1 00:00:00,025 --> 00:00:08,300 [music] Differentiating is a process really guaranteed to work. 2 00:00:08,300 --> 00:00:10,990 Alright, if somebody writes down a function and asks you to differentiate it. 3 00:00:10,990 --> 00:00:13,270 You can. It might be painful, alright, it might 4 00:00:13,270 --> 00:00:16,813 involve a lot of steps but it's just a matter of applying those steps carefully. 5 00:00:16,813 --> 00:00:20,870 In contrast antidifferentiation might be really hard. 6 00:00:20,870 --> 00:00:25,762 Let's try a really hard example now. For instance, can you find some function 7 00:00:25,762 --> 00:00:30,065 so if I differentiate that function, I get e to the negative x squared? 8 00:00:30,065 --> 00:00:34,446 Which is just another way of saying, I want to anti-differentiate e to the 9 00:00:34,446 --> 00:00:36,816 negative x squared. Right, what is that? 10 00:00:36,816 --> 00:00:40,638 Well, let's make a guess and see if our guess is correct. 11 00:00:40,638 --> 00:00:47,506 I mean maybe, an antiderivative of e to the negative x squared is I don't know, e 12 00:00:47,506 --> 00:00:52,062 to the negative x squared, over minus 2 x plus c, right. 13 00:00:52,062 --> 00:00:55,367 I mean I'm not saying this is correct, but it could be true. 14 00:00:55,368 --> 00:00:57,533 If this were the case, I can check it, right. 15 00:00:57,533 --> 00:01:00,517 I can differentiate this side and see if I get this. 16 00:01:00,518 --> 00:01:07,487 So let's differentiate e to the minus x squared over minus 2 x. 17 00:01:07,487 --> 00:01:12,719 That's sort of the same thing as differentiating e to the minus x squared 18 00:01:12,719 --> 00:01:18,181 times minus 2 x to the negative 1st power. So that's a derivative of a product, which 19 00:01:18,181 --> 00:01:20,954 is assuming the product rule's good for, right? 20 00:01:20,954 --> 00:01:26,449 So first I'll differentiate e to the minus x squared. 21 00:01:26,450 --> 00:01:32,324 And I'll multiply that by the second term minus two x to the minus 1st power and 22 00:01:32,324 --> 00:01:37,784 I'll add to that the first term, e to the minus x squared times the derivative of 23 00:01:37,784 --> 00:01:41,958 the second term, which is minus two x to the minus 1st power. 24 00:01:41,958 --> 00:01:44,442 But what's the derivative of e to the minus x squared? 25 00:01:44,442 --> 00:01:48,900 That's e to the minus x square cause the derivative of e to the is e to the. 26 00:01:48,900 --> 00:01:52,483 Times the derivative of minus x squared which is minus 2 x. 27 00:01:52,483 --> 00:02:00,013 And then it's times negative 2 x to the negative 1st power, right? 28 00:02:00,013 --> 00:02:05,306 Plus e to the minus x squared times the derivative of minus 2 x to the minus 1st 29 00:02:05,306 --> 00:02:07,820 power. I'll just copy that down and we'll deal 30 00:02:07,820 --> 00:02:10,911 with it in a second. Because this is pretty exciting right here 31 00:02:10,911 --> 00:02:13,588 right. This term cancels, this term cancels, this 32 00:02:13,588 --> 00:02:17,215 is looking pretty good right. Maybe the antiderivative of e to the minus 33 00:02:17,215 --> 00:02:20,682 x squared really is this. Because when I differentiate this thing, 34 00:02:20,682 --> 00:02:24,973 at least I get a e to the minus x squared. The bad news is that I also get this term, 35 00:02:24,973 --> 00:02:28,090 right. And this terms out to be e to the minus x 36 00:02:28,090 --> 00:02:33,105 squared, now I gotta deal with this. It's again e to the minus x squared and 37 00:02:33,105 --> 00:02:38,406 then times the derivative of this. Well, when I differentiate this, I get a 38 00:02:38,406 --> 00:02:43,250 negative sign times this thing to the negative 2nd power, right? 39 00:02:43,251 --> 00:02:48,567 And then by the chain rule, the derivative of the inside, which is negative 2. 40 00:02:48,567 --> 00:02:52,840 And yeah, I mean this is, this is bad right. 41 00:02:52,840 --> 00:02:57,396 I mean I differentiated this thing right here and I got back some of that included 42 00:02:57,396 --> 00:03:01,070 in the e to the minus x squared but also includes non zero term. 43 00:03:01,070 --> 00:03:07,305 So as result this is not the case. Well that didn't work but it wasn't from 44 00:03:07,305 --> 00:03:10,878 lack trying. In fact, an antiderivative for e to the 45 00:03:10,878 --> 00:03:14,930 negative x squared cannot be expressed using elementary functions. 46 00:03:14,930 --> 00:03:18,975 What do I mean by elementary functions? I mean things like polynomials, trig 47 00:03:18,975 --> 00:03:21,983 functions, e to the x, log, things like that, right? 48 00:03:21,983 --> 00:03:24,385 I mean, this is really a surprising result. 49 00:03:24,385 --> 00:03:28,416 I mean there is a function whose derivative is e to the negative x squared, 50 00:03:28,416 --> 00:03:33,180 but it's not a function I can write down. Using the functions that I already have in 51 00:03:33,180 --> 00:03:36,226 hand. May be e to the negative x squared just an 52 00:03:36,226 --> 00:03:41,096 isolated terrible example. I can't answer differentiate e to the e to 53 00:03:41,096 --> 00:03:45,586 the x using elementary functions. I can't answer differentiate log, log x 54 00:03:45,586 --> 00:03:49,157 using elementary functions. I can't even answer differentiate this 55 00:03:49,157 --> 00:03:51,480 very reasonable looking algebraic function. 56 00:03:51,480 --> 00:03:56,759 The square root of 1 plus x to the 4th. I can't find an anti-derivative of this 57 00:03:56,759 --> 00:04:00,000 just using elementary functions. So, what does this mean? 58 00:04:00,000 --> 00:04:04,270 Let's try to summarize the situation. It's not just that many functions are hard 59 00:04:04,270 --> 00:04:08,144 to antidifferentiate. It's that many functions are impossible to 60 00:04:08,144 --> 00:04:12,354 antidifferentiate, not in the sense that they don't have an antiderivative. 61 00:04:12,354 --> 00:04:16,106 But in the sense that we're not going to succeed in writing down that 62 00:04:16,106 --> 00:04:19,377 antiderivative using the functions that we have at hand. 63 00:04:19,378 --> 00:04:23,550 In light of the difficulties of antidifferentiating, the fact that there's 64 00:04:23,550 --> 00:04:27,840 no guaranteed answer, it means that we should be happy that we can ever evaluate 65 00:04:27,840 --> 00:04:32,111 an antiderivative problem right. It also reflects the fact that some real 66 00:04:32,111 --> 00:04:38,002 creativity is needed. You know, if an answer's not guaranteed 67 00:04:38,002 --> 00:04:46,913 then potentially we're going to require more creativity to cook up the answers, 68 00:04:46,913 --> 00:04:49,333 even when they exist.