1 00:00:00,040 --> 00:00:01,940 ANNOUNCER: The following content is provided under a 2 00:00:01,940 --> 00:00:03,690 Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:15,300 hundreds of MIT courses, visit mitopencourseware@ocw.mit.edu. 7 00:00:32,840 --> 00:00:33,430 PROFESSOR: Hi. 8 00:00:33,430 --> 00:00:36,300 Welcome once again to our lectures in Calculus 9 00:00:36,300 --> 00:00:39,440 Revisited, where today we are going to talk about the 10 00:00:39,440 --> 00:00:42,350 calculus of composite functions. 11 00:00:42,350 --> 00:00:46,120 Now, recall that we have already mentioned in previous 12 00:00:46,120 --> 00:00:49,420 lectures the notion of a composite function. 13 00:00:49,420 --> 00:00:53,490 And what we're going to do today is to emphasize the idea 14 00:00:53,490 --> 00:00:58,250 as to how often we are called upon to find functional 15 00:00:58,250 --> 00:01:01,500 relationships, where the first variable is given in terms of 16 00:01:01,500 --> 00:01:04,730 a second variable, and the second variable, say, is given 17 00:01:04,730 --> 00:01:06,590 in terms of the third variable. 18 00:01:06,590 --> 00:01:09,210 And we wish to find, say, the first variable in 19 00:01:09,210 --> 00:01:10,630 terms of the third. 20 00:01:10,630 --> 00:01:13,950 Fact here is where the name "the chain rule" seems to come 21 00:01:13,950 --> 00:01:17,770 from, a chain reaction where the variables are related in a 22 00:01:17,770 --> 00:01:18,940 chain this way. 23 00:01:18,940 --> 00:01:22,570 Now, we can see this quite easily in terms of a diagram. 24 00:01:22,570 --> 00:01:25,930 Suppose, for example, that I have a graph 25 00:01:25,930 --> 00:01:28,330 of 'x' versus 'y'. 26 00:01:28,330 --> 00:01:33,970 And suppose also, I have a graph of 't' versus 'x'. 27 00:01:33,970 --> 00:01:35,850 Without any reference to calculus-- 28 00:01:35,850 --> 00:01:37,210 and this is rather important-- 29 00:01:37,210 --> 00:01:40,470 without any reference to calculus, notice that these 30 00:01:40,470 --> 00:01:45,010 two graphs together allow me to visualize 31 00:01:45,010 --> 00:01:47,330 'y' in terms of 't'. 32 00:01:47,330 --> 00:01:50,430 For example, given a particular value of 't'-- 33 00:01:50,430 --> 00:01:52,210 let's call it 't sub 1'-- 34 00:01:52,210 --> 00:01:57,570 given a value of 't', from that value of 't', I can find 35 00:01:57,570 --> 00:01:59,710 the corresponding value of 'x'. 36 00:01:59,710 --> 00:02:01,780 Let's call that 'x sub 1'. 37 00:02:01,780 --> 00:02:06,330 Now, knowing 'x sub 1', I can come to this diagram. 38 00:02:06,330 --> 00:02:10,539 Knowing what 'x sub 1' is, I can find 'y sub 1'. 39 00:02:10,539 --> 00:02:14,590 And so you see in this chain of two diagrams, a particular 40 00:02:14,590 --> 00:02:20,670 value of 't' allows me to find a particular value of 'y'. 41 00:02:20,670 --> 00:02:25,560 And in this particular way, I can visualize 'y' as a 42 00:02:25,560 --> 00:02:27,420 function of 't'. 43 00:02:27,420 --> 00:02:30,550 And you see at this stage of the game, there is absolutely 44 00:02:30,550 --> 00:02:34,200 no need to have to have any knowledge of calculus to 45 00:02:34,200 --> 00:02:37,960 understand what it is that we're discussing. 46 00:02:37,960 --> 00:02:41,840 The place that calculus comes in is in the following way. 47 00:02:41,840 --> 00:02:45,400 Let's suppose it happened in this first diagram that the 48 00:02:45,400 --> 00:02:48,290 graph of 'y' versus 'x' was smooth. 49 00:02:48,290 --> 00:02:51,430 In other words, let's assume that 'y' is a differentiable 50 00:02:51,430 --> 00:02:52,580 function of 'x'. 51 00:02:52,580 --> 00:02:55,690 In particular, the way I've drawn this diagram here, we're 52 00:02:55,690 --> 00:03:01,810 saying suppose 'dy dx' evaluated at 'x' equals 'x1' 53 00:03:01,810 --> 00:03:03,850 happens to exist. 54 00:03:03,850 --> 00:03:09,220 And suppose also that this graph of 'x' versus 't', this 55 00:03:09,220 --> 00:03:12,460 also happens to be a smooth curve-- in other words, that 56 00:03:12,460 --> 00:03:15,100 'x' is a differentiable function of 't'. 57 00:03:15,100 --> 00:03:18,370 Again, in the language of calculus, what we're saying is 58 00:03:18,370 --> 00:03:21,340 the slope of this curve exists at this particular point, and 59 00:03:21,340 --> 00:03:26,710 it's given by the 'x dt' evaluated at 't' equals 't1'. 60 00:03:26,710 --> 00:03:28,980 Now, without going into a proof at this stage, all we're 61 00:03:28,980 --> 00:03:30,030 saying is this. 62 00:03:30,030 --> 00:03:33,990 We suspect that if 'y' is a differentiable function of 63 00:03:33,990 --> 00:03:36,930 'x', and 'x' is a differentiable function of 64 00:03:36,930 --> 00:03:40,880 't', that therefore, 'y' should also be a 65 00:03:40,880 --> 00:03:43,310 differentiable function of 't'. 66 00:03:43,310 --> 00:03:46,400 Notice it's not a conjecture at all that if 'y' is a 67 00:03:46,400 --> 00:03:49,750 function of 'x' and 'x' is a function of 't', that 'y' is a 68 00:03:49,750 --> 00:03:50,870 function of 't'. 69 00:03:50,870 --> 00:03:52,350 That part is clear. 70 00:03:52,350 --> 00:03:55,370 The conjecture is that we suspect that if 'y' is a 71 00:03:55,370 --> 00:03:58,420 differentiable function of 'x', and 'x' is a 72 00:03:58,420 --> 00:04:01,150 differentiable function of 't', that 'y' will be a 73 00:04:01,150 --> 00:04:03,310 differentiable function of 't'. 74 00:04:03,310 --> 00:04:06,870 In still other words, our suspicion is perhaps that a 75 00:04:06,870 --> 00:04:10,850 differentiable function of a differentiable function is 76 00:04:10,850 --> 00:04:13,490 again a differentiable function. 77 00:04:13,490 --> 00:04:17,120 But even more to the point, not only do we suspect, for 78 00:04:17,120 --> 00:04:22,250 example, that the 'dy dt' exists here when 't' equals 79 00:04:22,250 --> 00:04:27,590 't1', but in line with our lecture of last time, we might 80 00:04:27,590 --> 00:04:31,500 even begin to suspect, in terms of this fractional 81 00:04:31,500 --> 00:04:35,400 notation, that not only does the 'dy dt' exist at 't' 82 00:04:35,400 --> 00:04:39,930 equals 't1', but it can be found by multiplying 'dy dx' 83 00:04:39,930 --> 00:04:45,000 evaluated at 'x' equals 'x1' by the 'x dt' evaluated at 't' 84 00:04:45,000 --> 00:04:46,060 equals 't1'. 85 00:04:46,060 --> 00:04:51,190 Again, almost as if the 'dx' from the numerator here 86 00:04:51,190 --> 00:04:54,700 canceled with the 'dx' from the denominator here, the same 87 00:04:54,700 --> 00:04:56,430 as what we hope our differential 88 00:04:56,430 --> 00:04:58,120 notation would be. 89 00:04:58,120 --> 00:05:01,640 The question is granted that we would like a result like 90 00:05:01,640 --> 00:05:05,530 this to hold true, in a course such as calculus, where we're 91 00:05:05,530 --> 00:05:08,900 working with very tiny numbers and quotients of small 92 00:05:08,900 --> 00:05:12,620 numbers, places where we've seen that our intuition often 93 00:05:12,620 --> 00:05:16,390 leads us astray, it becomes fairly apparent that we had 94 00:05:16,390 --> 00:05:20,120 better have something stronger than just intuition in helping 95 00:05:20,120 --> 00:05:23,580 us derive certain results, no matter how natural these 96 00:05:23,580 --> 00:05:25,620 results might look. 97 00:05:25,620 --> 00:05:28,600 Now, the way we proceed here is as follows again, and 98 00:05:28,600 --> 00:05:30,970 notice again the building blocks of calculus. 99 00:05:30,970 --> 00:05:35,280 We go back to the fundamental result of last time. 100 00:05:35,280 --> 00:05:39,750 You see, after all, to find 'dy dt', we want 'delta y' 101 00:05:39,750 --> 00:05:42,950 divided by 'delta t', and then we'll take the limit as 'delta 102 00:05:42,950 --> 00:05:44,340 t' approaches 0. 103 00:05:44,340 --> 00:05:47,450 The question is, first of all, do we have a nice expression 104 00:05:47,450 --> 00:05:48,880 for 'delta y'? 105 00:05:48,880 --> 00:05:52,720 And in terms of the lecture of last time, we saw that if 'y' 106 00:05:52,720 --> 00:05:56,290 was a differentiable function of 'x', that 'x' equals 'x1', 107 00:05:56,290 --> 00:06:00,490 that 'delta y' was given by ''dy dx', evaluated 'x' equals 108 00:06:00,490 --> 00:06:06,330 'x1' times 'delta x'' plus 'k times delta x'-- 109 00:06:06,330 --> 00:06:09,320 and this is crucial now-- where the limit of 'k' as 110 00:06:09,320 --> 00:06:12,270 'delta x' approaches 0 was 0. 111 00:06:12,270 --> 00:06:16,170 Now you see, this recipe here is ironclad. 112 00:06:16,170 --> 00:06:18,780 I emphasized it from a geometric point of view last 113 00:06:18,780 --> 00:06:22,250 time, but you may recall that I proved it from an analytical 114 00:06:22,250 --> 00:06:22,930 point of view. 115 00:06:22,930 --> 00:06:25,360 In other words, whether you want to visualize this or 116 00:06:25,360 --> 00:06:27,240 derive it, it makes no difference. 117 00:06:27,240 --> 00:06:32,030 The key factors that this statement here is ironclad. 118 00:06:32,030 --> 00:06:34,710 It's something that we now know to be true in our 119 00:06:34,710 --> 00:06:36,510 so-called game of calculus. 120 00:06:36,510 --> 00:06:41,480 The point is, again, now how do we use this to check over 121 00:06:41,480 --> 00:06:43,550 our conjectured result? 122 00:06:43,550 --> 00:06:46,100 Again, the answer is almost straightforward. 123 00:06:46,100 --> 00:06:48,320 If you keep track of these things, you'll notice that 124 00:06:48,320 --> 00:06:51,860 calculus is a one-step-at-a-time procedure. 125 00:06:51,860 --> 00:06:54,240 Namely, we want 'dy dt'. 126 00:06:54,240 --> 00:06:57,710 That suggests we first want 'delta y' divided by 'delta 127 00:06:57,710 --> 00:07:00,340 t', and then we'll take the limit as 'delta 128 00:07:00,340 --> 00:07:02,280 t' approaches 0. 129 00:07:02,280 --> 00:07:04,010 So first, we do this. 130 00:07:04,010 --> 00:07:07,630 Namely, starting with our known recipe, we divide 131 00:07:07,630 --> 00:07:10,300 through by 'delta t', and why can we do this? 132 00:07:10,300 --> 00:07:14,120 We can do this because, of course, 'delta t' is not 0. 133 00:07:14,120 --> 00:07:17,810 Now we take the limit of both sides of the equality as 134 00:07:17,810 --> 00:07:19,630 'delta t' approaches 0. 135 00:07:19,630 --> 00:07:22,690 We observe that on the left-hand side, the limit of 136 00:07:22,690 --> 00:07:26,640 'delta y' divided by 'delta t', as 'delta t' approaches 0, 137 00:07:26,640 --> 00:07:30,780 is precisely 'dy dt', and in this particular case, 138 00:07:30,780 --> 00:07:33,090 evaluated at 't' equals 't1'. 139 00:07:33,090 --> 00:07:37,220 In other words, notice that the left-hand side here, as we 140 00:07:37,220 --> 00:07:41,420 let 'delta t' approach 0, becomes the left-hand side of 141 00:07:41,420 --> 00:07:43,220 our conjecture. 142 00:07:43,220 --> 00:07:46,070 Now we recall again that the limit of the sum is the sum of 143 00:07:46,070 --> 00:07:50,020 the limits, and we now take the limit of each of these 144 00:07:50,020 --> 00:07:52,570 terms separately, each term as a product. 145 00:07:52,570 --> 00:07:55,080 The limit of a product is the product of the limits. 146 00:07:55,080 --> 00:07:58,450 'dy dx' evaluated at 'x' equals 'x1' is a constant. 147 00:07:58,450 --> 00:08:01,450 In fact, that's just what, it's 'dy'. 148 00:08:01,450 --> 00:08:05,950 The limit of 'dy dx' evaluated 'x' equals 'x1', as 'delta t' 149 00:08:05,950 --> 00:08:11,530 approaches 0, is just 'dy dx' evaluated 'x' equals 'x1'. 150 00:08:11,530 --> 00:08:14,080 On the other hand, by definition, the limit of 151 00:08:14,080 --> 00:08:18,620 'delta x' divided by 'delta t', as 'delta t' approaches 0, 152 00:08:18,620 --> 00:08:21,460 is just 'dx dt'. 153 00:08:21,460 --> 00:08:25,380 And keeping track of the subscripts here, later on 154 00:08:25,380 --> 00:08:28,220 we'll become sloppy and leave the subscripts out. 155 00:08:28,220 --> 00:08:30,680 There really is no great harm done in 156 00:08:30,680 --> 00:08:32,610 calculus of a single variable. 157 00:08:32,610 --> 00:08:36,299 We shall find, in calculus of several variables, that it is 158 00:08:36,299 --> 00:08:41,980 extremely important to keep track of the subscripts and 159 00:08:41,980 --> 00:08:44,730 where the variables are being evaluated and things of this 160 00:08:44,730 --> 00:08:45,760 particular type. 161 00:08:45,760 --> 00:08:48,500 But I just want to get you used to the fact that these 162 00:08:48,500 --> 00:08:51,090 are specific numbers that we're using over here. 163 00:08:51,090 --> 00:08:54,220 Now let's continue. 164 00:08:54,220 --> 00:08:57,900 We take the limit of this term as 'delta t' approaches 0. 165 00:08:57,900 --> 00:09:04,620 We observe that this becomes 'dx dt', and the limit of 'k' 166 00:09:04,620 --> 00:09:06,550 as 'delta t' approaches 0-- 167 00:09:06,550 --> 00:09:09,770 well, as 'delta t' approaches 0, the fact that 'x' is a 168 00:09:09,770 --> 00:09:12,590 differentiable function of 't' means that 'delta x' 169 00:09:12,590 --> 00:09:13,820 approaches 0. 170 00:09:13,820 --> 00:09:19,310 And since the limit of 'k' as 'delta x' approaches 0 is 0, 171 00:09:19,310 --> 00:09:21,470 this term becomes 0. 172 00:09:21,470 --> 00:09:26,330 0 times anything is, any finite number, is 0. 173 00:09:26,330 --> 00:09:30,500 That means that this term here in the limit becomes 0, and 174 00:09:30,500 --> 00:09:35,700 we're left with the desired result. 175 00:09:35,700 --> 00:09:39,660 But notice that we did not arrive at this desired result 176 00:09:39,660 --> 00:09:40,830 by hand waving. 177 00:09:40,830 --> 00:09:45,140 We did not say this term 'delta x' is getting small, so 178 00:09:45,140 --> 00:09:46,510 it's becoming negligible. 179 00:09:46,510 --> 00:09:49,330 I can't emphasize this point enough that it is true that 180 00:09:49,330 --> 00:09:52,960 'delta x' is becoming small here, but so is 'delta t', and 181 00:09:52,960 --> 00:09:58,480 that indicates, essentially, you're 0 over 0 form. 182 00:09:58,480 --> 00:10:01,540 And the thing that saves us, the thing that makes this 183 00:10:01,540 --> 00:10:05,730 whole term drop out, is the key fact that 'k' itself goes 184 00:10:05,730 --> 00:10:08,800 to 0, as 'delta x' goes to 0. 185 00:10:08,800 --> 00:10:12,390 By the way, there are easier ways of intuitively trying to 186 00:10:12,390 --> 00:10:13,830 remember the chain rule. 187 00:10:13,830 --> 00:10:17,670 For example, one way that people often try to visualize 188 00:10:17,670 --> 00:10:19,160 the chain rule is this. 189 00:10:19,160 --> 00:10:21,930 They'll say, OK, we want 'dy dt'. 190 00:10:21,930 --> 00:10:25,100 So let's take 'delta y' divided by 'delta t', and then 191 00:10:25,100 --> 00:10:27,700 we'll take the limit as 'delta t' approaches 0. 192 00:10:27,700 --> 00:10:30,660 Now, you see in this notation here, 'delta y' and 'delta t' 193 00:10:30,660 --> 00:10:32,200 are actually numbers. 194 00:10:32,200 --> 00:10:36,130 As numbers, we can write these things in fractional notation, 195 00:10:36,130 --> 00:10:39,930 and we could write, what, that 'delta y' divided by 'delta t' 196 00:10:39,930 --> 00:10:44,150 is ''delta y' divided by 'delta x'' times ''delta x' 197 00:10:44,150 --> 00:10:45,750 divided by 'delta t''. 198 00:10:45,750 --> 00:10:49,550 Then we could take the limit, as delta t approaches 0, and 199 00:10:49,550 --> 00:10:51,640 we would arrive at the same result. 200 00:10:51,640 --> 00:10:55,710 But again, without trying to make this thing too 201 00:10:55,710 --> 00:10:59,130 obnoxiously long here, the thing to keep in mind is that 202 00:10:59,130 --> 00:11:00,990 'x' is a function of 't'. 203 00:11:00,990 --> 00:11:03,760 And from a rigorous point of view, the danger with this 204 00:11:03,760 --> 00:11:05,230 shortcut technique-- 205 00:11:05,230 --> 00:11:07,920 and it can be patched up but requires a great deal of 206 00:11:07,920 --> 00:11:09,420 mathematical analysis-- 207 00:11:09,420 --> 00:11:13,730 the danger here is that as 'delta t' approaches 0, it's 208 00:11:13,730 --> 00:11:16,530 quite possible that 'delta x' will be 0. 209 00:11:16,530 --> 00:11:18,750 In other words, it's possible that for a given change in 210 00:11:18,750 --> 00:11:20,810 't', there is no change in 'x'. 211 00:11:20,810 --> 00:11:25,030 Now, if 'delta x' happens to equal 0, then we're in 212 00:11:25,030 --> 00:11:26,900 trouble over here. 213 00:11:26,900 --> 00:11:30,560 In other words, in many cases, this shortened version gives 214 00:11:30,560 --> 00:11:32,650 us an idea as to what's going on. 215 00:11:32,650 --> 00:11:37,610 But our so-called longer method has no pitfalls to it. 216 00:11:37,610 --> 00:11:42,210 But enough said for what this recipe is. 217 00:11:42,210 --> 00:11:48,570 This result is known as the chain rule, and this will be 218 00:11:48,570 --> 00:11:51,640 the topic of the rest of today's lecture. 219 00:11:51,640 --> 00:11:53,480 Now, let's take a look at some of these things 220 00:11:53,480 --> 00:11:55,580 in a bit more detail. 221 00:11:55,580 --> 00:11:59,610 For example, let's look at an illustration. 222 00:11:59,610 --> 00:12:03,290 Suppose we want to find 'dy dx', if 'y' is equal to ''x 223 00:12:03,290 --> 00:12:05,630 squared + 1' squared'. 224 00:12:05,630 --> 00:12:08,030 Let me first do this problem the wrong way. 225 00:12:11,860 --> 00:12:14,570 Let's put a question mark over here. 226 00:12:14,570 --> 00:12:18,110 People learn things like, what, bring the exponent down 227 00:12:18,110 --> 00:12:22,240 and replace it by one less. 228 00:12:22,240 --> 00:12:25,350 Now certainly, if I bring the exponent down here, and 229 00:12:25,350 --> 00:12:29,302 replace it by one less, this is the answer that I get. 230 00:12:29,302 --> 00:12:31,920 Of course the question is, is this the right answer? 231 00:12:31,920 --> 00:12:36,620 Well, you see, notice one very nice way about finding out 232 00:12:36,620 --> 00:12:39,650 whether an answer is wrong, is to first find out by another 233 00:12:39,650 --> 00:12:41,740 way which is the right answer. 234 00:12:41,740 --> 00:12:45,020 For example, if 'y' equals ''x squared + 1' squared', it 235 00:12:45,020 --> 00:12:47,000 happens that we know how to square this thing. 236 00:12:47,000 --> 00:12:49,360 We can find directly that another way of 237 00:12:49,360 --> 00:12:50,980 expressing 'y' is what? 238 00:12:50,980 --> 00:12:55,740 It's 'x' to the fourth plus '2x squared' plus 1, but we 239 00:12:55,740 --> 00:12:59,800 have previously learned how to differentiate a polynomial. 240 00:12:59,800 --> 00:13:01,500 Through the polynomial is what, this is going to be 241 00:13:01,500 --> 00:13:08,220 what? '4 x cubed' plus '4x'. 242 00:13:08,220 --> 00:13:15,170 And you see somehow or other, this does not seem to give-- 243 00:13:15,170 --> 00:13:16,410 well, for one thing, we see that these are 244 00:13:16,410 --> 00:13:17,580 two different answers. 245 00:13:17,580 --> 00:13:19,630 For another thing, if this is the one that happens to be the 246 00:13:19,630 --> 00:13:22,860 right answer, this is the one that is the wrong answer. 247 00:13:22,860 --> 00:13:25,490 And since we know from previous material this is the 248 00:13:25,490 --> 00:13:28,130 right answer, there is something wrong with this 249 00:13:28,130 --> 00:13:30,390 regardless of how right it might look. 250 00:13:30,390 --> 00:13:33,200 In fact, how much are we off by over here? 251 00:13:37,010 --> 00:13:38,660 If we factor this thing out, what can we do? 252 00:13:38,660 --> 00:13:44,700 We can write this as '4x' times 'x squared + 1'. 253 00:13:44,700 --> 00:13:47,000 And what we really had over here was twice 254 00:13:47,000 --> 00:13:48,420 'x squared + 1'. 255 00:13:48,420 --> 00:13:52,630 It seems that the correction factor is '2x'. 256 00:13:52,630 --> 00:13:55,910 Now again, notice that the derivative of what's inside 257 00:13:55,910 --> 00:14:00,360 the parentheses over here just happens to be exactly '2x'. 258 00:14:00,360 --> 00:14:03,420 Now how does the chain rule come into play in a 259 00:14:03,420 --> 00:14:05,140 problem of this type? 260 00:14:05,140 --> 00:14:08,810 You see, the thing is, that what we should do over here is 261 00:14:08,810 --> 00:14:10,100 rewrite this. 262 00:14:10,100 --> 00:14:14,970 Namely, for example, let 'u' equal 'x squared + 1'. 263 00:14:14,970 --> 00:14:21,150 Then what this says is what? 'y' is equal to 'u squared', 264 00:14:21,150 --> 00:14:25,780 where 'u' is equal to 'x squared + 1'. 265 00:14:25,780 --> 00:14:28,000 This is just another way of writing this, and in this 266 00:14:28,000 --> 00:14:30,760 particular form, the chain rule seems to 267 00:14:30,760 --> 00:14:32,470 be emphasized more. 268 00:14:32,470 --> 00:14:36,540 You see, 'y' is a function of 'u', 'u' is a function of 'x'. 269 00:14:36,540 --> 00:14:39,630 Notice that from the first equation, it is relatively 270 00:14:39,630 --> 00:14:42,440 easy to find 'dy du'. 271 00:14:42,440 --> 00:14:45,760 In fact, it's just what, '2u'. 272 00:14:45,760 --> 00:14:47,950 We'll write that down later. 273 00:14:47,950 --> 00:14:51,640 From the second equation, it's easy to find 'du dx'. 274 00:14:51,640 --> 00:14:54,940 And by the chain rule, all we're saying is that 'dy du' 275 00:14:54,940 --> 00:14:58,140 times 'du dx' is 'dy dx'. 276 00:14:58,140 --> 00:14:59,960 See, what will that give us in this case? 277 00:14:59,960 --> 00:15:02,130 'dy du' is '2u'. 278 00:15:02,130 --> 00:15:05,610 'du dx' is '2x'. 279 00:15:05,610 --> 00:15:09,460 That gives us '4x' times 'u'. 280 00:15:09,460 --> 00:15:16,570 'u' is 'x squared + 1', and so this becomes '4x' times 'x 281 00:15:16,570 --> 00:15:18,170 squared + 1'. 282 00:15:18,170 --> 00:15:21,180 And if we now compare this with what was the correct 283 00:15:21,180 --> 00:15:26,250 answer, we see that in this case, 284 00:15:26,250 --> 00:15:28,060 everything worked out fine. 285 00:15:28,060 --> 00:15:31,750 I suppose what we should do here is to comment now on the 286 00:15:31,750 --> 00:15:35,280 danger of memorizing recipes without thoroughly 287 00:15:35,280 --> 00:15:36,580 understanding them. 288 00:15:36,580 --> 00:15:39,790 The idea, that said when you want to differentiate 289 00:15:39,790 --> 00:15:43,150 something raised to a power that you bring the power down 290 00:15:43,150 --> 00:15:47,450 and replace it by one less, hinged on the fact that the 291 00:15:47,450 --> 00:15:52,650 thing that was being raised to the power was the same 292 00:15:52,650 --> 00:15:55,830 variable with respect to which you were doing the 293 00:15:55,830 --> 00:15:57,380 differentiation. 294 00:15:57,380 --> 00:16:02,780 You see, for example, when we had 'y' equaled 'x squared', 295 00:16:02,780 --> 00:16:07,730 and then we wrote that 'dy dx' is '2x', the thing that was 296 00:16:07,730 --> 00:16:10,540 important over here was the fact that what? 297 00:16:10,540 --> 00:16:13,640 The thing that was being raised to the second power is 298 00:16:13,640 --> 00:16:17,320 precisely the variable with respect to which we were doing 299 00:16:17,320 --> 00:16:19,180 the differentiation. 300 00:16:19,180 --> 00:16:24,400 You see, in the problem 'y' equals 'x squared + 1' 301 00:16:24,400 --> 00:16:28,220 squared, the thing that was being raised to the second 302 00:16:28,220 --> 00:16:31,190 power was 'x squared + 1'. 303 00:16:31,190 --> 00:16:33,170 The variable with respect to which we were 304 00:16:33,170 --> 00:16:36,020 differentiating was 'x'. 305 00:16:36,020 --> 00:16:40,060 In other words, to write this thing more symbolically, if 306 00:16:40,060 --> 00:16:47,730 'y' is equal to something, square it, then the derivative 307 00:16:47,730 --> 00:16:51,390 that's equal to twice that something is the derivative of 308 00:16:51,390 --> 00:16:54,870 'y' with respect to that something. 309 00:16:54,870 --> 00:16:57,560 You see, the place the chain rule comes in is when the 310 00:16:57,560 --> 00:17:01,830 variable which appears here, is not the same as the 311 00:17:01,830 --> 00:17:06,150 variable which appears here, and we'll see this in greater 312 00:17:06,150 --> 00:17:08,390 detail as we go along. 313 00:17:08,390 --> 00:17:12,339 By the way, the chain rule comes up in another form known 314 00:17:12,339 --> 00:17:16,750 as parametric equations, and this form comes up very often. 315 00:17:16,750 --> 00:17:19,980 It's a twist of what we were talking about before. 316 00:17:19,980 --> 00:17:24,210 This is the situation in which frequently we want to compare 317 00:17:24,210 --> 00:17:25,589 two variables. 318 00:17:25,589 --> 00:17:28,580 Let's call them 'x' and 'y', all right? 319 00:17:28,580 --> 00:17:32,730 And it happens that both variables, 'x' and 'y', can be 320 00:17:32,730 --> 00:17:36,760 expressed more simply in terms of a third variable, 't'. 321 00:17:36,760 --> 00:17:40,040 And frequently, what one does is try to talk about the 322 00:17:40,040 --> 00:17:43,650 relationship that exists between 'y' and 'x' in terms 323 00:17:43,650 --> 00:17:47,120 of eliminating t between these two equations. 324 00:17:47,120 --> 00:17:50,390 By the way, in terms of differential language, there 325 00:17:50,390 --> 00:17:52,960 seems to be an easier way of handling this. 326 00:17:52,960 --> 00:17:56,380 Namely, you see, if we differentiate the first 327 00:17:56,380 --> 00:18:01,860 equation, we get what, that 'dy dt' is 'f prime of t'. 328 00:18:01,860 --> 00:18:06,610 If we differentiate the second equation, we get that 'dx dt' 329 00:18:06,610 --> 00:18:08,660 is 'g prime of t'. 330 00:18:08,660 --> 00:18:13,590 Now if, as we said in our last lecture, we can pretend that 331 00:18:13,590 --> 00:18:15,460 this is really a fraction, that it's 332 00:18:15,460 --> 00:18:17,550 'dy' divided by 'dt'-- 333 00:18:17,550 --> 00:18:21,090 in other words, if we think of 'dy' as being 'delta y-tan', 334 00:18:21,090 --> 00:18:25,430 of 'dx' as being 'delta x-tan', and 'dt' as being 335 00:18:25,430 --> 00:18:29,090 'delta t', it would appear that we could say, what, that 336 00:18:29,090 --> 00:18:36,980 'dy dt' divided by 'dx dt' would just be what? 337 00:18:36,980 --> 00:18:38,020 'dy dx'. 338 00:18:38,020 --> 00:18:42,890 In other words, ''dy' divided by 'dt'' divided by ''dx' 339 00:18:42,890 --> 00:18:45,600 divided by 'dt'', which is what this would say if this 340 00:18:45,600 --> 00:18:49,670 was in differential form, would just be 'dy dx'. 341 00:18:49,670 --> 00:18:52,570 In other words, we get the feeling that to find the 342 00:18:52,570 --> 00:18:56,580 derivative here, all we have to do is differentiate 'y' 343 00:18:56,580 --> 00:18:59,870 with respect to 't', and divide that by the derivative 344 00:18:59,870 --> 00:19:01,910 of 'x' with respect to 't'. 345 00:19:04,440 --> 00:19:07,110 And by the way, you see, this becomes a particularly 346 00:19:07,110 --> 00:19:11,590 powerful tool in those computational cases where we 347 00:19:11,590 --> 00:19:16,090 do not know how to eliminate 't', and to solve specifically 348 00:19:16,090 --> 00:19:17,860 for 'y' in terms of 'x'. 349 00:19:17,860 --> 00:19:20,700 You see, in terms of this particular recipe over here, 350 00:19:20,700 --> 00:19:24,890 we are allowed to leave 'x' and 'y' in terms of 't'. 351 00:19:24,890 --> 00:19:28,500 Again, the same old bugaboo comes up to plague us. 352 00:19:28,500 --> 00:19:32,420 The fact that something seems natural is not enough to allow 353 00:19:32,420 --> 00:19:35,060 us to believe that it's actually correct. 354 00:19:35,060 --> 00:19:38,930 Is there a more rigorous way of obtaining the same result? 355 00:19:38,930 --> 00:19:41,070 Again, the answer is yes. 356 00:19:41,070 --> 00:19:43,920 And not only is the answer yes, but it goes back to the 357 00:19:43,920 --> 00:19:46,370 fundamental recipe that we were discussing in our 358 00:19:46,370 --> 00:19:47,690 previous lecture. 359 00:19:47,690 --> 00:19:52,980 Namely, we know that 'delta y' is ''f prime of t' times 360 00:19:52,980 --> 00:20:00,850 'delta t'' plus 'k1 delta t', and the 'delta x' is ''g prime 361 00:20:00,850 --> 00:20:06,320 of t' times 'delta t'' plus 'k2 delta t', where both the 362 00:20:06,320 --> 00:20:13,260 limit of 'k1' and 'k2' as 'delta t' approach 0. 363 00:20:13,260 --> 00:20:15,490 And this is a notation, I think, that takes a while to 364 00:20:15,490 --> 00:20:16,510 get used to. 365 00:20:16,510 --> 00:20:19,570 We're used to seeing letters like 'k' stand for constants, 366 00:20:19,570 --> 00:20:22,650 but it's important over here to understand that 'k1' and 367 00:20:22,650 --> 00:20:26,670 'k2' are functions of 'delta t', that the difference 368 00:20:26,670 --> 00:20:30,440 between 'delta y' and 'delta y-tan', 'delta x' and 'delta 369 00:20:30,440 --> 00:20:35,260 x-tan', that difference, which is 'k delta x' or 'k delta y', 370 00:20:35,260 --> 00:20:37,820 depending on which problem we're dealing with that's 371 00:20:37,820 --> 00:20:44,670 certainly 'k' in that case, does depend on how big 'delta 372 00:20:44,670 --> 00:20:46,030 t' happens to be. 373 00:20:46,030 --> 00:20:49,430 At any rate, the important thing is that as 'delta t' 374 00:20:49,430 --> 00:20:52,430 approaches 0, these go to 0 also. 375 00:20:52,430 --> 00:20:56,570 Now you see if we take this, and actually compute 'delta y' 376 00:20:56,570 --> 00:20:58,005 divided by 'delta x'-- 377 00:21:01,010 --> 00:21:06,940 and we'll write this a little bit more suggestively, factor 378 00:21:06,940 --> 00:21:10,325 out 'delta t' from both numerator and denominator-- 379 00:21:14,020 --> 00:21:16,740 it rigorously tells us what 'delta y' divided 380 00:21:16,740 --> 00:21:18,400 by 'delta x' is. 381 00:21:18,400 --> 00:21:23,890 Now we take the limit, as 'delta x' approaches 0. 382 00:21:23,890 --> 00:21:26,300 That, by definition, is what? 383 00:21:26,300 --> 00:21:28,180 That's by definition 'dy dx'. 384 00:21:28,180 --> 00:21:33,460 Well, you see, first of all, we cancel out the 'delta t' 385 00:21:33,460 --> 00:21:37,510 over here, see, 'delta t' is not 0, we're assuming. 386 00:21:37,510 --> 00:21:40,370 Since it's not 0 it can be canceled out, and once we've 387 00:21:40,370 --> 00:21:43,950 canceled out 'delta t', notice that as 'delta t' approaches 388 00:21:43,950 --> 00:21:47,320 0, so does 'delta x'. 389 00:21:47,320 --> 00:21:49,130 As 'delta x' approaches 0, so does 'delta t'. 390 00:21:49,130 --> 00:21:52,050 That makes 'k1' and 'k2' go to 0. 391 00:21:52,050 --> 00:21:54,660 And then since the limit of a quotient is the quotient of 392 00:21:54,660 --> 00:21:58,530 the limits, provided only the 'g prime of t' is not 0, we 393 00:21:58,530 --> 00:22:01,330 see that in the eliminating process, we 394 00:22:01,330 --> 00:22:04,150 get the same answer. 395 00:22:04,150 --> 00:22:07,170 And by the way, see, once we get the same answer, as we 396 00:22:07,170 --> 00:22:11,050 would have got the short way, then we can use the 397 00:22:11,050 --> 00:22:13,500 convenience of the short recipe. 398 00:22:13,500 --> 00:22:17,430 However, the fact that the short recipe was nice is not 399 00:22:17,430 --> 00:22:19,620 enough of a guarantee that it was giving 400 00:22:19,620 --> 00:22:21,050 us the correct answer. 401 00:22:21,050 --> 00:22:25,570 As a case in point, it's rather interesting to point 402 00:22:25,570 --> 00:22:28,200 out that if you want the second derivative-- 403 00:22:28,200 --> 00:22:29,940 in other words, let's recall what we have here. 404 00:22:29,940 --> 00:22:36,770 We have what? 'y' was given to 'b', say 'f of t'. 405 00:22:36,770 --> 00:22:39,960 'x' was given by 'g of t'. 406 00:22:39,960 --> 00:22:43,100 And you see from these two equations, what we 407 00:22:43,100 --> 00:22:44,450 could do is find what? 408 00:22:44,450 --> 00:22:48,660 We could find the second derivative of 'y with respect 409 00:22:48,660 --> 00:22:52,310 to t', and we could also find from this equation the second 410 00:22:52,310 --> 00:22:54,450 derivative of 'x' with respect to 't'. 411 00:22:54,450 --> 00:22:56,280 This we could certainly do. 412 00:22:56,280 --> 00:22:59,680 And mechanically, we could certainly say, let's cancel 413 00:22:59,680 --> 00:23:01,770 the common denominator. 414 00:23:01,770 --> 00:23:04,490 The interesting thing is that when you form that quotient, 415 00:23:04,490 --> 00:23:08,170 whatever that quotient is, it does not come out to be the 416 00:23:08,170 --> 00:23:10,900 second derivative of 'y' with respect to 'x'. 417 00:23:10,900 --> 00:23:13,990 And there is an interesting piece of folklore over here. 418 00:23:13,990 --> 00:23:17,040 I don't know if this ever bothered you or not, but it 419 00:23:17,040 --> 00:23:18,170 used to bother me. 420 00:23:18,170 --> 00:23:20,720 I never understood why, when you talk about the second 421 00:23:20,720 --> 00:23:24,130 derivative, that the exponent was written between the 'd' 422 00:23:24,130 --> 00:23:28,965 and the variable in one case, but written at the end in the 423 00:23:28,965 --> 00:23:29,270 other case. 424 00:23:29,270 --> 00:23:32,010 In other words, notice that the 2 here appears between the 425 00:23:32,010 --> 00:23:34,340 'd' and the 'y', but in the denominator, 426 00:23:34,340 --> 00:23:35,990 the 'd' appears outside. 427 00:23:35,990 --> 00:23:39,510 And again, it was the foresight of the fathers of 428 00:23:39,510 --> 00:23:42,950 differential calculus who noticed rather interestingly 429 00:23:42,950 --> 00:23:47,190 that if mechanically you did agree to cancel the common 430 00:23:47,190 --> 00:23:51,560 denominator here, that what you would wind up with is not 431 00:23:51,560 --> 00:23:56,900 'd2y dx squared', but rather what? 432 00:23:56,900 --> 00:24:00,860 'd2y d2x'. 433 00:24:00,860 --> 00:24:03,280 In other words, if you mechanically carried this out, 434 00:24:03,280 --> 00:24:06,390 notice that the notation would be incorrect. 435 00:24:06,390 --> 00:24:10,390 The 2 comes out to be in the wrong place over here. 436 00:24:10,390 --> 00:24:14,750 You see, again, the interesting point is we don't 437 00:24:14,750 --> 00:24:17,800 have to rely on taking my word for it. 438 00:24:17,800 --> 00:24:21,220 Somebody might say to me, now look, all you've told me is 439 00:24:21,220 --> 00:24:25,120 that I get the wrong answer solving this problem this 440 00:24:25,120 --> 00:24:26,600 particular way. 441 00:24:26,600 --> 00:24:29,040 And you've given me a nice lecture about how the 2's come 442 00:24:29,040 --> 00:24:30,410 out the wrong way and everything. 443 00:24:30,410 --> 00:24:33,456 How do I know that this is the wrong answer? 444 00:24:33,456 --> 00:24:35,930 See, and again, everything comes back to 445 00:24:35,930 --> 00:24:37,360 fundamentals again. 446 00:24:37,360 --> 00:24:42,500 To find 'd2y dx squared', observe that by definition, 447 00:24:42,500 --> 00:24:46,210 that's just 'd dx' of 'dy dx'. 448 00:24:49,220 --> 00:24:51,470 That definition doesn't depend on what functions we're 449 00:24:51,470 --> 00:24:51,940 dealing with. 450 00:24:51,940 --> 00:24:54,300 The second derivative with respect to 'x' is the 451 00:24:54,300 --> 00:24:57,450 derivative with respect to 'x' of the first derivative. 452 00:24:57,450 --> 00:25:00,730 Now, once we have this, you see, knowing from our previous 453 00:25:00,730 --> 00:25:03,620 case, that what? 454 00:25:03,620 --> 00:25:10,550 'dy dx' was 'f prime of t' divided by 'g prime of t'. 455 00:25:10,550 --> 00:25:12,580 We can now do what? 456 00:25:12,580 --> 00:25:13,890 Take this derivative. 457 00:25:13,890 --> 00:25:16,240 By the way, again, notice how the chain 458 00:25:16,240 --> 00:25:18,330 rule comes up in practice. 459 00:25:18,330 --> 00:25:20,220 It's not always dictated to us. 460 00:25:20,220 --> 00:25:24,000 If you look at the expression inside the parentheses, what 461 00:25:24,000 --> 00:25:25,110 do we have? 462 00:25:25,110 --> 00:25:28,400 Inside the parentheses, we have a function of 't' only. 463 00:25:28,400 --> 00:25:29,830 This is a function of 't'. 464 00:25:29,830 --> 00:25:32,780 We want to differentiate it with respect to 'x'. 465 00:25:32,780 --> 00:25:35,880 The most natural variable to differentiate a function of 466 00:25:35,880 --> 00:25:39,210 't' with respect to is 't' itself. 467 00:25:39,210 --> 00:25:42,370 In other words, what would've been nice is if this was the 468 00:25:42,370 --> 00:25:47,780 derivative of 'f prime of t' over 'g prime of t', with 469 00:25:47,780 --> 00:25:49,970 respect to 't'. 470 00:25:49,970 --> 00:25:51,760 See, this would be easier to handle. 471 00:25:51,760 --> 00:25:53,330 We would then use the quotient rule, et cetera. 472 00:25:53,330 --> 00:25:55,930 You see, we can differentiate a function of 't' 473 00:25:55,930 --> 00:25:57,380 with respect to 't'. 474 00:25:57,380 --> 00:25:59,380 The trouble is we have the derivative 475 00:25:59,380 --> 00:26:00,570 with respect to 'x'. 476 00:26:00,570 --> 00:26:03,885 And if we just change this to a 't', that's cheating. 477 00:26:03,885 --> 00:26:06,600 See, I mean, you pretend you copy it wrong, because it's an 478 00:26:06,600 --> 00:26:08,610 easier problem to solve that way. 479 00:26:08,610 --> 00:26:12,010 The beauty of the chain rule is that it allows us to do the 480 00:26:12,010 --> 00:26:15,900 problem the easier way, and to doctor up the resulting 481 00:26:15,900 --> 00:26:18,300 incorrect answer by the right answer. 482 00:26:18,300 --> 00:26:22,320 Namely, you see what we wanted to wind up with here is what, 483 00:26:22,320 --> 00:26:24,860 the derivative not with respect to 't', but with 484 00:26:24,860 --> 00:26:26,810 respect to 'x'. 485 00:26:26,810 --> 00:26:30,730 And so, by using the chain rule, you see we do what? 486 00:26:30,730 --> 00:26:34,410 We take the derivative with respect to 't', multiply that 487 00:26:34,410 --> 00:26:35,520 by 'dt dx'-- 488 00:26:35,520 --> 00:26:38,950 again, mechanically, almost as if these canceled. 489 00:26:38,950 --> 00:26:42,320 But this is the way the chain rule works, and now, you see, 490 00:26:42,320 --> 00:26:45,450 I can work this out by the regular quotient 491 00:26:45,450 --> 00:26:46,890 rule, which says what? 492 00:26:46,890 --> 00:26:53,460 It's the denominator times the derivative of the numerator. 493 00:26:53,460 --> 00:26:55,530 See, and I am differentiating out respect to 't', the 494 00:26:55,530 --> 00:27:02,530 natural variable, minus the numerator times the derivative 495 00:27:02,530 --> 00:27:08,695 of the denominator over the square of the denominator. 496 00:27:12,790 --> 00:27:16,220 Now, that's a mess by itself, meaning, what, 497 00:27:16,220 --> 00:27:17,930 computationally, it's not that obvious. 498 00:27:17,930 --> 00:27:20,280 I mean, it's quite a bit of work to do here, and then that 499 00:27:20,280 --> 00:27:27,370 whole thing must be multiplied by 'dt dx'. 500 00:27:27,370 --> 00:27:31,500 And this, you see, is how one goes around finding the second 501 00:27:31,500 --> 00:27:34,010 derivative of 'y' with respect to 'x' in terms 502 00:27:34,010 --> 00:27:35,620 of parametric equations. 503 00:27:35,620 --> 00:27:38,630 And more than once, if you're not careful, you're going to 504 00:27:38,630 --> 00:27:42,150 find yourself making serious mistakes, by forgetting to put 505 00:27:42,150 --> 00:27:44,740 in this factor of 'dt dx'. 506 00:27:44,740 --> 00:27:48,450 By the way, an interesting point is that we have not 507 00:27:48,450 --> 00:27:49,910 computed 'dt dx'. 508 00:27:49,910 --> 00:27:52,325 We have computed 'dx dt'. 509 00:27:55,220 --> 00:27:56,680 Let's go back here. 510 00:27:56,680 --> 00:27:59,310 See, 'x' was 'g of t'. 511 00:27:59,310 --> 00:28:06,210 So from that, 'dx dt' is 'g prime of t'. 512 00:28:06,210 --> 00:28:09,730 And the question is if 'dx dt' is 'g prime of t', how does 513 00:28:09,730 --> 00:28:12,170 one find 'dt dx'? 514 00:28:12,170 --> 00:28:15,140 And again, I think your intuition is going to tell you 515 00:28:15,140 --> 00:28:17,490 to just take reciprocals. 516 00:28:17,490 --> 00:28:21,230 And again, the question is it's true that this suggests 517 00:28:21,230 --> 00:28:24,880 taking reciprocals, but how do we know that we can do this, 518 00:28:24,880 --> 00:28:27,950 and if we can do this, what does it really mean? 519 00:28:27,950 --> 00:28:30,730 You see, what this is leading into is what's going to be the 520 00:28:30,730 --> 00:28:34,720 subject of our lecture next time, called 'Inverse 521 00:28:34,720 --> 00:28:35,820 Functions'. 522 00:28:35,820 --> 00:28:38,550 And just to give you a preview of what that lecture is about, 523 00:28:38,550 --> 00:28:41,450 and how we work things like this, let's take a look at 524 00:28:41,450 --> 00:28:43,790 what we mean by inverse functions. 525 00:28:43,790 --> 00:28:46,300 Well, we won't even mention it in much detail. 526 00:28:46,300 --> 00:28:49,770 But let's take a look and see what's going on over here. 527 00:28:49,770 --> 00:28:51,360 Let's suppose that the first-- 528 00:28:51,360 --> 00:28:55,320 and by the way, I've started to abandon using the 't' over 529 00:28:55,320 --> 00:28:56,450 here all the time. 530 00:28:56,450 --> 00:28:59,310 I think those of us in engineering work primarily 531 00:28:59,310 --> 00:29:02,500 keep thinking of 't' as being time, and you may get the 532 00:29:02,500 --> 00:29:06,940 mistaken notion that if the variable isn't time, the thing 533 00:29:06,940 --> 00:29:09,000 doesn't work this way. 534 00:29:09,000 --> 00:29:11,870 In most cases, physically, the variable that we're interested 535 00:29:11,870 --> 00:29:13,030 in will be time. 536 00:29:13,030 --> 00:29:15,780 But just for the idea of getting you used to the fact 537 00:29:15,780 --> 00:29:17,780 that it makes no difference what the name of the variable 538 00:29:17,780 --> 00:29:20,320 is, I've taken the liberty of writing this slightly 539 00:29:20,320 --> 00:29:21,290 differently. 540 00:29:21,290 --> 00:29:24,570 Namely, I now assume that y is a differentiable function of 541 00:29:24,570 --> 00:29:26,200 'u', and that 'u' is a 542 00:29:26,200 --> 00:29:28,130 differentiable function of 'x'. 543 00:29:28,130 --> 00:29:31,940 By the chain rule, I now know that 'y' is a differentiable 544 00:29:31,940 --> 00:29:37,580 function of 'x', and that 'dy dx' is 'dy du' times 'du dx'. 545 00:29:37,580 --> 00:29:40,990 The interesting thing here is, is that there is nothing in 546 00:29:40,990 --> 00:29:44,080 the statement of the chain rule that says that the first 547 00:29:44,080 --> 00:29:47,030 variable in the third that 'x' and 'y' must 548 00:29:47,030 --> 00:29:48,280 be different variables. 549 00:29:48,280 --> 00:29:51,820 In fact, it might happen that 'x' and 'y' are synonyms for 550 00:29:51,820 --> 00:29:53,100 one another. 551 00:29:53,100 --> 00:29:55,500 If 'x' and 'y' happen to be synonyms-- 552 00:29:55,500 --> 00:29:57,600 suppose 'x' and 'y' are synonyms-- 553 00:29:57,600 --> 00:29:59,460 look what happens over here. 554 00:29:59,460 --> 00:30:04,740 'dy dx' is then just 'dy dy', which is 1. 555 00:30:04,740 --> 00:30:05,480 See, let's write that down. 556 00:30:05,480 --> 00:30:07,140 That's 'dy dy'. 557 00:30:07,140 --> 00:30:10,410 This would be 'dy du', and if 'x' is equal to 558 00:30:10,410 --> 00:30:13,040 'y', this is 'du dy'. 559 00:30:13,040 --> 00:30:18,700 And if this is equal to 1, and this is 'dy du', and this is 560 00:30:18,700 --> 00:30:23,160 'du dy', what does this tell us about the relationship 561 00:30:23,160 --> 00:30:26,110 between 'dy du' and 'du dy'? 562 00:30:26,110 --> 00:30:28,840 It says their product is 1. 563 00:30:28,840 --> 00:30:32,250 And if the product is 1, that by definition means that the 564 00:30:32,250 --> 00:30:35,330 two factors are reciprocals. 565 00:30:35,330 --> 00:30:38,440 Now, what I want you to observe over here is what this 566 00:30:38,440 --> 00:30:39,990 whole thing means. 567 00:30:39,990 --> 00:30:44,450 Namely, if 'y' happens to equal 'x', do you see what 568 00:30:44,450 --> 00:30:45,610 this thing says? 569 00:30:45,610 --> 00:30:48,090 It says that 'y' is a differentiable function of 570 00:30:48,090 --> 00:30:50,740 'u', and 'u' in turn is a 571 00:30:50,740 --> 00:30:53,050 differentiable function of 'y'. 572 00:30:53,050 --> 00:30:55,880 That's precisely what we meant when we talked 573 00:30:55,880 --> 00:30:57,480 about inverse functions. 574 00:30:57,480 --> 00:31:00,440 We don't know when an inverse function exists. 575 00:31:00,440 --> 00:31:03,960 All we're saying is, is that if 'f inverse' happens to 576 00:31:03,960 --> 00:31:09,480 exist over here, to find 'du dy', all we have to do is take 577 00:31:09,480 --> 00:31:12,560 the reciprocal of 'dy du'. 578 00:31:12,560 --> 00:31:15,240 Now again, this is going to be the 579 00:31:15,240 --> 00:31:17,230 subject of our next lecture. 580 00:31:17,230 --> 00:31:20,550 All I wanted to do was to make this aside for the time being. 581 00:31:20,550 --> 00:31:23,470 What I want to do to complete today's lecture is to get to 582 00:31:23,470 --> 00:31:24,910 something more tangible. 583 00:31:24,910 --> 00:31:27,140 See, now that we've talked about the chain rule, we've 584 00:31:27,140 --> 00:31:30,390 talked about inverse functions a little bit, and talked about 585 00:31:30,390 --> 00:31:33,100 these things from a highly theoretical point of view, 586 00:31:33,100 --> 00:31:34,980 let's go ahead and try to solve a 587 00:31:34,980 --> 00:31:36,550 particularly simple problem. 588 00:31:36,550 --> 00:31:38,830 By particularly simple, I mean this. 589 00:31:38,830 --> 00:31:42,230 I have chosen the numbers to come out in a very, very easy 590 00:31:42,230 --> 00:31:45,580 way, so we don't get lost in the maze of details. 591 00:31:45,580 --> 00:31:48,110 In other words, there was a danger that we will confuse 592 00:31:48,110 --> 00:31:51,450 the computational details with the theory. 593 00:31:51,450 --> 00:31:53,560 So to emphasize the theory, I've tried to pick a 594 00:31:53,560 --> 00:31:55,820 straightforward simple problem, but let's see how 595 00:31:55,820 --> 00:31:57,770 this thing works out. 596 00:31:57,770 --> 00:32:01,570 Let's suppose that we're given that 'y' is equal to 't to the 597 00:32:01,570 --> 00:32:06,010 fourth power', and 'x' is equal to 't squared'. 598 00:32:06,010 --> 00:32:08,850 What we would like to do-- and by the way, notice what this 599 00:32:08,850 --> 00:32:13,050 thing says, a given value of 't' determines both an 'x' and 600 00:32:13,050 --> 00:32:17,530 the 'y', so that makes 'x' and 'y' functionally related. 601 00:32:17,530 --> 00:32:21,200 Notice that from the first equation, we can find that 'dy 602 00:32:21,200 --> 00:32:25,350 dt' is '4 t cubed'. 603 00:32:25,350 --> 00:32:30,900 From the second equation, we can find that 'dx dt' is '2t'. 604 00:32:30,900 --> 00:32:36,280 And if we now use the chain rule, 'dy dx' will be what? 605 00:32:36,280 --> 00:32:42,580 It'll be 'dy dt' divided by 'dx dt', and 606 00:32:42,580 --> 00:32:48,140 that's just '2 t squared'. 607 00:32:48,140 --> 00:32:52,890 By the way, as a check, notice this. 608 00:32:52,890 --> 00:32:56,150 If 'y' is equal to 't to the fourth', and 'x' is equal to 609 00:32:56,150 --> 00:32:59,970 't squared', since 't to the fourth' is the square of 't 610 00:32:59,970 --> 00:33:04,280 squared', that says 'y' is equal to ''t squared' 611 00:33:04,280 --> 00:33:08,900 squared', 'y' is equal to 'x squared'. 612 00:33:08,900 --> 00:33:12,180 And if 'y' is equal to 'x squared', in this case, it's 613 00:33:12,180 --> 00:33:16,800 very easy to see that 'dy dx' is equal to '2x'. 614 00:33:16,800 --> 00:33:20,530 By the way, when we try to compare these two answers, 615 00:33:20,530 --> 00:33:22,340 they look different, but that's because they're 616 00:33:22,340 --> 00:33:24,750 expressed in terms of different variables. 617 00:33:24,750 --> 00:33:30,430 If we return to our original equations, and we see that 'x' 618 00:33:30,430 --> 00:33:32,530 is equal to 't squared'-- 619 00:33:32,530 --> 00:33:34,470 'x' is a synonym for 't squared'-- 620 00:33:34,470 --> 00:33:38,080 this is the check that we have received the right answer. 621 00:33:38,080 --> 00:33:41,400 By the way, before I conclude today's lecture, I would like 622 00:33:41,400 --> 00:33:43,630 to make a rather important aside 623 00:33:43,630 --> 00:33:46,060 about parametric equations. 624 00:33:46,060 --> 00:33:49,500 After one works the problem this way, and comes down to 625 00:33:49,500 --> 00:33:53,300 the check, and says, hey, after all of this mess over 626 00:33:53,300 --> 00:33:56,610 here, I could have replaced it by just 'y' equals 'x 627 00:33:56,610 --> 00:33:59,000 squared', why did I have to work with this 628 00:33:59,000 --> 00:34:00,220 in the first place? 629 00:34:00,220 --> 00:34:02,770 We are going to have many, many examples throughout the 630 00:34:02,770 --> 00:34:04,770 course that will illustrate this. 631 00:34:04,770 --> 00:34:07,440 But at least once in a lecture, I would like to go on 632 00:34:07,440 --> 00:34:12,380 record as pointing out that this pair of equations tells 633 00:34:12,380 --> 00:34:15,250 you much more than this equation here. 634 00:34:15,250 --> 00:34:17,909 This equation simply tells you this. 635 00:34:17,909 --> 00:34:21,880 If a particle were moving along a curve with respect to 636 00:34:21,880 --> 00:34:26,139 time according to these equations, this equation here 637 00:34:26,139 --> 00:34:29,050 simply tells you what path the particle would follow. 638 00:34:29,050 --> 00:34:34,480 Namely, the parabola 'y' equals 'x squared'. 639 00:34:34,480 --> 00:34:38,100 On the other hand, these two equations tell you 640 00:34:38,100 --> 00:34:39,420 much more than that. 641 00:34:39,420 --> 00:34:43,719 These not only tell you that the particle moved along the 642 00:34:43,719 --> 00:34:47,310 parabola 'y' equals 'x squared', but rather, it tells 643 00:34:47,310 --> 00:34:53,020 you at a particular time the point on the parabola that the 644 00:34:53,020 --> 00:34:54,630 particle was at. 645 00:34:54,630 --> 00:34:56,130 What I mean is this. 646 00:34:56,130 --> 00:35:00,480 As another example, suppose we had 'y' equals 't squared,' 647 00:35:00,480 --> 00:35:02,920 and 'x' equals 't'. 648 00:35:02,920 --> 00:35:07,090 If we eliminate 't' from these two equations, we also find 649 00:35:07,090 --> 00:35:09,960 that 'y' is equal to 'x squared'. 650 00:35:09,960 --> 00:35:15,050 Yet notice that this is not the same as our original set 651 00:35:15,050 --> 00:35:16,040 of equations. 652 00:35:16,040 --> 00:35:20,340 For example, here, when 't' is 2, when 't' is 2 over here, 653 00:35:20,340 --> 00:35:24,260 what point are we on as far as the parabola is concerned? 654 00:35:24,260 --> 00:35:27,930 When 't' is 2, this is 2, and this is 4. 655 00:35:27,930 --> 00:35:30,760 That would be the point 2 comma 4. 656 00:35:30,760 --> 00:35:33,960 On the other hand, with respect to this equation, when 657 00:35:33,960 --> 00:35:42,230 't' is 2, 'x' is 4, and 'y' is 16, you see both of these 658 00:35:42,230 --> 00:35:47,940 particles would follow the same curve, but they are at 659 00:35:47,940 --> 00:35:50,500 different points at different times. 660 00:35:50,500 --> 00:35:53,360 So don't belittle the parametric approach. 661 00:35:53,360 --> 00:35:56,980 Having the parameter 't' in there tells you more than just 662 00:35:56,980 --> 00:35:59,680 what the path of the motion is. 663 00:35:59,680 --> 00:36:02,380 It tells you at what time a particle was at what 664 00:36:02,380 --> 00:36:03,700 particular point. 665 00:36:03,700 --> 00:36:05,550 Well, enough about that. 666 00:36:05,550 --> 00:36:08,870 Let's go ahead and find the second derivative now. 667 00:36:08,870 --> 00:36:12,770 You see, we already know that 'dy dx' is '2 t squared'. 668 00:36:12,770 --> 00:36:18,230 Now what we'd like to find is 'd2y dx squared'. 669 00:36:18,230 --> 00:36:20,710 Again, the same basic definition. 670 00:36:20,710 --> 00:36:22,260 'd2y dx squared'. 671 00:36:22,260 --> 00:36:24,930 The second derivative is the derivative of the first 672 00:36:24,930 --> 00:36:26,970 derivative. 673 00:36:26,970 --> 00:36:30,600 The first derivative we saw was '2 t squared', so this is 674 00:36:30,600 --> 00:36:33,200 the derivative of '2 t squared'. 675 00:36:33,200 --> 00:36:35,470 Again, and this is where most of the mistakes are made, 676 00:36:35,470 --> 00:36:37,000 people get sloppy. 677 00:36:37,000 --> 00:36:38,720 They forget the 'x' is in here. 678 00:36:38,720 --> 00:36:41,710 They say I know the derivative of this, it's '4t'. 679 00:36:41,710 --> 00:36:45,600 Well, the derivative of this is '4t' with respect to 't'. 680 00:36:45,600 --> 00:36:49,560 We want to differentiate with respect to 'x'. 681 00:36:49,560 --> 00:36:54,020 And the way the chain rule comes in, we say OK, since 't' 682 00:36:54,020 --> 00:36:56,660 is the natural variable with respect to which to 683 00:36:56,660 --> 00:36:58,470 differentiate, let's do it. 684 00:36:58,470 --> 00:37:00,820 We'll differentiate in respect to 't'. 685 00:37:00,820 --> 00:37:04,270 But since the final answer has to be with respect to 'x', our 686 00:37:04,270 --> 00:37:08,590 correction factor by the chain rule will be 'dt dx'. 687 00:37:08,590 --> 00:37:11,710 Well, the derivative of '2 t squared' with respect to 't' 688 00:37:11,710 --> 00:37:13,750 is clearly '4t'. 689 00:37:13,750 --> 00:37:16,860 The derivative of 't' with respect to 'x', assuming that 690 00:37:16,860 --> 00:37:19,470 we know something about inverse functions, that's the 691 00:37:19,470 --> 00:37:21,680 reciprocal of 'dx dt'. 692 00:37:21,680 --> 00:37:25,270 We just saw that 'dx dt' was '2t'. 693 00:37:25,270 --> 00:37:28,950 Therefore, 'dt dx' is 1 over '2t', and therefore, the 694 00:37:28,950 --> 00:37:31,870 correct answer appears to be 2. 695 00:37:31,870 --> 00:37:35,330 Again, this is why I picked the simple case. 696 00:37:35,330 --> 00:37:38,800 Given that 'y' equals 'x squared', we see at a glance 697 00:37:38,800 --> 00:37:42,520 that 'dy dx' is equal to '2x', and also at a glance, 698 00:37:42,520 --> 00:37:48,330 therefore, that 'd2y dx squared' is equal to 2. 699 00:37:48,330 --> 00:37:51,520 By the way, that's exactly what this is equal to. 700 00:37:51,520 --> 00:37:54,660 You see, had we forgot the chain rule, and had we left 701 00:37:54,660 --> 00:37:57,610 this factor out, this would have given us-- 702 00:37:57,610 --> 00:37:59,530 in other words, to simply write down that the answer was 703 00:37:59,530 --> 00:38:02,510 '4t', which is the most common mistake that's made, would 704 00:38:02,510 --> 00:38:04,300 have given us the wrong answer. 705 00:38:04,300 --> 00:38:06,300 That's why I put such an easy problem. 706 00:38:06,300 --> 00:38:09,150 You see, if I had picked a tougher computational problem, 707 00:38:09,150 --> 00:38:10,940 the theory would have remained the same. 708 00:38:10,940 --> 00:38:13,190 But when I got two different answers, it would have been 709 00:38:13,190 --> 00:38:17,380 difficult to determine which was the correct answer, and 710 00:38:17,380 --> 00:38:19,420 which was the incorrect answer. 711 00:38:19,420 --> 00:38:22,140 But again, to summarize today's lecture, it was a 712 00:38:22,140 --> 00:38:25,510 continuation in a way of the lecture of last time, when we 713 00:38:25,510 --> 00:38:29,370 developed the primary recipe involving differentials. 714 00:38:29,370 --> 00:38:31,440 Now we applied that to find something 715 00:38:31,440 --> 00:38:33,190 called the chain rule. 716 00:38:33,190 --> 00:38:37,130 In the process of emphasizing the chain rule, we talked 717 00:38:37,130 --> 00:38:39,320 about the necessity of knowing something 718 00:38:39,320 --> 00:38:41,200 about inverse functions. 719 00:38:41,200 --> 00:38:45,440 Consequently, that dictates what our next lecture will be 720 00:38:45,440 --> 00:38:49,100 concerned with, namely inverse functions. 721 00:38:49,100 --> 00:38:51,390 And so until next time, goodbye. 722 00:38:54,730 --> 00:38:57,270 ANNOUNCER: Funding for the publication of this video was 723 00:38:57,270 --> 00:39:01,980 provided by the Gabriella and Paul Rosenbaum Foundation. 724 00:39:01,980 --> 00:39:06,160 Help OCW continue to provide free and open access to MIT 725 00:39:06,160 --> 00:39:10,350 courses by making a donation at ocw.mit.edu/donate.