:59,910 I hope so. 279 00:15:59,910 --> 00:16:02,720 I'm certainly seeing the 1/2. 280 00:16:02,720 --> 00:16:04,490 So the 1/2-- 281 00:16:04,490 --> 00:16:05,290 that's the n. 282 00:16:05,290 --> 00:16:06,980 It's supposed to show up here. 283 00:16:06,980 --> 00:16:09,380 And then what do I look for here? 284 00:16:09,380 --> 00:16:13,770 One lower power than 1/2, which will be x 285 00:16:13,770 --> 00:16:16,050 to the minus 1/2. 286 00:16:16,050 --> 00:16:17,290 And is that what I have? 287 00:16:17,290 --> 00:16:19,520 Yes. 288 00:16:19,520 --> 00:16:20,970 You see the 1/2. 289 00:16:20,970 --> 00:16:24,820 And that square root of x, that's x to the 1/2, but it's 290 00:16:24,820 --> 00:16:26,410 down in the denominator. 291 00:16:26,410 --> 00:16:30,030 And things in the denominator-- 292 00:16:30,030 --> 00:16:35,150 the exponent for those, there's a minus sign. 293 00:16:35,150 --> 00:16:36,620 We'll come back to that. 294 00:16:36,620 --> 00:16:40,490 That's a crucial fact, going back to algebra. 295 00:16:40,490 --> 00:16:45,970 But, you know, calculus is now using all that-- 296 00:16:45,970 --> 00:16:48,230 I won't say stuff. 297 00:16:48,230 --> 00:16:53,710 All those good things that we learned in 298 00:16:53,710 --> 00:16:57,410 algebra, like exponents. 299 00:16:57,410 --> 00:17:00,980 So that was a good example. 300 00:17:00,980 --> 00:17:03,670 OK. 301 00:17:03,670 --> 00:17:08,560 So my pattern held for n equals 1/2. 302 00:17:08,560 --> 00:17:15,710 And maybe I'll just say that it also would hold for cube 303 00:17:15,710 --> 00:17:23,660 roots, and any root, and other powers. 304 00:17:23,660 --> 00:17:26,089 In other words, I get this formula. 305 00:17:26,089 --> 00:17:31,610 This is the handy formula that we're trying to get. 306 00:17:31,610 --> 00:17:38,150 We got it very directly for positive whole numbers. 307 00:17:38,150 --> 00:17:44,250 Now I'm getting it for n equals 1 over any-- 308 00:17:44,250 --> 00:17:47,510 now I'm getting it for capital Nth roots, like 1/2. 309 00:17:51,340 --> 00:17:55,910 Then I could go on to get it for-- 310 00:17:55,910 --> 00:18:00,630 I could take then the n-th power of the n-th root. 311 00:18:00,630 --> 00:18:05,540 I could even stretch this to get up to m over n. 312 00:18:05,540 --> 00:18:09,510 Any fraction, I can get to. 313 00:18:09,510 --> 00:18:15,080 But I can't get to negative exponents yet, because those 314 00:18:15,080 --> 00:18:16,380 are divisions. 315 00:18:16,380 --> 00:18:19,060 Negative exponent is a division, and I'm going to 316 00:18:19,060 --> 00:18:20,820 need the quotient rule, which is right 317 00:18:20,820 --> 00:18:22,840 now still a big blank. 318 00:18:22,840 --> 00:18:23,560 OK. 319 00:18:23,560 --> 00:18:26,360 Pause for a moment. 320 00:18:26,360 --> 00:18:29,120 We've used the product rule. 321 00:18:29,120 --> 00:18:30,720 I haven't explained it, though. 322 00:18:30,720 --> 00:18:35,640 Let me, so, explain the product rule. 323 00:18:35,640 --> 00:18:37,720 Where did it come from? 324 00:18:37,720 --> 00:18:44,080 I'm going back before the examples, and before that 325 00:18:44,080 --> 00:18:48,140 board full of chalk, back to that formula and just think, 326 00:18:48,140 --> 00:18:50,300 where did it come from? 327 00:18:50,300 --> 00:18:54,560 How did we find the derivative of f times g, 328 00:18:54,560 --> 00:18:55,740 of the product p? 329 00:18:55,740 --> 00:19:00,600 So we needed delta p, right? 330 00:19:00,600 --> 00:19:02,940 And then I'm going to divide by delta x. 331 00:19:02,940 --> 00:19:03,340 OK. 332 00:19:03,340 --> 00:19:05,140 So let me try to make-- 333 00:19:05,140 --> 00:19:08,760 what's the delta p when p is-- 334 00:19:08,760 --> 00:19:10,955 remember, p is f times g. 335 00:19:14,850 --> 00:19:19,730 Thinking about f times g, maybe let's make it visual. 336 00:19:19,730 --> 00:19:23,200 Let's make it like a rectangle, where this side is 337 00:19:23,200 --> 00:19:27,880 f of x and this side is g of x. 338 00:19:27,880 --> 00:19:33,740 Then this area is f times g, right? 339 00:19:33,740 --> 00:19:35,420 The area of a rectangle. 340 00:19:35,420 --> 00:19:37,460 And that's our p. 341 00:19:37,460 --> 00:19:41,580 OK, that's sitting there at x. 342 00:19:41,580 --> 00:19:44,760 Now move it a little. 343 00:19:44,760 --> 00:19:47,290 Move x a little bit. 344 00:19:47,290 --> 00:19:52,930 Move x a little and figure out, how much does p change? 345 00:19:52,930 --> 00:19:53,700 That's our goal. 346 00:19:53,700 --> 00:19:56,280 We need the change in p. 347 00:19:56,280 --> 00:20:02,925 If I move x by a little bit, then f changes a little, by a 348 00:20:02,925 --> 00:20:05,650 little amount, delta f, right? 349 00:20:05,650 --> 00:20:11,100 And g changes a little, by a little amount, delta g. 350 00:20:11,100 --> 00:20:13,860 You remember those deltas? 351 00:20:13,860 --> 00:20:15,380 So it's the change in f. 352 00:20:15,380 --> 00:20:20,070 There's a delta x in here. 353 00:20:20,070 --> 00:20:24,490 x is the starting point. 354 00:20:24,490 --> 00:20:26,520 It's the thing we move a little. 355 00:20:26,520 --> 00:20:32,700 When we move x a little, by delta x, f will move a little, 356 00:20:32,700 --> 00:20:35,000 g will move a little, and their 357 00:20:35,000 --> 00:20:36,640 product will move a little. 358 00:20:36,640 --> 00:20:40,300 And now, can you see, in the picture, where is the product? 359 00:20:40,300 --> 00:20:44,130 Well, this is where f moved to. 360 00:20:44,130 --> 00:20:45,550 This is where g moved to. 361 00:20:45,550 --> 00:20:51,840 The product is this, that bigger area. 362 00:20:51,840 --> 00:20:53,540 So where is delta p? 363 00:20:53,540 --> 00:20:56,120 Where is the change between the bigger area and the 364 00:20:56,120 --> 00:20:56,900 smaller area? 365 00:20:56,900 --> 00:20:58,100 It's this. 366 00:20:58,100 --> 00:21:03,630 I have to figure out, what's that new area? 367 00:21:03,630 --> 00:21:07,240 The delta p is in here. 368 00:21:07,240 --> 00:21:09,460 OK, can you see what that area-- 369 00:21:09,460 --> 00:21:11,330 well, look, here's the way to do it. 370 00:21:11,330 --> 00:21:14,540 Cut it up into little three pieces. 371 00:21:17,090 --> 00:21:20,460 Because now they're little rectangles, and we know the 372 00:21:20,460 --> 00:21:20,942 area of rectangles. 373 00:21:20,942 --> 00:21:22,050 Right? 374 00:21:22,050 --> 00:21:23,760 So help me out here. 375 00:21:23,760 --> 00:21:26,850 What is the area of that rectangle? 376 00:21:26,850 --> 00:21:33,120 Well, its base is f, and its height is delta g. 377 00:21:33,120 --> 00:21:36,055 So that is f times delta g. 378 00:21:38,630 --> 00:21:41,150 What about this one? 379 00:21:41,150 --> 00:21:46,140 That has height g and base delta f. 380 00:21:46,140 --> 00:21:51,080 So here I'm seeing a g times delta f, for that area. 381 00:21:54,750 --> 00:22:02,400 And what about this little corner piece? 382 00:22:02,400 --> 00:22:07,210 Well, its height is just delta g, its width is delta f. 383 00:22:07,210 --> 00:22:10,970 This is delta g times delta f. 384 00:22:16,600 --> 00:22:21,290 And it's going to disappear. 385 00:22:21,290 --> 00:22:25,580 This is like a perfect place to recognize that an 386 00:22:25,580 --> 00:22:27,300 expression-- 387 00:22:27,300 --> 00:22:30,570 that's sort of like second order. 388 00:22:30,570 --> 00:22:33,240 Let me use words without trying to 389 00:22:33,240 --> 00:22:35,460 pin them down perfectly. 390 00:22:35,460 --> 00:22:40,410 Here is a zero-order, an f, a real number, times 391 00:22:40,410 --> 00:22:42,530 a small delta g. 392 00:22:42,530 --> 00:22:43,740 So that's first order. 393 00:22:43,740 --> 00:22:46,370 That's going to show up-- 394 00:22:46,370 --> 00:22:47,620 you'll see it disappear. 395 00:22:51,000 --> 00:22:53,890 These three pieces, remember, were the delta p. 396 00:22:53,890 --> 00:22:55,490 So what have I got here? 397 00:22:55,490 --> 00:22:59,890 I've got this piece, f delta g, and I'm always 398 00:22:59,890 --> 00:23:01,970 dividing by delta x. 399 00:23:01,970 --> 00:23:06,640 And then I have this piece, which is the g times the delta 400 00:23:06,640 --> 00:23:09,530 f, and I divide by the delta x. 401 00:23:09,530 --> 00:23:13,330 And then this piece that I'm claiming I don't have to worry 402 00:23:13,330 --> 00:23:17,400 much about, because I divide that by delta x. 403 00:23:17,400 --> 00:23:19,580 So that was the third piece. 404 00:23:19,580 --> 00:23:20,830 This is it, now. 405 00:23:23,310 --> 00:23:30,770 The picture has led to the algebra, the formula for delta 406 00:23:30,770 --> 00:23:36,090 p, the change in the product divided by delta x. 407 00:23:36,090 --> 00:23:38,030 That's what calculus says-- 408 00:23:38,030 --> 00:23:42,760 OK, look at that, and then take the tricky step, the 409 00:23:42,760 --> 00:23:47,740 calculus step, which is let delta x get smaller and 410 00:23:47,740 --> 00:23:51,680 smaller and smaller, approaching 0. 411 00:23:51,680 --> 00:23:57,320 So what do those three terms do as delta x gets smaller? 412 00:23:57,320 --> 00:24:00,980 Well, all the deltas get smaller. 413 00:24:00,980 --> 00:24:08,430 So what happens to this term as delta x goes to 0? 414 00:24:08,430 --> 00:24:13,920 As the change in x is just tiny, tiny, tiny? 415 00:24:13,920 --> 00:24:19,970 That term is the one that gives the delta g over delta 416 00:24:19,970 --> 00:24:25,510 x, in the limit when delta x goes to 0, is that one, right? 417 00:24:25,510 --> 00:24:31,730 And this guy is giving my g. 418 00:24:31,730 --> 00:24:34,020 That ratio is familiar, df dx. 419 00:24:34,020 --> 00:24:40,370 You see, the cool thing about splitting it into these pieces 420 00:24:40,370 --> 00:24:44,090 was that we got this piece by itself, which was 421 00:24:44,090 --> 00:24:46,370 just the f delta g. 422 00:24:46,370 --> 00:24:48,425 And we know what that does. 423 00:24:48,425 --> 00:24:50,130 It goes here. 424 00:24:50,130 --> 00:24:51,930 And this piece-- 425 00:24:51,930 --> 00:24:53,190 we know what that does. 426 00:24:53,190 --> 00:24:57,600 And now, what about this dumb piece? 427 00:24:57,600 --> 00:25:00,890 Well, as delta x goes to 0, this would go 428 00:25:00,890 --> 00:25:05,310 to df dx, all right. 429 00:25:05,310 --> 00:25:06,780 But what would delta g do? 430 00:25:06,780 --> 00:25:08,350 It'll go to 0. 431 00:25:08,350 --> 00:25:12,190 You see, we have two little things divided by only one 432 00:25:12,190 --> 00:25:12,760 little thing. 433 00:25:12,760 --> 00:25:18,780 This ratio is sensible, it gives df dx, but this ratio is 434 00:25:18,780 --> 00:25:19,980 going to 0. 435 00:25:19,980 --> 00:25:22,060 So forget it. 436 00:25:22,060 --> 00:25:27,650 And now the two pieces that we have are the two pieces of the 437 00:25:27,650 --> 00:25:30,150 product rule. 438 00:25:30,150 --> 00:25:30,590 OK. 439 00:25:30,590 --> 00:25:38,520 Product rule sort of visually makes sense. 440 00:25:38,520 --> 00:25:39,670 OK. 441 00:25:39,670 --> 00:25:42,960 I'm ready to go to the quotient rule. 442 00:25:42,960 --> 00:25:46,690 OK, so how am I going to deal, now, with a ratio of 443 00:25:46,690 --> 00:25:48,310 f divided by g? 444 00:25:50,930 --> 00:25:51,140 OK. 445 00:25:51,140 --> 00:25:54,330 Let's put that on a fourth board. 446 00:25:54,330 --> 00:26:01,910 How to deal then with the ratio of f over g. 447 00:26:01,910 --> 00:26:08,230 Well, what I know is the product rule, right? 448 00:26:08,230 --> 00:26:14,290 So let me multiply both sides by g of x and get a product. 449 00:26:18,580 --> 00:26:21,160 There, that looks better. 450 00:26:21,160 --> 00:26:24,950 Of course the part that I don't know is in here, but 451 00:26:24,950 --> 00:26:26,450 just fire away. 452 00:26:26,450 --> 00:26:27,935 Take the derivative of both sides. 453 00:26:27,935 --> 00:26:29,060 OK. 454 00:26:29,060 --> 00:26:33,810 The derivative of the left side is df dx, of course. 455 00:26:33,810 --> 00:26:35,990 Now I can use the product rule. 456 00:26:35,990 --> 00:26:39,730 It's g of x, dq dx. 457 00:26:39,730 --> 00:26:42,950 That's the very, very thing I'm wanting. 458 00:26:42,950 --> 00:26:43,590 dq dx-- 459 00:26:43,590 --> 00:26:47,860 that's my big empty space. 460 00:26:47,860 --> 00:26:49,770 That's going to be the quotient rule. 461 00:26:49,770 --> 00:26:56,730 And then the second one is q of x times dg dx. 462 00:26:59,900 --> 00:27:03,040 That's the product rule applied to this. 463 00:27:03,040 --> 00:27:04,360 Now I have it. 464 00:27:04,360 --> 00:27:06,060 I've got dq dx. 465 00:27:06,060 --> 00:27:09,440 Well, I've got to get it by itself. 466 00:27:09,440 --> 00:27:12,010 I want to get dq dx by itself. 467 00:27:12,010 --> 00:27:16,200 So I'm going to move this part over there. 468 00:27:16,200 --> 00:27:18,840 Let me, even, multiply both sides-- 469 00:27:18,840 --> 00:27:23,980 this q, of course, I recognize as f times g. 470 00:27:23,980 --> 00:27:27,080 This is f of x times g of x. 471 00:27:27,080 --> 00:27:29,350 That's what q was. 472 00:27:29,350 --> 00:27:32,362 Now I'm going to-- 473 00:27:32,362 --> 00:27:35,360 oh, was not. 474 00:27:35,360 --> 00:27:37,500 It was f of x over g of x. 475 00:27:37,500 --> 00:27:38,750 Good Lord. 476 00:27:41,900 --> 00:27:43,900 You would never have allowed me to go on. 477 00:27:43,900 --> 00:27:45,420 OK. 478 00:27:45,420 --> 00:27:46,270 Good. 479 00:27:46,270 --> 00:27:49,920 This is came from the product rule, and now my final job is 480 00:27:49,920 --> 00:27:55,610 just to isolate dq dx and see what I've got. 481 00:27:55,610 --> 00:27:59,820 What I'll have will be the quotient rule. 482 00:27:59,820 --> 00:28:04,460 One good way is if I multiply both sides by g. 483 00:28:04,460 --> 00:28:11,410 So I multiply everything by g, so here's a g, df dx. 484 00:28:11,410 --> 00:28:14,890 And now this guy I'm going to bring over to the other side. 485 00:28:14,890 --> 00:28:17,870 When I multiply that by g, that just knocks that out. 486 00:28:17,870 --> 00:28:19,620 When I bring it over, it comes over with a 487 00:28:19,620 --> 00:28:22,480 minus sign, f dg dx. 488 00:28:26,440 --> 00:28:31,040 And this one got multiplied by g, so right now I'm looking at 489 00:28:31,040 --> 00:28:36,830 g squared, dq dx. 490 00:28:36,830 --> 00:28:38,080 The guy I want. 491 00:28:44,190 --> 00:28:46,850 Again, just algebra. 492 00:28:46,850 --> 00:28:49,170 Moving stuff from one side to the other 493 00:28:49,170 --> 00:28:50,940 produced the minus sign. 494 00:28:50,940 --> 00:28:54,310 Multiplying by g, you see what happened. 495 00:28:54,310 --> 00:28:56,420 So what do I now finally do? 496 00:28:56,420 --> 00:28:58,380 I'm ready to write this formula in. 497 00:29:02,220 --> 00:29:03,380 I've got it there. 498 00:29:03,380 --> 00:29:06,690 I've got dq dx, just as soon as I divide 499 00:29:06,690 --> 00:29:10,410 both sides by g squared. 500 00:29:10,410 --> 00:29:12,470 So let me write that left-hand side. 501 00:29:12,470 --> 00:29:23,910 g df dx minus f dg dx, and I have to divide everything-- 502 00:29:23,910 --> 00:29:26,240 this g squared has got to come down here. 503 00:29:26,240 --> 00:29:28,060 It's a little bit messier formula but 504 00:29:28,060 --> 00:29:29,540 you get used to it. 505 00:29:29,540 --> 00:29:30,790 g squared. 506 00:29:34,210 --> 00:29:37,380 That's the quotient rule. 507 00:29:37,380 --> 00:29:38,750 Can I say it in words? 508 00:29:38,750 --> 00:29:41,390 Because I actually say those words to myself 509 00:29:41,390 --> 00:29:46,680 every time I use it. 510 00:29:46,680 --> 00:29:50,790 So here are the words I say, because that's a kind of 511 00:29:50,790 --> 00:29:52,060 messy-looking expression. 512 00:29:52,060 --> 00:29:54,270 But if you just think about words-- 513 00:29:54,270 --> 00:29:59,280 so for me, remember we're dealing with f over g. f is 514 00:29:59,280 --> 00:30:01,170 the top, g at the bottom. 515 00:30:01,170 --> 00:30:04,830 So I say to myself, the bottom times the derivative of the 516 00:30:04,830 --> 00:30:10,160 top minus the top times the derivative of the bottom, 517 00:30:10,160 --> 00:30:14,630 divided by the bottom squared. 518 00:30:14,630 --> 00:30:16,910 That wasn't brilliant, but anyway, I 519 00:30:16,910 --> 00:30:18,900 remember it that way. 520 00:30:18,900 --> 00:30:19,730 OK. 521 00:30:19,730 --> 00:30:25,645 so now, finally, I'm ready to go further with this pattern. 522 00:30:29,500 --> 00:30:33,010 I still like that pattern. 523 00:30:33,010 --> 00:30:37,330 We've got the quotient rule, so the two rules are now set, 524 00:30:37,330 --> 00:30:40,810 and I want to do one last example before stopping. 525 00:30:40,810 --> 00:30:44,550 And that example is going to be a quotient, of course. 526 00:30:44,550 --> 00:30:52,330 And it might as well be a negative power of x. 527 00:30:52,330 --> 00:30:56,650 So now my example-- 528 00:30:56,650 --> 00:30:59,460 last example for today-- 529 00:30:59,460 --> 00:31:03,690 my quotient is going to be 1. 530 00:31:03,690 --> 00:31:09,380 The f of x will be 1 and the g of x-- so this is my f. 531 00:31:09,380 --> 00:31:11,360 This is my g. 532 00:31:11,360 --> 00:31:13,220 I have a ratio of two things. 533 00:31:13,220 --> 00:31:21,285 And as I've said, this is x to the minus n. 534 00:31:21,285 --> 00:31:23,570 Right? 535 00:31:23,570 --> 00:31:25,650 That's what we mean. 536 00:31:25,650 --> 00:31:29,800 We can think again about exponents. 537 00:31:29,800 --> 00:31:37,520 A negative exponent becomes positive when it's in the 538 00:31:37,520 --> 00:31:38,930 denominator. 539 00:31:38,930 --> 00:31:43,880 And we want it in the denominator so we can use this 540 00:31:43,880 --> 00:31:45,820 crazy quotient rule. 541 00:31:45,820 --> 00:31:46,280 All right. 542 00:31:46,280 --> 00:31:49,120 So let me think through the quotient rule. 543 00:31:49,120 --> 00:31:58,580 So the derivative of this ratio, which is x to the minus 544 00:31:58,580 --> 00:32:02,750 n That's the q, is 1 over x to the n. 545 00:32:02,750 --> 00:32:04,260 The derivative is-- 546 00:32:04,260 --> 00:32:06,800 OK, ready for the quotient rule? 547 00:32:06,800 --> 00:32:11,950 Bottom times the derivative of the top-- 548 00:32:11,950 --> 00:32:14,350 ah, but the top's just a constant, so its 549 00:32:14,350 --> 00:32:16,350 derivative is 0-- 550 00:32:16,350 --> 00:32:17,390 minus-- 551 00:32:17,390 --> 00:32:21,210 remembering that minus-- the top times the derivative of 552 00:32:21,210 --> 00:32:22,460 the bottom. 553 00:32:24,260 --> 00:32:24,445 Ha. 554 00:32:24,445 --> 00:32:27,190 Now we have a chance to use our 555 00:32:27,190 --> 00:32:30,550 pattern with a plus exponent. 556 00:32:30,550 --> 00:32:36,620 The derivative of the bottom is nx to the n minus 1. 557 00:32:36,620 --> 00:32:40,230 So it's two terms, again, but with a minus sign. 558 00:32:40,230 --> 00:32:44,830 And then the other thing I must remember is, divide by g 559 00:32:44,830 --> 00:32:51,520 squared, x to the n twice squared. 560 00:32:51,520 --> 00:32:52,790 OK. 561 00:32:52,790 --> 00:32:54,730 That's it. 562 00:32:54,730 --> 00:32:57,440 Of course, I'm going to simplify it, 563 00:32:57,440 --> 00:32:59,430 and then I'm done. 564 00:32:59,430 --> 00:33:01,280 So this is 0. 565 00:33:01,280 --> 00:33:02,680 Gone. 566 00:33:02,680 --> 00:33:07,170 This is minus n, which I like. 567 00:33:07,170 --> 00:33:09,690 I like to see minus n come down. 568 00:33:09,690 --> 00:33:12,620 That's my pattern, that this exponent should come down. 569 00:33:12,620 --> 00:33:15,450 Minus n, and then I want to see-- oh, what 570 00:33:15,450 --> 00:33:16,480 else do I have here? 571 00:33:16,480 --> 00:33:18,050 What's the power of x? 572 00:33:18,050 --> 00:33:20,440 Well, here I have an x to the n-th. 573 00:33:20,440 --> 00:33:25,930 And here I have, twice, so can I cancel this one and just 574 00:33:25,930 --> 00:33:28,540 keep this one? 575 00:33:28,540 --> 00:33:30,930 So I still have an x to the minus 1. 576 00:33:30,930 --> 00:33:33,530 I don't let him go. 577 00:33:33,530 --> 00:33:36,430 Actually the pattern's here. 578 00:33:36,430 --> 00:33:43,980 The answer is minus n minus capital N, which was the 579 00:33:43,980 --> 00:33:49,700 exponent, times x to one smaller power. 580 00:33:49,700 --> 00:33:52,690 This is x to the minus n, and then there's another x 581 00:33:52,690 --> 00:33:54,060 to the minus 1. 582 00:33:54,060 --> 00:34:01,770 The final result was that the derivative is minus nx to the 583 00:34:01,770 --> 00:34:04,950 minus n, minus 1. 584 00:34:04,950 --> 00:34:11,360 And that's the good pattern that matches here. 585 00:34:11,360 --> 00:34:17,460 When little n matches minus big N, that pattern is the 586 00:34:17,460 --> 00:34:18,199 same as that. 587 00:34:18,199 --> 00:34:24,860 So we now have the derivatives of powers of x as an example 588 00:34:24,860 --> 00:34:28,830 from the quotient rule and the product rule. 589 00:34:28,830 --> 00:34:30,407 Well, I just have to say one thing. 590 00:34:30,407 --> 00:34:31,920 We haven't got-- 591 00:34:31,920 --> 00:34:38,820 We've fractions, we've got negative numbers, but we don't 592 00:34:38,820 --> 00:34:42,469 have a whole lot of other numbers, like pi. 593 00:34:42,469 --> 00:34:45,989 We don't know what is, for example, the 594 00:34:45,989 --> 00:34:49,110 derivative of x to the pi. 595 00:34:49,110 --> 00:34:51,150 Because pi isn't-- 596 00:34:51,150 --> 00:34:54,909 pi is positive, so we're OK in the product rule, but it's not 597 00:34:54,909 --> 00:34:56,820 a fraction and we haven't got it yet. 598 00:34:59,370 --> 00:35:02,400 What do you think it is? 599 00:35:02,400 --> 00:35:03,490 You're right-- 600 00:35:03,490 --> 00:35:09,520 it is pi x to the pi minus 1. 601 00:35:09,520 --> 00:35:13,340 Well, actually I never met x to the pi in my life, until 602 00:35:13,340 --> 00:35:21,340 just there, but I've certainly met all kinds of powers of x 603 00:35:21,340 --> 00:35:25,310 and this is just one more example. 604 00:35:25,310 --> 00:35:25,350 OK. 605 00:35:25,350 --> 00:35:26,600 So that's quotient rule-- 606 00:35:28,150 --> 00:35:36,270 first came product rule, power rule, and then quotient rule, 607 00:35:36,270 --> 00:35:39,770 leading to this calculation. 608 00:35:39,770 --> 00:35:42,200 Now, the quotient rule I can use for other things, like 609 00:35:42,200 --> 00:35:46,330 sine x over cosine x. 610 00:35:46,330 --> 00:35:50,400 We're far along, and one more big rule will 611 00:35:50,400 --> 00:35:52,440 be the chain rule. 612 00:35:52,440 --> 00:35:54,770 OK, that's for another time. 613 00:35:54,770 --> 00:35:55,976 Thank you. 614 00:35:55,976 --> 00:35:56,410 [NARRATOR:] 615 00:35:56,410 --> 00:35:57,770 This has been a production of MIT 616 00:35:57,770 --> 00:36:00,600 OpenCourseWare and Gilbert Strang. 617 00:36:00,600 --> 00:36:02,870 Funding for this video was provided by the Lord 618 00:36:02,870 --> 00:36:04,090 Foundation. 619 00:36:04,090 --> 00:36:07,220 To help OCW continue to provide free and open access 620 00:36:07,220 --> 00:36:10,300 to MIT courses, please make a donation at 621 00:36:10,300 --> 00:36:11,860 ocw.mit.edu/donate.