1 00:00:00,040 --> 00:00:01,940 NARRATOR: 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 view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:31,700 --> 00:00:32,710 PROFESSOR: Hi. 9 00:00:32,710 --> 00:00:36,890 Our lecture today is entitled Implicit Differentiation. 10 00:00:36,890 --> 00:00:40,440 And aside from other considerations, this lecture 11 00:00:40,440 --> 00:00:43,200 bares a very strong relationship to many of the 12 00:00:43,200 --> 00:00:47,460 comments that we've made about single-valued functions and 13 00:00:47,460 --> 00:00:49,190 one-to-one functions. 14 00:00:49,190 --> 00:00:53,020 Let's get right into the topic by talking about one 15 00:00:53,020 --> 00:00:56,430 particular phase that somehow or other we may have taken for 16 00:00:56,430 --> 00:01:01,610 granted, but which really is a lot more sophisticated than we 17 00:01:01,610 --> 00:01:03,750 really imagine at first glance. 18 00:01:03,750 --> 00:01:08,330 Let's look, for example, at an expression such as 'y' 19 00:01:08,330 --> 00:01:09,820 equals 'f of x'. 20 00:01:09,820 --> 00:01:12,810 A rather harmless expression, we write it down quite 21 00:01:12,810 --> 00:01:15,270 frequently, and we say things like what? 22 00:01:15,270 --> 00:01:19,470 Given that 'y' equals 'f of x', find 'dy dx'. 23 00:01:22,030 --> 00:01:24,840 And you see the point that we've made here is we have 24 00:01:24,840 --> 00:01:28,360 assumed that 'y' can be solved for 25 00:01:28,360 --> 00:01:30,710 explicitly in terms of 'x'. 26 00:01:30,710 --> 00:01:33,130 In other words, in terms of our function machine, we 27 00:01:33,130 --> 00:01:35,040 visualize an 'f' machine. 28 00:01:35,040 --> 00:01:39,480 'x' goes in as the input, 'y' comes out as the output, and 29 00:01:39,480 --> 00:01:41,180 everything is fine. 30 00:01:41,180 --> 00:01:44,200 Now, the interesting point is simply this. 31 00:01:44,200 --> 00:01:45,810 Let's look at a slightly more 32 00:01:45,810 --> 00:01:48,110 complicated algebraic equation. 33 00:01:48,110 --> 00:01:52,100 For example, 'x to the eighth' plus ''x to the sixth' 'y to 34 00:01:52,100 --> 00:01:56,460 the fourth'' plus 'y to the sixth' equals 3. 35 00:01:56,460 --> 00:02:01,830 Now, in this particular example, notice that 'x' and 36 00:02:01,830 --> 00:02:05,190 'y' are not given at random. 37 00:02:05,190 --> 00:02:09,370 For example, if I say let me pick 'x' to be 2 and 'y' to be 38 00:02:09,370 --> 00:02:14,620 2, and I put 2 in for 'x' and 2 in for 'y', just look at 39 00:02:14,620 --> 00:02:17,040 what happens to the left-hand side here. 40 00:02:17,040 --> 00:02:19,280 It makes, in particular, 3 a pretty big number. 41 00:02:19,280 --> 00:02:23,950 In other words, when 'x' is 2 and 'y' is 2, this equation-- 42 00:02:23,950 --> 00:02:25,930 and we'll talk about the meaning of the equation more 43 00:02:25,930 --> 00:02:28,460 in just a few minutes, but this equation does not 44 00:02:28,460 --> 00:02:30,230 balance, so to speak. 45 00:02:30,230 --> 00:02:33,750 In other words, while we cannot pick 'x' at random for 46 00:02:33,750 --> 00:02:38,440 a given value of 'y', once 'x' is given, notice that we wind 47 00:02:38,440 --> 00:02:42,100 up with a sixth degree polynomial equation in 'y', 48 00:02:42,100 --> 00:02:45,290 which determines at most six values of 'y'. 49 00:02:45,290 --> 00:02:48,980 And if we're given a value of 'y' as a fixed number, we wind 50 00:02:48,980 --> 00:02:52,490 up with an eighth degree polynomial equation in 'x', 51 00:02:52,490 --> 00:02:53,880 which means that what? 52 00:02:53,880 --> 00:02:56,650 'x' is no longer random either. 53 00:02:56,650 --> 00:03:00,800 The point is it is very, very difficult, if not impossible, 54 00:03:00,800 --> 00:03:05,230 to try to solve this particular equation for 'y' 55 00:03:05,230 --> 00:03:07,790 explicitly in terms of 'x' or for 'x' 56 00:03:07,790 --> 00:03:10,330 explicitly in terms of 'y'. 57 00:03:10,330 --> 00:03:13,590 And the question is how do we tackle 58 00:03:13,590 --> 00:03:15,540 something along these lines? 59 00:03:15,540 --> 00:03:19,290 And we make a very tacit assumption. 60 00:03:19,290 --> 00:03:21,490 And by the way, you'll notice that when you read this 61 00:03:21,490 --> 00:03:25,430 section in the text, there are several remarks saying that 62 00:03:25,430 --> 00:03:29,590 many of the proofs are beyond the scope of the textbook at 63 00:03:29,590 --> 00:03:30,970 this particular point. 64 00:03:30,970 --> 00:03:33,145 And they'll say we'll talk about this later, and there 65 00:03:33,145 --> 00:03:35,890 are references to more advanced textbooks. 66 00:03:35,890 --> 00:03:37,520 The point is that from a more rigorous point of 67 00:03:37,520 --> 00:03:39,740 view, this is true. 68 00:03:39,740 --> 00:03:43,400 However, from a geometric intuitive point of view, it's 69 00:03:43,400 --> 00:03:46,660 quite easy to see what's really going on here, as we 70 00:03:46,660 --> 00:03:48,470 shall do with as the lecture progresses. 71 00:03:48,470 --> 00:03:51,280 But for the time being, let's review what that tacit 72 00:03:51,280 --> 00:03:52,400 assumption is. 73 00:03:52,400 --> 00:03:55,930 Essentially what we say is let's assume that 'y' is a 74 00:03:55,930 --> 00:03:57,720 particular function of 'x'. 75 00:03:57,720 --> 00:04:00,060 What particular function of 'x' is it? 76 00:04:00,060 --> 00:04:05,730 Well, it's that function of 'x' such that when you replace 77 00:04:05,730 --> 00:04:10,450 'y' by that function of 'x', this becomes an identity. 78 00:04:10,450 --> 00:04:12,050 And this is what I want to talk about next. 79 00:04:12,050 --> 00:04:15,820 You see, assume that 'y' is that function of 'x' such that 80 00:04:15,820 --> 00:04:20,180 when you replace 'y' in terms of 'x', 'x to the eighth' plus 81 00:04:20,180 --> 00:04:23,420 ''x to the sixth' 'y the fourth'' plus 'y to the sixth' 82 00:04:23,420 --> 00:04:24,980 is identically 3. 83 00:04:24,980 --> 00:04:29,120 And notice the use of the three lines here to indicate 84 00:04:29,120 --> 00:04:32,520 the identity as opposed to the two lines to 85 00:04:32,520 --> 00:04:34,720 indicate the equality. 86 00:04:34,720 --> 00:04:38,070 Now, obviously, there's a more important difference than to 87 00:04:38,070 --> 00:04:40,640 say that one is indicated by three lines and 88 00:04:40,640 --> 00:04:41,770 the other by two. 89 00:04:41,770 --> 00:04:46,030 Let me take a few moments to digress on a very important 90 00:04:46,030 --> 00:04:49,600 topic, namely, the difference between what we call an 91 00:04:49,600 --> 00:04:53,860 equation, or perhaps more appropriately, a conditional 92 00:04:53,860 --> 00:04:58,800 equality, and that which we call an absolute equality or 93 00:04:58,800 --> 00:04:59,950 an identity. 94 00:04:59,950 --> 00:05:03,920 For example, let's take a look at an expression such as 'x 95 00:05:03,920 --> 00:05:06,310 squared' equals 4. 96 00:05:06,310 --> 00:05:09,290 You say, ah, these two things are equals, and if I do the 97 00:05:09,290 --> 00:05:11,800 same thing to equals, I get equals. 98 00:05:11,800 --> 00:05:14,630 So the fellow says I think I'll differentiate both sides 99 00:05:14,630 --> 00:05:15,970 of this equality. 100 00:05:15,970 --> 00:05:19,540 If he differentiates 'x squared', he gets '2x', and if 101 00:05:19,540 --> 00:05:22,260 he differentiates 4, he gets 0. 102 00:05:22,260 --> 00:05:24,760 Now notice that if you solve the equation 'x squared' 103 00:05:24,760 --> 00:05:27,180 equals 4, you get what? 104 00:05:27,180 --> 00:05:30,790 'x' equals minus 2, or 2. 105 00:05:30,790 --> 00:05:33,600 On the other hand, if you solve the equation '2x' equals 106 00:05:33,600 --> 00:05:37,490 0, you get x equals 0, and there seems to be no 107 00:05:37,490 --> 00:05:40,510 correlation between these two. 108 00:05:40,510 --> 00:05:44,990 Now, you see the answer to this thing is this. 109 00:05:44,990 --> 00:05:47,660 'x squared' is not a synonym. 110 00:05:47,660 --> 00:05:51,310 'x squared' is not another way of saying 4. 111 00:05:51,310 --> 00:05:54,720 To use the language of the new mathematics, the solution set 112 00:05:54,720 --> 00:05:58,070 for 'x squared' equals 4 is what? 113 00:05:58,070 --> 00:06:03,100 The set of all 'x' such that 'x squared' equals 4. 114 00:06:03,100 --> 00:06:05,660 That's another way of saying what? 115 00:06:05,660 --> 00:06:10,330 The set whose only two members are 2 and negative 2. 116 00:06:12,920 --> 00:06:15,760 In other words, we cannot say that 'x squared' is 117 00:06:15,760 --> 00:06:16,900 a synonym for 4. 118 00:06:16,900 --> 00:06:22,000 All we can say is given the condition if 'x' is either 2 119 00:06:22,000 --> 00:06:25,040 or minus 2, then 'x squared' equals 4. 120 00:06:25,040 --> 00:06:27,190 Otherwise, this is not the case. 121 00:06:27,190 --> 00:06:31,590 For example, if you subtract 4 from 'x squared', you do not 122 00:06:31,590 --> 00:06:33,020 get identically 0. 123 00:06:33,020 --> 00:06:35,610 In other words, 'x squared' minus 4 is not another 124 00:06:35,610 --> 00:06:37,210 way of saying 0. 125 00:06:37,210 --> 00:06:39,320 For example, if I put 'x' equals 3 in 126 00:06:39,320 --> 00:06:40,650 here, this says what? 127 00:06:40,650 --> 00:06:43,580 3 squared equals 4, which is certainly a meaningful 128 00:06:43,580 --> 00:06:47,250 statement, but nonetheless a false statement. 129 00:06:47,250 --> 00:06:50,290 You see, somehow or other, an equation is something which 130 00:06:50,290 --> 00:06:55,850 can be true for a certain values of your variable but 131 00:06:55,850 --> 00:06:57,160 false for others. 132 00:06:57,160 --> 00:07:00,110 On the other hand, an identity, as the name may 133 00:07:00,110 --> 00:07:03,200 imply, is something that's true for all 134 00:07:03,200 --> 00:07:04,690 values of the variable. 135 00:07:04,690 --> 00:07:06,780 For example, let's go back to something we learned in 136 00:07:06,780 --> 00:07:11,850 elementary school algebra: 'x squared minus 1' equals 'x + 137 00:07:11,850 --> 00:07:14,150 1' times 'x - 1'. 138 00:07:14,150 --> 00:07:16,520 Notice I wrote this with the three lines here. 139 00:07:16,520 --> 00:07:21,720 It's my way of saying that for any number 'x' whatsoever, 'x 140 00:07:21,720 --> 00:07:26,500 squared - 1' always names the same number as 'x + 141 00:07:26,500 --> 00:07:28,660 1' times 'x - 1'. 142 00:07:28,660 --> 00:07:31,860 In other words, the solution set of this particular 143 00:07:31,860 --> 00:07:35,860 equation includes all real numbers. 144 00:07:35,860 --> 00:07:39,150 Or another way of saying it is that if you were to subtract 145 00:07:39,150 --> 00:07:43,940 'x + 1' times 'x - 1' from the 'x squared minus 1', the 146 00:07:43,940 --> 00:07:46,890 result would be 0 independently of what the 147 00:07:46,890 --> 00:07:49,010 value of 'x' was. 148 00:07:49,010 --> 00:07:52,530 In more colloquial terms, what we're saying is that 'x 149 00:07:52,530 --> 00:07:56,830 squared minus 1' and 'x + 1' times 'x - 1' are two 150 00:07:56,830 --> 00:07:59,510 different ways of saying the same thing. 151 00:07:59,510 --> 00:08:03,330 Now, the beauty of an identity versus a 152 00:08:03,330 --> 00:08:05,430 conditional equality is this. 153 00:08:05,430 --> 00:08:08,970 That if two expressions are just two different names for 154 00:08:08,970 --> 00:08:12,820 the same thing, then whatever is true for one expression 155 00:08:12,820 --> 00:08:15,300 will be true for the other. 156 00:08:15,300 --> 00:08:17,750 You see, in other words, if the concept is the same but 157 00:08:17,750 --> 00:08:20,430 only the names are different, then certainly anything that 158 00:08:20,430 --> 00:08:23,050 depends on the concept will not depend on the particular 159 00:08:23,050 --> 00:08:23,970 name that's involved. 160 00:08:23,970 --> 00:08:27,710 Well, as a case in point, let's suppose now I take 'x 161 00:08:27,710 --> 00:08:30,270 squared minus 1' and I differentiate it. 162 00:08:30,270 --> 00:08:32,580 The result is '2x'. 163 00:08:32,580 --> 00:08:35,980 On the other hand, if I take 'x + 1' times 'x - 1' and 164 00:08:35,980 --> 00:08:37,780 differentiate it, I get what? 165 00:08:37,780 --> 00:08:41,140 By use of the product rule, it's the first factor times 166 00:08:41,140 --> 00:08:45,260 the derivative of the second plus the derivative of the 167 00:08:45,260 --> 00:08:47,660 first factor times the second. 168 00:08:47,660 --> 00:08:52,120 That's what? 'x + 1' plus 'x - 1', and that 169 00:08:52,120 --> 00:08:55,420 comes out to be '2x'. 170 00:08:55,420 --> 00:08:58,050 In other words, since these two expressions were just two 171 00:08:58,050 --> 00:09:01,360 different names for the same thing, the derivative of one 172 00:09:01,360 --> 00:09:04,740 of the expressions must equal the derivative of the other 173 00:09:04,740 --> 00:09:06,230 because it's still the same function that you're 174 00:09:06,230 --> 00:09:07,360 differentiating. 175 00:09:07,360 --> 00:09:11,130 And this is what we mean when we say let's assume that 'y' 176 00:09:11,130 --> 00:09:14,420 is that function of 'x' that makes the resulting equation 177 00:09:14,420 --> 00:09:15,640 an identity. 178 00:09:15,640 --> 00:09:18,450 Let's look at a particularly simple illustration, and we'll 179 00:09:18,450 --> 00:09:20,330 do this quite often in this course. 180 00:09:20,330 --> 00:09:24,240 Namely, whenever we want to illustrate a new topic, we 181 00:09:24,240 --> 00:09:29,270 will always, when possible, pick an illustrative example 182 00:09:29,270 --> 00:09:33,170 that could have been solved by a previous method. 183 00:09:33,170 --> 00:09:37,040 As a case in point, let's look at the identity 'x' 184 00:09:37,040 --> 00:09:40,010 times 'y' is 1. 185 00:09:40,010 --> 00:09:42,970 See, in other words, 'y' is that particular function of 186 00:09:42,970 --> 00:09:48,270 'x', such that whenever you replace 'y' by that function, 187 00:09:48,270 --> 00:09:49,365 this becomes an identity. 188 00:09:49,365 --> 00:09:53,920 Now, if that seems hard, notice that we could turn this 189 00:09:53,920 --> 00:09:57,650 into an explicit relationship just by dividing both sides of 190 00:09:57,650 --> 00:10:00,310 this equation, our identity by 'x'. 191 00:10:00,310 --> 00:10:03,670 Of course, that assumes 'x' is not equal to 0. 192 00:10:03,670 --> 00:10:06,430 But if we do that, 'y' becomes '1/x'. 193 00:10:06,430 --> 00:10:10,600 Now notice that '1/x' has the property that as long as 'x' 194 00:10:10,600 --> 00:10:15,490 is not 0, if you multiply that by 'x', you get identically 1. 195 00:10:15,490 --> 00:10:17,650 'x' times '1/x' is 1. 196 00:10:17,650 --> 00:10:20,630 Two different ways of saying 1. 197 00:10:20,630 --> 00:10:22,440 Now, how does the method of implicit 198 00:10:22,440 --> 00:10:24,060 differentiation proceed? 199 00:10:24,060 --> 00:10:28,940 We say OK, since this is an identity, if I differentiate 200 00:10:28,940 --> 00:10:32,270 both sides with respect to 'x', the derivative of the 201 00:10:32,270 --> 00:10:34,910 left-hand side should equal the derivative of the 202 00:10:34,910 --> 00:10:36,220 right-hand side. 203 00:10:36,220 --> 00:10:39,090 Now, how do we differentiate 'x' times 'y'? 204 00:10:39,090 --> 00:10:42,470 Observe that we're assuming that 'y' is a function of 'x'. 205 00:10:42,470 --> 00:10:45,490 Therefore, 'x' times 'y' is a product of two 206 00:10:45,490 --> 00:10:46,820 functions of 'x'. 207 00:10:46,820 --> 00:10:49,650 To differentiate a product, we use the product rule. 208 00:10:49,650 --> 00:10:53,670 Namely, we will take the first factor times the derivative of 209 00:10:53,670 --> 00:10:55,940 the second with respect to 'x'. 210 00:10:55,940 --> 00:11:01,030 That's 'dy dx', because 'y' is the second factor, plus the 211 00:11:01,030 --> 00:11:03,900 derivative of the first factor with respect to 'x'. 212 00:11:03,900 --> 00:11:07,570 Well, the derivative of 'x' with respect to 'x' is 1 times 213 00:11:07,570 --> 00:11:10,550 the second factor, which is 'y', that's now the derivative 214 00:11:10,550 --> 00:11:11,810 of the left-hand side. 215 00:11:11,810 --> 00:11:14,870 That must equal identically the derivative of the 216 00:11:14,870 --> 00:11:16,120 right-hand side. 217 00:11:16,120 --> 00:11:17,710 The right-hand side is 1. 218 00:11:17,710 --> 00:11:19,530 The derivative of a constant is 0. 219 00:11:19,530 --> 00:11:23,110 So we wind up with the relationship that 'x' times 220 00:11:23,110 --> 00:11:27,190 'dy dx' plus 'y' must be identically 0. 221 00:11:27,190 --> 00:11:31,810 And now solving for 'dy dx' in terms of 'x' and 'y', we find 222 00:11:31,810 --> 00:11:36,970 that 'dy dx' is equal to minus 'y/x'. 223 00:11:36,970 --> 00:11:40,480 By the way, notice that by picking a problem that could 224 00:11:40,480 --> 00:11:44,120 be solved explicitly for 'y' in terms of 'x', we have a 225 00:11:44,120 --> 00:11:46,920 very simple check on this particular problem. 226 00:11:46,920 --> 00:11:50,980 Namely, we know that 'y', if 'x' is not 0, that y is 227 00:11:50,980 --> 00:11:52,820 another name for '1/x'. 228 00:11:52,820 --> 00:11:56,540 Therefore, minus 'y/x' is another name for ''minus 1' 229 00:11:56,540 --> 00:11:57,900 over 'x squared''. 230 00:11:57,900 --> 00:12:01,930 But we already know that if 'y' equals 'x to the minus 1' 231 00:12:01,930 --> 00:12:06,480 by another method, we know that 'dy dx' is 'minus x to 232 00:12:06,480 --> 00:12:11,390 the minus 2', which is also ''minus 1' over 'x squared''. 233 00:12:11,390 --> 00:12:14,820 And so we see that the new method does give us the same 234 00:12:14,820 --> 00:12:17,530 answer as the old method. 235 00:12:17,530 --> 00:12:19,560 By the way, let me make a rather 236 00:12:19,560 --> 00:12:22,440 important aside over here. 237 00:12:22,440 --> 00:12:25,250 And that is when you look at something like this, you might 238 00:12:25,250 --> 00:12:28,990 say something like I wonder what happens if I try to 239 00:12:28,990 --> 00:12:35,640 compute 'dy dx', say, when 'x' equals 2 and 'y' equals 3? 240 00:12:35,640 --> 00:12:37,300 Now, you see if you mechanically plug into 241 00:12:37,300 --> 00:12:38,880 something like this, you get what? 242 00:12:38,880 --> 00:12:42,710 Minus 3/2, which is minus three-halves. 243 00:12:42,710 --> 00:12:45,810 The point that I want to bring out, and we'll come to this at 244 00:12:45,810 --> 00:12:49,590 the conclusion of today's lecture also, is the concept 245 00:12:49,590 --> 00:12:53,720 of related rates and related variables. 246 00:12:53,720 --> 00:12:57,470 Notice that whereas from this equation it looks as if we can 247 00:12:57,470 --> 00:13:01,550 let 'y' equal 3 and 'x' equal 2, notice that if we go back 248 00:13:01,550 --> 00:13:07,000 to our basic definition, 'x' and 'y' are related so that 249 00:13:07,000 --> 00:13:08,290 they are not independent. 250 00:13:08,290 --> 00:13:12,720 Notice that as soon as we say let 'x' equal 2, we have what? 251 00:13:12,720 --> 00:13:17,030 That '2y' is 1, and 'y' must be 1/2. 252 00:13:17,030 --> 00:13:20,740 In other words, when you use implicit differentiation, 253 00:13:20,740 --> 00:13:24,150 never forget that whenever you're going to compare 'x' 254 00:13:24,150 --> 00:13:29,260 and 'y', you must go back to the equation or the identity 255 00:13:29,260 --> 00:13:32,520 which implicitly relates 'x' to 'y'. 256 00:13:32,520 --> 00:13:35,690 Well, so far this may look rather easy and 257 00:13:35,690 --> 00:13:36,760 straightforward. 258 00:13:36,760 --> 00:13:39,360 But the fact remains that there are certain subtleties 259 00:13:39,360 --> 00:13:41,650 here which we have not hit yet. 260 00:13:41,650 --> 00:13:45,040 So what I'd like to do now is pick a second example, 261 00:13:45,040 --> 00:13:48,230 slightly more complicated than this one, which can still be 262 00:13:48,230 --> 00:13:51,950 solved explicitly but which leads to a wrinkle which we 263 00:13:51,950 --> 00:13:54,220 may not have observed before. 264 00:13:54,220 --> 00:13:55,850 With this in mind, what I would like 265 00:13:55,850 --> 00:13:58,680 to do is the following. 266 00:13:58,680 --> 00:14:04,170 Let's consider the relation 'x squared' plus 'y squared' 267 00:14:04,170 --> 00:14:09,310 equals 25 and ask the question how do you find 'dy dx' in 268 00:14:09,310 --> 00:14:10,830 this particular case? 269 00:14:10,830 --> 00:14:15,450 Again what we do is we assume, and this is the big word here. 270 00:14:15,450 --> 00:14:17,860 There are lots of things that you can assume, but whether 271 00:14:17,860 --> 00:14:20,570 they exist or not is another question. 272 00:14:20,570 --> 00:14:22,770 That's the part that the textbook means is more 273 00:14:22,770 --> 00:14:24,970 advanced and is hard to justify. 274 00:14:24,970 --> 00:14:25,920 But let's take a look here. 275 00:14:25,920 --> 00:14:30,310 We'll assume that 'y' is a particular function of 'x' 276 00:14:30,310 --> 00:14:34,380 with the property that when 'y' is that function of 'x', 277 00:14:34,380 --> 00:14:39,970 'x squared' plus 'y squared' is identically 25, OK? 278 00:14:39,970 --> 00:14:43,040 And what that means is that whatever 'x' and 'y' are, 279 00:14:43,040 --> 00:14:45,760 they're related in such a way that 'x squared' plus 'y 280 00:14:45,760 --> 00:14:48,780 squared' is a synonym for 25. 281 00:14:48,780 --> 00:14:52,930 If we now proceed by implicit differentiation here, you see 282 00:14:52,930 --> 00:14:57,310 the left-hand side is a function which is the sum of 283 00:14:57,310 --> 00:14:59,080 two functions of 'x'. 284 00:14:59,080 --> 00:15:01,370 The derivative of a sum is the sum of the derivatives. 285 00:15:01,370 --> 00:15:04,890 The derivative of 'x squared' with respect to 'x' is '2x'. 286 00:15:04,890 --> 00:15:09,440 The derivative of 'y squared' with respect 287 00:15:09,440 --> 00:15:11,840 to 'x' is not '2y'. 288 00:15:11,840 --> 00:15:15,320 The derivative of 'y squared' with respect to 'y' is '2y'. 289 00:15:15,320 --> 00:15:18,130 By the chain rule, the derivative of 'y squared' with 290 00:15:18,130 --> 00:15:20,080 respect to 'x' is what? 291 00:15:20,080 --> 00:15:22,480 The derivative of 'y squared' with respect to 292 00:15:22,480 --> 00:15:24,760 'y' times 'dy dx'. 293 00:15:24,760 --> 00:15:26,450 In other words, this is simply what? 294 00:15:26,450 --> 00:15:28,860 ''2y' 'dy dx''. 295 00:15:28,860 --> 00:15:31,720 So the derivative of the left-hand side is '2x' plus 296 00:15:31,720 --> 00:15:33,160 ''2y' 'dy dx''. 297 00:15:33,160 --> 00:15:35,800 The derivative of the right-hand side, the 298 00:15:35,800 --> 00:15:38,020 right-hand side being a constant, is 0. 299 00:15:38,020 --> 00:15:41,490 And if we now solve for 'dy dx' in terms of 'x' and 'y', 300 00:15:41,490 --> 00:15:47,180 we find that 'dy dx' is 'minus x/y', OK? 301 00:15:47,180 --> 00:15:50,210 This is all there is to this thing mechanically. 302 00:15:50,210 --> 00:15:53,390 By the way, of course, it happens as you probably 303 00:15:53,390 --> 00:15:56,770 remember, that 'x squared' plus 'y squared' equals 25 is 304 00:15:56,770 --> 00:16:00,850 a circle centered at the origin with radius equal to 5. 305 00:16:00,850 --> 00:16:02,520 OK, so far so good. 306 00:16:02,520 --> 00:16:05,270 We'll come back to this diagram in a little while. 307 00:16:05,270 --> 00:16:08,770 But the thing now is could we have solved the same problem 308 00:16:08,770 --> 00:16:13,250 by solving for 'y' explicitly in terms of 'x'? 309 00:16:13,250 --> 00:16:15,310 The answer in this case, of course, is yes. 310 00:16:15,310 --> 00:16:19,220 Namely, if 'x squared' plus 'y squared' is 25, that says that 311 00:16:19,220 --> 00:16:22,760 'y squared' is 25 minus 'x squared', and therefore, the 312 00:16:22,760 --> 00:16:24,040 desired function of 'x'. 313 00:16:24,040 --> 00:16:25,740 'y' is what function of 'x'? 314 00:16:25,740 --> 00:16:28,810 It's plus or minus the square root of 315 00:16:28,810 --> 00:16:31,070 '25 minus 'x squared''. 316 00:16:31,070 --> 00:16:32,210 In other words, that's what? 317 00:16:32,210 --> 00:16:36,900 It's the positive '25 minus 'x squared'' to the 1/2 power. 318 00:16:36,900 --> 00:16:38,640 That's one of the solutions. 319 00:16:38,640 --> 00:16:44,190 And the other solution is the negative '25 minus 'x 320 00:16:44,190 --> 00:16:45,940 squared'' to the 1/2 power. 321 00:16:45,940 --> 00:16:48,730 I simply use the exponent notation because it's more 322 00:16:48,730 --> 00:16:51,560 familiar to us in terms of differentiation to 323 00:16:51,560 --> 00:16:54,570 differentiate the exponent rather than the radical sign. 324 00:16:54,570 --> 00:16:59,290 But the idea is this: Notice now that we begin to see a 325 00:16:59,290 --> 00:17:02,320 multivalued function creeping in over here. 326 00:17:02,320 --> 00:17:05,720 In other words, we now find that we want to solve for 'y' 327 00:17:05,720 --> 00:17:09,660 explicitly in terms of 'x', that we do get the problem 328 00:17:09,660 --> 00:17:12,960 that y might be a multivalued function of 'x'. 329 00:17:12,960 --> 00:17:14,900 Well, we'll look at that in a moment. 330 00:17:14,900 --> 00:17:17,980 For the time being, let's simply do a 331 00:17:17,980 --> 00:17:19,290 double check over here. 332 00:17:19,290 --> 00:17:22,569 Let's actually differentiate this thing explicitly and see 333 00:17:22,569 --> 00:17:24,599 what the derivative turns out to be. 334 00:17:24,599 --> 00:17:27,369 If we differentiate this, we bring the 1/2 down. 335 00:17:27,369 --> 00:17:30,050 We replace it to an exponent one less. 336 00:17:30,050 --> 00:17:35,010 And by the chain rule, we must multiply by the derivative of 337 00:17:35,010 --> 00:17:39,200 what's inside here, 25 minus x squared with respect to 'x'. 338 00:17:39,200 --> 00:17:40,730 That's 'minus 2x'. 339 00:17:40,730 --> 00:17:44,630 Collecting terms and simplifying, we get what? 340 00:17:44,630 --> 00:17:50,240 'Minus x' over ''25 minus 'x squared'' to the 1/2'. 341 00:17:50,240 --> 00:17:54,530 And recalling that ''25 minus 'x squared'' to the 1/2' is 342 00:17:54,530 --> 00:17:59,070 equal to the 'y' value in this case, we get the derivative of 343 00:17:59,070 --> 00:18:04,050 'y1' with respect to 'x' is 'minus x' over 'y1', just 344 00:18:04,050 --> 00:18:05,260 calling the function this. 345 00:18:05,260 --> 00:18:07,190 That at any rate shows what? 346 00:18:07,190 --> 00:18:11,030 The negative x-coordinate over the y-coordinate, and that 347 00:18:11,030 --> 00:18:15,840 certainly checks with the result that we had before. 348 00:18:15,840 --> 00:18:21,370 In a similar way, if we now differentiate 'y2' with 349 00:18:21,370 --> 00:18:25,870 respect to 'x', bringing down the exponent, multiplying by 350 00:18:25,870 --> 00:18:28,650 the derivative of what's inside with respect to 'x', we 351 00:18:28,650 --> 00:18:31,590 wind up with the same expression as we did before, 352 00:18:31,590 --> 00:18:37,040 only now we remember that 'y2' is defined to be negative ''25 353 00:18:37,040 --> 00:18:39,920 minus 'x squared'' to the 1/2', so this is really 354 00:18:39,920 --> 00:18:42,560 negative y2. 355 00:18:42,560 --> 00:18:46,990 By the way, notice again that we get the same answer here as 356 00:18:46,990 --> 00:18:48,770 we did here. 357 00:18:48,770 --> 00:18:52,650 I like to keep the minus sign with the appropriate term. 358 00:18:52,650 --> 00:18:55,540 In other words, notice it was the fact that 'y2' was 359 00:18:55,540 --> 00:19:00,570 multivalued that caused us to have two different 360 00:19:00,570 --> 00:19:02,700 possibilities here in the first place. 361 00:19:02,700 --> 00:19:06,530 Now, let's go back to our little graph over here and 362 00:19:06,530 --> 00:19:09,170 take a look to see just what happened here. 363 00:19:09,170 --> 00:19:11,930 Remember, all through this course, we've said what? 364 00:19:11,930 --> 00:19:15,360 Be aware of what happens when 'y' equals 0. 365 00:19:15,360 --> 00:19:19,320 Notice in terms of our picture and our related rates here, 366 00:19:19,320 --> 00:19:23,300 when 'y' is 0, 'x' is not some arbitrary number. 367 00:19:23,300 --> 00:19:26,430 When 'y' is 0, since 'x squared' plus 'y squared' 368 00:19:26,430 --> 00:19:32,200 equals 25 when 'y' is 0, 'x' must either be 5 or minus 5. 369 00:19:32,200 --> 00:19:36,880 So what's happening is the bad point when 'y' is 0, or the 370 00:19:36,880 --> 00:19:39,290 bad points, are right here. 371 00:19:39,290 --> 00:19:42,010 And notice the rather interesting thing here, and 372 00:19:42,010 --> 00:19:45,480 this is what comes up more analytically later. 373 00:19:45,480 --> 00:19:48,920 The only time you're in trouble when you assume that 374 00:19:48,920 --> 00:19:51,500 'y' is a function of 'x'-- remember, function mean 375 00:19:51,500 --> 00:19:52,960 single-valued-- 376 00:19:52,960 --> 00:19:57,690 is that if you happen to be in the neighborhood of 5 comma 0 377 00:19:57,690 --> 00:20:01,420 or minus 5 comma 0, notice that for any neighborhood 378 00:20:01,420 --> 00:20:04,080 surrounding this point, there is no way-- 379 00:20:04,080 --> 00:20:07,270 if this point here is included, there is no way of 380 00:20:07,270 --> 00:20:10,230 breaking up this function to be single valued. 381 00:20:10,230 --> 00:20:12,580 In other words, if you must have a small portion of the 382 00:20:12,580 --> 00:20:16,910 curve that includes 5 comma 0 as an interior point, no 383 00:20:16,910 --> 00:20:20,140 matter how you do it, the resulting curve is going to be 384 00:20:20,140 --> 00:20:24,190 multivalued, and then we're in the same predicament again. 385 00:20:24,190 --> 00:20:27,080 In other words, from a theoretical point of view, 386 00:20:27,080 --> 00:20:31,600 geometrically speaking, it's very easy to assume that 'y' 387 00:20:31,600 --> 00:20:34,080 is a function of 'x' that makes 388 00:20:34,080 --> 00:20:36,620 our equation an identity. 389 00:20:36,620 --> 00:20:40,370 But from a real point of view, what may frequently happen is 390 00:20:40,370 --> 00:20:45,050 that the only type of function that would work is what we 391 00:20:45,050 --> 00:20:46,610 call a multivalued function. 392 00:20:46,610 --> 00:20:48,740 That's exactly what's going on over here. 393 00:20:48,740 --> 00:20:51,750 And that's why a textbook, which is trying to be 394 00:20:51,750 --> 00:20:55,340 rigorous, has to be very, very careful in explaining what 395 00:20:55,340 --> 00:20:58,110 happens in neighborhoods of points like this. 396 00:20:58,110 --> 00:21:01,205 But again, we'll talk about that in more detail in a 397 00:21:01,205 --> 00:21:02,130 little while. 398 00:21:02,130 --> 00:21:04,880 I would like to make one aside over here. 399 00:21:04,880 --> 00:21:09,710 You may recall that we learned in one of our previous 400 00:21:09,710 --> 00:21:12,880 lectures that by inverse functions, we can 401 00:21:12,880 --> 00:21:16,840 differentiate things like 'x' to the 1/2, 'x' to the 1/3. 402 00:21:16,840 --> 00:21:20,830 This generalizes very nicely and very simply in terms of 403 00:21:20,830 --> 00:21:22,680 implicit differentiation. 404 00:21:22,680 --> 00:21:27,310 Namely, suppose you have 'y' equals 'x to the 'p/q'' power 405 00:21:27,310 --> 00:21:29,340 where 'p' and 'q' are integers. 406 00:21:29,340 --> 00:21:33,160 In other words, your exponent is now a fraction. 407 00:21:33,160 --> 00:21:38,160 Raise both sides to the q-th power, and you get 'y' to the 408 00:21:38,160 --> 00:21:40,560 'q' equals 'x to the p'. 409 00:21:40,560 --> 00:21:44,450 Now, we know how to differentiate powers of 'x' or 410 00:21:44,450 --> 00:21:47,720 'y' if the power happens to be an integer. 411 00:21:47,720 --> 00:21:50,950 So by implicit differentiation, assuming that 412 00:21:50,950 --> 00:21:53,960 this is now an identity, which it is if 'y' equals 'x to the 413 00:21:53,960 --> 00:21:57,370 'p/q'', if we differentiate both sides with respect to 414 00:21:57,370 --> 00:22:01,890 'x', the derivative of the left-hand side is 'qy to the 415 00:22:01,890 --> 00:22:05,670 'q - 1'' times the derivative of 'y' with respect to 'x'. 416 00:22:05,670 --> 00:22:08,110 That is, we're differentiating with respect to 'x'. 417 00:22:08,110 --> 00:22:10,180 And the derivative of the right-hand side is 418 00:22:10,180 --> 00:22:12,790 'px to the 'p - 1''. 419 00:22:12,790 --> 00:22:15,710 And since this is an identity, these two 420 00:22:15,710 --> 00:22:17,360 expressions must be equal. 421 00:22:17,360 --> 00:22:21,940 If we solve now for 'dy dx', we obtain this expression, 422 00:22:21,940 --> 00:22:27,780 replacing 'y' by 'x to the 'p/q'', and multiplying out 423 00:22:27,780 --> 00:22:31,980 here, and remembering that when you have an exponent in 424 00:22:31,980 --> 00:22:34,700 the denominator, it can come up into the numerator by 425 00:22:34,700 --> 00:22:36,540 changing the sign. 426 00:22:36,540 --> 00:22:41,060 A little bit of arithmetic shows that 'dy dx' is ''p/q' x 427 00:22:41,060 --> 00:22:47,780 to the ''p - 1' minus 'p' plus 'p/q'' power. 428 00:22:47,780 --> 00:22:51,180 The 'p' and the 'minus p' cancel out, and we find what? 429 00:22:51,180 --> 00:22:57,090 That 'dy dx' is 'p/q' times 'x to the ''p/q' - 1''. 430 00:22:57,090 --> 00:23:00,760 Remembering that 'p/q' was our fractional exponent, 431 00:23:00,760 --> 00:23:02,350 we again see what? 432 00:23:02,350 --> 00:23:05,150 That to differentiate 'x' to a fractional exponent-- 433 00:23:05,150 --> 00:23:06,260 'p/q'-- 434 00:23:06,260 --> 00:23:10,250 you bring the exponent down and replace the 2 by one less. 435 00:23:10,250 --> 00:23:13,380 So that gives us an alternative method for finding 436 00:23:13,380 --> 00:23:17,680 the derivative of a fractional exponent by namely using 437 00:23:17,680 --> 00:23:19,580 implicit differentiation. 438 00:23:19,580 --> 00:23:22,040 Now, the reason I put that aside in is we could certainly 439 00:23:22,040 --> 00:23:23,180 get along without it. 440 00:23:23,180 --> 00:23:27,170 But I felt that before we go any further, that I would like 441 00:23:27,170 --> 00:23:30,580 at least to add it on the more elementary levels to be sure 442 00:23:30,580 --> 00:23:34,110 that we understand explicitly what implicit 443 00:23:34,110 --> 00:23:36,720 differentiation means. 444 00:23:36,720 --> 00:23:41,000 With this in mind, let's now go back to the problem that we 445 00:23:41,000 --> 00:23:45,890 started off with in this lecture, namely, the curve 446 00:23:45,890 --> 00:23:48,760 whose equation is 'x to the eighth' plus ''x to the sixth' 447 00:23:48,760 --> 00:23:52,390 'y to the fourth'' plus 'y the sixth' equals 3. 448 00:23:52,390 --> 00:23:53,340 And we say what? 449 00:23:53,340 --> 00:23:56,630 Let's find the equation of the line tangent to this curve at 450 00:23:56,630 --> 00:23:58,470 the point 1 comma 1. 451 00:23:58,470 --> 00:24:00,790 By the way, I did something that teachers are 452 00:24:00,790 --> 00:24:01,950 allowed to do here. 453 00:24:01,950 --> 00:24:04,620 I rigged the problem to come out rather simply. 454 00:24:04,620 --> 00:24:08,250 You see, what I did was by picking 1 and 1 here, 455 00:24:08,250 --> 00:24:09,200 it turns out what? 456 00:24:09,200 --> 00:24:12,280 That if I made the right-hand side just equal to the number 457 00:24:12,280 --> 00:24:15,730 of terms here, I would get this thing to check out. 458 00:24:15,730 --> 00:24:18,160 But notice, by the way, if I had just picked the point at 459 00:24:18,160 --> 00:24:22,700 random to try to even check where that point is on this 460 00:24:22,700 --> 00:24:27,720 curve, or if it were on this curve, to find out what its 461 00:24:27,720 --> 00:24:31,500 coordinates are, this is a rather difficult problem. 462 00:24:31,500 --> 00:24:34,600 In other words, if I just pick a random value of 'x', as I 463 00:24:34,600 --> 00:24:37,560 said before, I get a sixth degree polynomial equation to 464 00:24:37,560 --> 00:24:42,260 solve, and if I pick a random value of 'y', an eighth degree 465 00:24:42,260 --> 00:24:43,770 polynomial equation to solve. 466 00:24:43,770 --> 00:24:46,480 You know, it might be nice to pretend that the 8 was a 2 or 467 00:24:46,480 --> 00:24:49,100 something like this and solve it as a quadratic equation. 468 00:24:49,100 --> 00:24:52,770 But quite frankly, solving polynomial equations, which 469 00:24:52,770 --> 00:24:56,940 are higher than the degree two is a very difficult task. 470 00:24:56,940 --> 00:25:00,500 In fact, if the degree is greater than four, it may even 471 00:25:00,500 --> 00:25:02,510 be an impossible task. 472 00:25:02,510 --> 00:25:04,140 I'm not going to go into that now because 473 00:25:04,140 --> 00:25:05,280 it's not that crucial. 474 00:25:05,280 --> 00:25:09,170 What is crucial is to observe that this particular problem 475 00:25:09,170 --> 00:25:10,870 makes sense. 476 00:25:10,870 --> 00:25:12,890 It's a meaningful problem. 477 00:25:12,890 --> 00:25:16,600 It would be difficult, if not impossible, to solve for 'y' 478 00:25:16,600 --> 00:25:19,690 explicitly in terms of 'x' here, OK? 479 00:25:19,690 --> 00:25:23,430 Yet to solve this problem, we can now use implicit 480 00:25:23,430 --> 00:25:27,030 differentiation except for the fact that we can no longer 481 00:25:27,030 --> 00:25:29,950 check back by another method to see if the answer is right. 482 00:25:29,950 --> 00:25:33,150 That's why I wanted to pick a few introductory examples that 483 00:25:33,150 --> 00:25:35,240 we could check by other means. 484 00:25:35,240 --> 00:25:37,800 Now that we have a feeling for this, maybe we will trust the 485 00:25:37,800 --> 00:25:40,880 technique in a case where we have no recourse other than to 486 00:25:40,880 --> 00:25:42,660 use the technique. 487 00:25:42,660 --> 00:25:46,660 So what we do now is this: We say OK, let's assume that 'y' 488 00:25:46,660 --> 00:25:50,540 is that function of 'x', that differentiable function of 'x' 489 00:25:50,540 --> 00:25:54,760 that makes this particular equation an identity. 490 00:25:54,760 --> 00:25:57,780 And assuming now that this is an identity, let me 491 00:25:57,780 --> 00:26:01,810 differentiate both sides with respect to 'x'. 492 00:26:01,810 --> 00:26:03,410 See, this is a sum. 493 00:26:03,410 --> 00:26:05,850 One of the terms in the sum happens to be a product. 494 00:26:05,850 --> 00:26:09,220 But by now, hopefully how we go about something like this 495 00:26:09,220 --> 00:26:10,330 will be old hat. 496 00:26:10,330 --> 00:26:13,180 Namely, to differentiate the left-hand side with respect to 497 00:26:13,180 --> 00:26:14,710 'x', we get what? 498 00:26:14,710 --> 00:26:18,750 '8x to the seventh' plus what? 499 00:26:18,750 --> 00:26:21,580 The derivative of 'x to the sixth', which is '6x to the 500 00:26:21,580 --> 00:26:24,230 fifth', times the second factor, which is 'y the 501 00:26:24,230 --> 00:26:27,050 fourth', plus the first factor, which is 'x to the 502 00:26:27,050 --> 00:26:30,090 sixth', times the derivative of 'y to the fourth' with 503 00:26:30,090 --> 00:26:31,150 respect to 'x'-- 504 00:26:31,150 --> 00:26:33,800 that's ''4y cubed' 'dy dx''-- 505 00:26:33,800 --> 00:26:36,060 plus the derivative of 'y to the sixth' with respect to 506 00:26:36,060 --> 00:26:39,700 'x', which is ''6y to the fifth' 'dy dx''. 507 00:26:39,700 --> 00:26:42,210 That must be identically 0. 508 00:26:42,210 --> 00:26:45,710 And if we now solve this for 'dy dx', just by transposing, 509 00:26:45,710 --> 00:26:50,320 we wind up again with a rather messy expression, but which 510 00:26:50,320 --> 00:26:55,770 does show what 'dy dx' looks like in terms of 'x' and 'y'. 511 00:26:55,770 --> 00:26:59,830 We were interested in knowing what the slope was not only 512 00:26:59,830 --> 00:27:03,900 for any old value of 'x' and 'y' but rather for what? 513 00:27:03,900 --> 00:27:06,860 When 'x' is 1 and 'y' is 1. 514 00:27:06,860 --> 00:27:09,280 And that works out very nicely computationally. 515 00:27:09,280 --> 00:27:11,490 It's just minus 14/10. 516 00:27:11,490 --> 00:27:14,960 In other words, at the point 1 comma 1, the slope of the 517 00:27:14,960 --> 00:27:17,370 curve is minus 7/5. 518 00:27:17,370 --> 00:27:20,240 The curve passes through the point (1, 1). 519 00:27:20,240 --> 00:27:24,460 Hence, it's equation is 'y - 1' over 'x - 1' equals minus 520 00:27:24,460 --> 00:27:30,660 7/5, or more explicitly, '7x + 5y' equals 12. 521 00:27:30,660 --> 00:27:33,660 And the point that I wanted to make here is notice that 522 00:27:33,660 --> 00:27:38,090 nothing changed in principle from our first few lectures. 523 00:27:38,090 --> 00:27:41,220 Notice that to find the equation of the line, we still 524 00:27:41,220 --> 00:27:44,090 use the recipe that we have to know a point on the 525 00:27:44,090 --> 00:27:45,760 line and the slope. 526 00:27:45,760 --> 00:27:48,920 The only thing that's changed with today's lesson is that we 527 00:27:48,920 --> 00:27:53,140 can now find the slope of a curve at a particular point 528 00:27:53,140 --> 00:27:56,160 that we could not find prior to today's lesson. 529 00:27:56,160 --> 00:27:59,560 All that has changed is that we have one more technique for 530 00:27:59,560 --> 00:28:03,950 finding the derivative of a particular type of function. 531 00:28:03,950 --> 00:28:06,710 I would like to analyze this problem in more detail. 532 00:28:06,710 --> 00:28:10,680 In particular, I would like to see where the numerator of 533 00:28:10,680 --> 00:28:13,780 this expression can be 0 and where the 534 00:28:13,780 --> 00:28:15,330 denominator can be 0. 535 00:28:15,330 --> 00:28:18,390 Because you see in terms of slopes, where the numerator is 536 00:28:18,390 --> 00:28:21,770 0, it means the slope will be 0. 537 00:28:21,770 --> 00:28:24,110 That means we have a horizontal tangent line. 538 00:28:24,110 --> 00:28:27,800 Where the denominator is 0, that means the slope is 539 00:28:27,800 --> 00:28:29,890 infinite, OK? 540 00:28:29,890 --> 00:28:31,820 And where the slope is infinite, that means you have 541 00:28:31,820 --> 00:28:34,770 a vertical line, and that means you have a vertical 542 00:28:34,770 --> 00:28:36,490 tangent, OK. 543 00:28:36,490 --> 00:28:38,450 So let's take a look and see what that means. 544 00:28:38,450 --> 00:28:42,420 Keep this particular equation in mind, because now you see 545 00:28:42,420 --> 00:28:45,900 on the next board, all I want to do is work with what this 546 00:28:45,900 --> 00:28:46,420 thing means. 547 00:28:46,420 --> 00:28:51,780 In other words, 'dy dx' will be 0 when my numerator is 0. 548 00:28:51,780 --> 00:28:54,810 My numerator can be written in this particular form. 549 00:28:54,810 --> 00:28:57,420 And by the way, here's again an interesting point. 550 00:28:57,420 --> 00:28:59,800 When will this expression be 0? 551 00:28:59,800 --> 00:29:03,810 And the answer is when either of these two factors is 0. 552 00:29:03,810 --> 00:29:07,260 Well, the first factor is 0 when 'x' is 0. 553 00:29:07,260 --> 00:29:11,630 And the second factor is 0, since these are even powers, 554 00:29:11,630 --> 00:29:14,460 only when 'x' and 'y' are both 0. 555 00:29:14,460 --> 00:29:18,870 Notice, however, that 'x' and 'y' cannot both be 0. 556 00:29:18,870 --> 00:29:22,420 Recall that the equation was what? 557 00:29:22,420 --> 00:29:25,900 Whatever it was, notice that (0, 0) is not a point which 558 00:29:25,900 --> 00:29:28,110 satisfies the equation. 559 00:29:28,110 --> 00:29:32,980 Remember, (0, 0) does not satisfy 'x to the eighth' plus 560 00:29:32,980 --> 00:29:35,760 ''x to the sixth' 'y squared'' plus 'y to the 561 00:29:35,760 --> 00:29:37,510 sixth' equals 3. 562 00:29:37,510 --> 00:29:38,830 It's equal to 0, you see. 563 00:29:38,830 --> 00:29:42,030 But at any rate, notice that the slope is 0 564 00:29:42,030 --> 00:29:43,990 only when 'x' is 0. 565 00:29:43,990 --> 00:29:47,970 And because of the particular equation, when 'x' is 0-- 566 00:29:47,970 --> 00:29:50,100 well, we'll go back to that in a minute. 567 00:29:50,100 --> 00:29:52,410 Let's just check to see what's happening here. 568 00:29:52,410 --> 00:29:53,760 The denominator will be 0. 569 00:29:53,760 --> 00:29:56,890 In other words, when 'dx dy' is 0-- '1 over 570 00:29:56,890 --> 00:29:58,650 'dy dx'', you see-- 571 00:29:58,650 --> 00:30:02,265 only when this factor here is 0, and that occurs again only 572 00:30:02,265 --> 00:30:03,697 when 'y' is 0. 573 00:30:03,697 --> 00:30:06,440 Now, the point to keep in mind is this. 574 00:30:06,440 --> 00:30:09,460 Remember, I put this down here so we could refer to it, and I 575 00:30:09,460 --> 00:30:11,690 forgot that I put it here, and that's why I didn't see it 576 00:30:11,690 --> 00:30:12,810 until just now. 577 00:30:12,810 --> 00:30:17,400 All I'm saying here is that notice that when 'x' is 0, 'y' 578 00:30:17,400 --> 00:30:20,880 must be plus or minus the sixth root of 3. 579 00:30:20,880 --> 00:30:24,570 And when 'y' is 0, these two terms drop out. 'x' must be 580 00:30:24,570 --> 00:30:27,740 plus or minus the eighth root of 3. 581 00:30:27,740 --> 00:30:35,190 Coming over to a graph here then, what we see is that the 582 00:30:35,190 --> 00:30:40,260 curve crosses the x-axis at this point 583 00:30:40,260 --> 00:30:41,880 with a vertical tangent. 584 00:30:41,880 --> 00:30:44,660 It crosses the y-axis at this point with 585 00:30:44,660 --> 00:30:47,910 a horizontal tangent. 586 00:30:47,910 --> 00:30:51,650 Notice, by the way, that this curve is also symmetric with 587 00:30:51,650 --> 00:30:55,470 respect to both the x- and the y-axes, because 588 00:30:55,470 --> 00:30:57,330 if I replace 'x'-- 589 00:30:57,330 --> 00:31:00,990 well, it's not important, and I don't want to obscure the 590 00:31:00,990 --> 00:31:05,870 lecture by taking time out for this now. 591 00:31:05,870 --> 00:31:09,100 But the point is a quick check shows that this curve is 592 00:31:09,100 --> 00:31:12,990 symmetric with respect to both the x- and the y-axes, that if 593 00:31:12,990 --> 00:31:15,650 I could plot this curve just in the first quadrant, the 594 00:31:15,650 --> 00:31:18,730 mirror image with respect to the y-axis would then show me 595 00:31:18,730 --> 00:31:19,970 the second quadrant. 596 00:31:19,970 --> 00:31:24,170 If I then took the mirror image of this upper half with 597 00:31:24,170 --> 00:31:27,180 respect to the x-axis, that would give me the lower 598 00:31:27,180 --> 00:31:28,440 portion of this curve. 599 00:31:28,440 --> 00:31:31,230 The curve tends to look something like this. 600 00:31:31,230 --> 00:31:34,530 And by the way, all we've done, if we look back over 601 00:31:34,530 --> 00:31:39,330 here if you can see this OK, the '7x + 5y' equals 12 simply 602 00:31:39,330 --> 00:31:40,630 turned out to be what? 603 00:31:40,630 --> 00:31:45,270 The equation of the line which was tangent to this curve at 604 00:31:45,270 --> 00:31:46,870 this particular point. 605 00:31:46,870 --> 00:31:49,850 What I'd like to show you in terms of one-to-oneness and 606 00:31:49,850 --> 00:31:53,530 single-valuedness is this. 607 00:31:53,530 --> 00:31:56,130 I told you I was interested in what was happening at this 608 00:31:56,130 --> 00:31:58,750 curve at the point 1 comma 1. 609 00:31:58,750 --> 00:32:02,690 Suppose I had said instead find the equation of the line 610 00:32:02,690 --> 00:32:07,130 tangent to this curve at the point whose x-coordinate is 1? 611 00:32:07,130 --> 00:32:09,850 Well, you see, there are two points on this curve whose 612 00:32:09,850 --> 00:32:12,700 x-coordinate is 1. 613 00:32:12,700 --> 00:32:16,170 You see, this is a double-valued curve. 614 00:32:16,170 --> 00:32:19,940 A given value of 'x' between these two extremes yields the 615 00:32:19,940 --> 00:32:21,300 two values of 'y'. 616 00:32:21,300 --> 00:32:23,910 I would have had no way of knowing which of the two 617 00:32:23,910 --> 00:32:26,580 y-values I meant, you see? 618 00:32:26,580 --> 00:32:30,090 And correspondingly, if somebody had said find the 619 00:32:30,090 --> 00:32:33,290 slope of the curve, of the equation of a line tangent to 620 00:32:33,290 --> 00:32:37,920 the curve, at the point whose y-coordinate is 1, notice that 621 00:32:37,920 --> 00:32:39,950 this is not 1 to 1. 622 00:32:39,950 --> 00:32:43,430 In other words, if the y-coordinate is 1, notice that 623 00:32:43,430 --> 00:32:47,980 I cannot distinguish between the point 1 comma 1 and the 624 00:32:47,980 --> 00:32:49,240 point what? 625 00:32:49,240 --> 00:32:51,810 Minus 1 comma 1. 626 00:32:51,810 --> 00:32:55,120 You see, there's again our problem with inverse functions 627 00:32:55,120 --> 00:32:56,770 and things of this particular type. 628 00:32:56,770 --> 00:32:59,830 If we're told the neighborhood of the point that we're 629 00:32:59,830 --> 00:33:01,760 interested in, we're fine. 630 00:33:01,760 --> 00:33:05,180 If all we're told are one of the coordinates and have to 631 00:33:05,180 --> 00:33:08,180 find the other, there is a certain amount of ambiguity. 632 00:33:08,180 --> 00:33:11,620 And keep in mind, by the way, that I deliberately rigged 633 00:33:11,620 --> 00:33:14,610 this problem to get something I could graph at least. 634 00:33:14,610 --> 00:33:18,010 In many cases, it's much more difficult to even visualize 635 00:33:18,010 --> 00:33:21,080 what the graph looks like, much more complicated. 636 00:33:21,080 --> 00:33:24,060 You see, computationally, this can be come quite a mess. 637 00:33:24,060 --> 00:33:27,370 The important point is that what we were doing implicitly 638 00:33:27,370 --> 00:33:33,280 here assumed on the explicit fact I could take this curve 639 00:33:33,280 --> 00:33:36,270 and break it down at the points where I have vertical 640 00:33:36,270 --> 00:33:40,380 tangents and look at this as two separate curves: 'c1', 641 00:33:40,380 --> 00:33:44,390 namely, the original curve, but restricted to 'y' being 642 00:33:44,390 --> 00:33:48,610 non-negative, and 'c2', the original curve, but restricted 643 00:33:48,610 --> 00:33:49,990 to 'y' being negative. 644 00:33:49,990 --> 00:33:54,880 In other words, I can look at 'c1' as being this piece and 645 00:33:54,880 --> 00:33:58,360 'c2' as being this piece. 646 00:33:58,360 --> 00:34:00,550 And again, the point is what? 647 00:34:00,550 --> 00:34:03,840 That whenever you're in the neighborhood of these points, 648 00:34:03,840 --> 00:34:04,840 you're in trouble. 649 00:34:04,840 --> 00:34:07,340 Because notice that no matter how small a neighborhood I 650 00:34:07,340 --> 00:34:12,219 pick, if I can't tell one of these branches from the other, 651 00:34:12,219 --> 00:34:15,170 no matter how I do this, I'm going to be caught on a 652 00:34:15,170 --> 00:34:18,409 multivalued part of the curve over here. 653 00:34:18,409 --> 00:34:22,310 Well, at any rate, I think this begins to show us what 654 00:34:22,310 --> 00:34:25,199 implicit differentiation means, why we have to be 655 00:34:25,199 --> 00:34:28,780 careful of points at which vertical tangent occur, but I 656 00:34:28,780 --> 00:34:31,949 would like before closing this lecture to generalize the 657 00:34:31,949 --> 00:34:35,650 concept of related rates and make this a little bit more 658 00:34:35,650 --> 00:34:38,719 applicable from a physical point of view. 659 00:34:38,719 --> 00:34:41,949 And that is when we say in something like this that let's 660 00:34:41,949 --> 00:34:44,449 assume that 'y' is a differentiable function of 661 00:34:44,449 --> 00:34:48,380 'x', there is no reason to have to assume that the 662 00:34:48,380 --> 00:34:51,929 variable you want to relate things to is 'x' itself. 663 00:34:51,929 --> 00:34:56,620 For example, let me do something which uses the same 664 00:34:56,620 --> 00:34:59,370 kind of an equation that we had before, but only from a 665 00:34:59,370 --> 00:35:00,760 different point of view. 666 00:35:00,760 --> 00:35:03,820 See, now instead of asking for the slope of this circle or 667 00:35:03,820 --> 00:35:06,560 what have you, let's work the question this way. 668 00:35:06,560 --> 00:35:08,450 Let's suppose we have a particle. 669 00:35:08,450 --> 00:35:11,130 The particle is moving along the curve 'x squared' plus 'y 670 00:35:11,130 --> 00:35:14,620 squared' equals 25 where for physical reasons we'll say 'x' 671 00:35:14,620 --> 00:35:18,330 and 'y' are in feet, and that we know at the point 3 comma 672 00:35:18,330 --> 00:35:20,940 4, 'dx dt'-- 673 00:35:20,940 --> 00:35:24,110 the horizontal component of the speed of the particle-- 674 00:35:24,110 --> 00:35:25,670 is 8 feet per second. 675 00:35:25,670 --> 00:35:28,550 And the question is we'd like to find 'dy dt', 676 00:35:28,550 --> 00:35:29,455 the vertical component. 677 00:35:29,455 --> 00:35:32,810 In other words, the particle is moving along the circle. 678 00:35:32,810 --> 00:35:37,620 We know that at the point 3 comma 4, 'x' is increasing at 679 00:35:37,620 --> 00:35:40,630 the rate of 6 feet per second while the increasing 680 00:35:40,630 --> 00:35:42,180 x-direction is this way. 681 00:35:42,180 --> 00:35:44,190 So somehow or other, we know that the particle is moving 682 00:35:44,190 --> 00:35:46,620 along the curve in this direction. 683 00:35:46,620 --> 00:35:49,340 You see, if it were moving in this direction, it's 684 00:35:49,340 --> 00:35:53,280 x-coordinate would be decreasing, not increasing. 685 00:35:53,280 --> 00:35:56,040 And the question that comes up is how do we find 686 00:35:56,040 --> 00:35:57,860 'dy dt' in this case? 687 00:35:57,860 --> 00:36:00,480 Well, see, all we do in this case is we say look-it, 688 00:36:00,480 --> 00:36:03,770 instead of assuming that 'y' is a differentiable function 689 00:36:03,770 --> 00:36:07,520 of 'x', why don't we assume that both 'x' and 'y' are 690 00:36:07,520 --> 00:36:09,860 differentiable functions of 't'? 691 00:36:09,860 --> 00:36:12,230 In other words, let's assume that 'x' and 'y' are 692 00:36:12,230 --> 00:36:21,100 differentiable functions of 't' that make this an identity 693 00:36:21,100 --> 00:36:22,540 in terms of 't'. 694 00:36:22,540 --> 00:36:26,190 Now again, as long as this is an identity and we're assuming 695 00:36:26,190 --> 00:36:28,070 that 'x' and 'y' are differentiable functions of 696 00:36:28,070 --> 00:36:30,990 't', there is absolutely no reason why we can't 697 00:36:30,990 --> 00:36:34,790 differentiate both sides of this equation with respect to 698 00:36:34,790 --> 00:36:37,120 't' instead of with respect to 'x'. 699 00:36:37,120 --> 00:36:38,940 If we do that, we get what? 700 00:36:38,940 --> 00:36:44,630 '2x' 'dx dt' plus '2y' 'dy dt' equals 0. 701 00:36:44,630 --> 00:36:47,610 You see the same principle as before even though we're now 702 00:36:47,610 --> 00:36:50,830 differentiating implicitly evidently what we could call 703 00:36:50,830 --> 00:36:52,160 parametric equations. 704 00:36:52,160 --> 00:36:54,880 We're assuming that 'x' and 'y' are differentiable 705 00:36:54,880 --> 00:36:56,700 functions of 't'. 706 00:36:56,700 --> 00:37:00,320 You see, from this identity now, we can conclude that 'dy 707 00:37:00,320 --> 00:37:04,950 dt' is 'minus x/y' times 'dx dt'. 708 00:37:04,950 --> 00:37:09,610 And now, you see, to polish off our particular problem, 709 00:37:09,610 --> 00:37:11,300 namely, we're interested in what? 710 00:37:11,300 --> 00:37:17,240 When 'x' was 3, 'y' was 4, and 'dx dt' is 8. 711 00:37:17,240 --> 00:37:21,190 In other words, I find out from this that 'dy dt' is 712 00:37:21,190 --> 00:37:24,310 minus 6 feet per second. 713 00:37:24,310 --> 00:37:29,240 You see, the idea was that 'x' and 'y' as positions of the 714 00:37:29,240 --> 00:37:33,070 point were related by the fact that 'x squared' plus 'y 715 00:37:33,070 --> 00:37:35,710 squared' had to equal 25. 716 00:37:35,710 --> 00:37:37,960 In other words, this is what we physically call a 717 00:37:37,960 --> 00:37:39,020 constraint. 718 00:37:39,020 --> 00:37:41,860 And by the way, this is what makes calculus such an 719 00:37:41,860 --> 00:37:44,370 important, powerful, analytical tool. 720 00:37:44,370 --> 00:37:47,320 Notice that in doing this particular problem, I never 721 00:37:47,320 --> 00:37:50,500 had to know explicitly what functions 'x' 722 00:37:50,500 --> 00:37:52,000 and 'y' were of 't'. 723 00:37:52,000 --> 00:37:53,600 All I had to know was what? 724 00:37:53,600 --> 00:37:56,140 That 'x' and 'y' were differentiable 725 00:37:56,140 --> 00:37:57,770 functions of 't'. 726 00:37:57,770 --> 00:38:01,030 I don't care how the particle was moving away from the point 727 00:38:01,030 --> 00:38:04,160 3 comma 4 as long as 'x' and 'y' were differentiable 728 00:38:04,160 --> 00:38:09,130 functions of 't', this is how the rates 'dy dt' and 'dx dt' 729 00:38:09,130 --> 00:38:10,300 had to be related. 730 00:38:10,300 --> 00:38:13,220 Well, let me summarize what was really important 731 00:38:13,220 --> 00:38:15,890 conceptually about today's lecture. 732 00:38:15,890 --> 00:38:18,340 The most important thing conceptually was what? 733 00:38:18,340 --> 00:38:22,110 With all of our talk about explicitly writing an output 734 00:38:22,110 --> 00:38:25,400 and an input, in many important mathematical 735 00:38:25,400 --> 00:38:29,810 relationships, the variables that we're concerned with are 736 00:38:29,810 --> 00:38:31,670 implicitly related. 737 00:38:31,670 --> 00:38:34,170 In other words, we are not told what 'y' looks like in 738 00:38:34,170 --> 00:38:37,880 terms of 'x', but rather how 'x' and 'y' are interrelated, 739 00:38:37,880 --> 00:38:40,680 and from here we have to find a derivative. 740 00:38:40,680 --> 00:38:44,190 And the way we do this is that we make the assumption that an 741 00:38:44,190 --> 00:38:47,910 appropriate identity exists, that 'y' is an appropriate 742 00:38:47,910 --> 00:38:51,310 function of 'x' that makes the relationship an identity. 743 00:38:51,310 --> 00:38:54,350 The validity of when you can do this, for example, 744 00:38:54,350 --> 00:38:57,660 geometrically, when do you wind up not having a point 745 00:38:57,660 --> 00:39:00,730 that includes a vertical tangent, a point where you 746 00:39:00,730 --> 00:39:02,910 can't separate the curve doubling back? 747 00:39:02,910 --> 00:39:05,780 That turns out to be a rather difficult point from an 748 00:39:05,780 --> 00:39:08,465 analytical point of view, a point that we will return to 749 00:39:08,465 --> 00:39:11,860 in a more advanced context when we deal with functions of 750 00:39:11,860 --> 00:39:13,030 several variables. 751 00:39:13,030 --> 00:39:15,850 But for the time being, all I want you to be left with in 752 00:39:15,850 --> 00:39:20,020 this lecture is the feeling that we can, given an implicit 753 00:39:20,020 --> 00:39:23,770 relationship under the proper conditions, assume that the 754 00:39:23,770 --> 00:39:26,870 appropriate explicit relationship exists and that 755 00:39:26,870 --> 00:39:30,540 we can differentiate both sides as an identity. 756 00:39:30,540 --> 00:39:33,700 Well, enough said for today, and until next time, goodbye. 757 00:39:36,740 --> 00:39:39,270 NARRATOR: Funding for the publication of this video was 758 00:39:39,270 --> 00:39:43,990 provided by the Gabriella and Paul Rosenbaum Foundation. 759 00:39:43,990 --> 00:39:48,170 Help OCW continue to provide free and open access to MIT 760 00:39:48,170 --> 00:39:52,360 courses by making a donation at ocw.mit.edu/donate.