1 00:00:00,012 --> 00:00:05,209 [MUSIC] Up until now, we've been considering the functions that you can 2 00:00:05,209 --> 00:00:10,680 get by starting with variables and numbers, and combining them using sums, 3 00:00:10,680 --> 00:00:16,533 products, quotients, and differences. So we can write down, you know, functions 4 00:00:16,533 --> 00:00:21,442 like f(x)=x2+x/(x+1^)^10+x, all of this, -1/x. 5 00:00:21,442 --> 00:00:27,372 But there's more things in heaven and earth that are dreamt of in your rational 6 00:00:27,372 --> 00:00:31,582 functions. For instance, can you imagine a function 7 00:00:31,582 --> 00:00:37,137 f, which is its own derivative? I'm looking for a functions, that if I 8 00:00:37,137 --> 00:00:42,670 differentiate it, I get back itself. Now, if you're thinking cleverly, you 9 00:00:42,670 --> 00:00:46,006 might be able to cook up such a function very quickly. 10 00:00:46,006 --> 00:00:50,364 What if f is just the zero function? Or if I differentiate the zero function, 11 00:00:50,364 --> 00:00:53,255 differentiate a constant function, that's zero. 12 00:00:53,255 --> 00:00:57,072 So this would be an example of function in its own deriviative. 13 00:00:57,072 --> 00:01:02,092 But, that's not a very exciting example. [SOUND] So let's try to think of a 14 00:01:02,092 --> 00:01:05,372 nonzero function, which is its own derivative. 15 00:01:05,372 --> 00:01:11,006 How might we try to find such a function? So to make this concrete, I'm looking for 16 00:01:11,006 --> 00:01:16,486 a function f, so if I differentiate it, I get itself and just make sure that it's 17 00:01:16,486 --> 00:01:21,972 not the zero function. Let's have this function output one if I plug in zero. 18 00:01:21,972 --> 00:01:27,617 Now, how could I rig this function to have the correct derivative at zero? 19 00:01:27,617 --> 00:01:33,497 If the derivative of this function itself, the derivative of this function 20 00:01:33,497 --> 00:01:38,282 at zero should also be one. Can you think of a function whose value 21 00:01:38,282 --> 00:01:41,412 at zero is one and whose derivative at zero is one? 22 00:01:41,412 --> 00:01:44,488 Yes. Here is a function, f(x)=1+x. 23 00:01:44,488 --> 00:01:50,315 This function's value with zero is one, and this function's derivative at zero is 24 00:01:50,315 --> 00:01:54,237 also one. But if the derivative of f is f, then the 25 00:01:54,237 --> 00:02:00,490 derivative of the derivative of f is also the derivative of f, which is also f. 26 00:02:00,490 --> 00:02:04,252 So, the second derivative must be f as well. 27 00:02:04,252 --> 00:02:08,571 So, if this function is its own derivative, the second derivative of f 28 00:02:08,571 --> 00:02:12,031 would also be equal to f. Now, specifically, at the point zero, 29 00:02:12,031 --> 00:02:16,622 that means the second derivative of the function at the point zero would be the 30 00:02:16,622 --> 00:02:19,370 function's value with zero which should be equal to one. 31 00:02:19,370 --> 00:02:22,872 is this function's second derivative at zero equal to one? 32 00:02:22,872 --> 00:02:26,554 No. If I differentiate this function twice, I 33 00:02:26,554 --> 00:02:31,442 just get the zero function, but I can fix this at least to the point 34 00:02:31,442 --> 00:02:36,732 zero. If I add on x^/2, now, this function's derivative at zero 35 00:02:36,732 --> 00:02:41,378 is one and this function's second derivative at zero is one. 36 00:02:41,378 --> 00:02:47,939 Since f is its own derivative, the third derivative of f must also be f. 37 00:02:47,939 --> 00:02:50,506 No worries. If the thid derivative of f is also equal 38 00:02:50,506 --> 00:02:56,375 to f, which is a consequence of the derivative of f being equal to f. 39 00:02:56,375 --> 00:03:00,550 That means the third derivative of f at zero is equal to one, but this thing's 40 00:03:00,550 --> 00:03:05,128 third derivative is just zero. But if I add on x^3/6, now, if I take the 41 00:03:05,128 --> 00:03:09,328 third derivative of this function and plug in zero, I get out one. 42 00:03:09,328 --> 00:03:12,306 The fourth derivative of f must also be f. 43 00:03:12,306 --> 00:03:15,075 Okay, yeah. I gotta deal with the fourth derivative. 44 00:03:15,075 --> 00:03:18,408 I'm out of space here, but no worries, I'll just get more paper. 45 00:03:18,408 --> 00:03:22,694 Here, I've written down a function whose value at zero is one, whose derivative at 46 00:03:22,694 --> 00:03:26,864 zero is one, whose second derivative at zero is one, whose third derivative at 47 00:03:26,864 --> 00:03:29,658 zero is one, whose fourth derivative at zero is one. 48 00:03:29,658 --> 00:03:33,947 And you can see, this is sort of building me closer and closer to a function which 49 00:03:33,947 --> 00:03:37,566 is its own derivative. If I try to differentiate this function, 50 00:03:37,566 --> 00:03:41,371 what do I get? Well, the derivative of one is zero, but the derivative of x is 51 00:03:41,371 --> 00:03:47,389 one, and the derivative of x^2/2 is x, and the derivative of x^3/6, well, that's 52 00:03:47,389 --> 00:03:51,777 x^2/2, and the derivative of x^4/24, well, that's x^3/6. 53 00:03:51,777 --> 00:03:54,568 And yeah, I mean, this function isn't its own 54 00:03:54,568 --> 00:03:57,946 derivative, but things are looking better and better. 55 00:03:57,946 --> 00:04:01,195 But the fifth derivative of f must also be equal to f. 56 00:04:01,195 --> 00:04:03,367 Okay, yeah. The fifth derivative. 57 00:04:03,367 --> 00:04:06,090 I'll just add on another term, x^5/120. 58 00:04:06,090 --> 00:04:10,112 And if you check, take the fifth derivative now of this function, 59 00:04:10,112 --> 00:04:13,429 its value at zero is one. I've written down a function, 60 00:04:13,429 --> 00:04:16,362 so that if I take its fifth derivative at zero, 61 00:04:16,362 --> 00:04:19,314 I get one. The sixth derivative of f must be equal 62 00:04:19,314 --> 00:04:22,213 to f. The sixth derivative I am out of room, 63 00:04:22,213 --> 00:04:26,048 but here we, go. Here is a polynomial whose value first, 64 00:04:26,048 --> 00:04:31,334 second, third, fourth, fifth and sixth derivative at the point zero are all one. 65 00:04:31,334 --> 00:04:36,517 And you can see how this is edging us a little bit closer still to a function 66 00:04:36,517 --> 00:04:41,185 which is its own derivative, because if I differentiate this function, 67 00:04:41,185 --> 00:04:45,742 yeah, the one goes away, but the x gives me the one back, and the x^2/2, when I 68 00:04:45,742 --> 00:04:51,257 differentiate that, gives me the x. X^3/6, when I differentiate that, gives 69 00:04:51,257 --> 00:04:55,858 me x^2/2. X^4/24, when I differentiate that gives 70 00:04:55,858 --> 00:05:00,602 me x^3/6. x^5/120, when I differentiate that, gives 71 00:05:00,602 --> 00:05:09,222 me x^4/24. X^6/720, when I differentiate that, I've got x^5/120. And now, of 72 00:05:09,222 --> 00:05:11,866 course, these aren't the same, but I'm doing better. 73 00:05:11,866 --> 00:05:14,086 The seventh derivative must be equal to f. 74 00:05:14,086 --> 00:05:19,982 To get the seventh derivative at zero to be correct, I'll add on x^7/5040. 75 00:05:19,982 --> 00:05:24,897 The eighth derivative, I'll add on x^8/40,320. 76 00:05:24,897 --> 00:05:29,722 The ninth derivative, I'll add on x^9/362,880. 77 00:05:29,722 --> 00:05:32,647 Okay, okay. This is, isn't working out. 78 00:05:32,647 --> 00:05:38,192 We're not really succeeding in writing down a function which is its own 79 00:05:38,192 --> 00:05:42,492 derivative. Let's introduce a new friend, the number 80 00:05:42,492 --> 00:05:46,977 e to help us. Here is how we're going to get to the 81 00:05:46,977 --> 00:05:48,567 number e. This limit, 82 00:05:48,567 --> 00:05:54,687 the limit of 2^h-1/h as h approaches zero is about 0.69, a little bit more. 83 00:05:54,687 --> 00:05:58,422 On the other hand, this limit, the limit of three to the h minus one 84 00:05:58,422 --> 00:06:02,712 over h as h approaches zero is a little bit more than one, 85 00:06:02,712 --> 00:06:07,211 it's about 1.099. If you think of this as a function that 86 00:06:07,211 --> 00:06:11,860 depends not on two or three, you could define a function g(x), right? 87 00:06:11,860 --> 00:06:14,811 The limit as h approaches zero of x to the h minus one over h. In that case, 88 00:06:14,811 --> 00:06:18,088 this first statement, the statement about the limit of two to the h minus one over 89 00:06:18,088 --> 00:06:23,192 h, that's really saying that g(2) is a bit less than one. 90 00:06:23,192 --> 00:06:27,119 And, this statement over here, and if you think of this as a function g, this 91 00:06:27,119 --> 00:06:30,123 statement is really saying that g(3) is a bit more than 1. 92 00:06:30,123 --> 00:06:34,260 Now, if you're also willing to concede that this function g is continuous, which 93 00:06:34,260 --> 00:06:37,631 is a huge assumption to make, but let's suppose that's the case. 94 00:06:37,631 --> 00:06:41,672 If that's the case, I've got a continuous function, let's say, and if I plug in 95 00:06:41,672 --> 00:06:44,188 two, I get a value that's a little bit less than one, 96 00:06:44,188 --> 00:06:47,842 and if I plug in three, I get a value that's a little bit more than one. 97 00:06:47,842 --> 00:06:54,194 Well, by the intermediate value theorem, that would tell me there must be some 98 00:06:54,194 --> 00:07:00,116 input so that the output is exactly one. I'm going to call that input e. 99 00:07:00,116 --> 00:07:06,835 In other words, e is the number, so that the limit of e^h-1/h as h approaches zero 100 00:07:06,835 --> 00:07:11,235 is equal to one, and this number is about 2.7183 blah, 101 00:07:11,235 --> 00:07:16,583 blah, blah. Now lets consider the function f(x)=e^x. 102 00:07:16,583 --> 00:07:20,816 So let's think about this function f(x)=e^x. 103 00:07:20,816 --> 00:07:27,438 Now, what's the derivative of this function? Well, from the definition, 104 00:07:27,438 --> 00:07:31,309 that's the limit as h approaches zero of f of x plus h minus f of x over h. 105 00:07:31,309 --> 00:07:38,675 Now, in this case f is just e^x, so this is the limit as h approaches zero 106 00:07:38,675 --> 00:07:41,725 of e to the x plus h minus e to the x over h. 107 00:07:41,725 --> 00:07:43,662 And this is e to the x plus h minus e to the x over h, 108 00:07:43,662 --> 00:07:47,593 so I can write this as e to the x times e to the h. 109 00:07:47,593 --> 00:07:54,021 This is the limit then as h goes to zero of e to the x, e to the h minus e to the 110 00:07:54,021 --> 00:07:57,709 x over h, Now I've got a common factor of e to the 111 00:07:57,709 --> 00:08:00,679 x. So I'll pull out that common factor and 112 00:08:00,679 --> 00:08:05,081 I've got the limit as h approaches zero of e to the x times e to the h minus one 113 00:08:05,081 --> 00:08:08,330 over h. Now, as far as h is concerned, e to the x 114 00:08:08,330 --> 00:08:12,043 is a constant, and this is the limit of a constant times 115 00:08:12,043 --> 00:08:15,065 something, so I can pull that constant out. 116 00:08:15,065 --> 00:08:20,922 This is e to the x time the limit as h goes to zero of e to the h minus one over 117 00:08:20,922 --> 00:08:24,142 h. But I picked the number e precisely, so 118 00:08:24,142 --> 00:08:29,067 that this limit was eqal to one. And consequently, this is e to the x 119 00:08:29,067 --> 00:08:31,732 times one, this is just e to the x. Look, 120 00:08:31,732 --> 00:08:36,582 I've got a function whose derivative is the same function. 121 00:08:36,582 --> 00:08:40,032 We've done it. We've found a function which is its own 122 00:08:40,032 --> 00:08:43,772 derivative. The derivative of e^x is e^x. 123 00:08:43,772 --> 00:08:48,107 E^x is honestly different from this polynomials and rational functions. 124 00:08:48,107 --> 00:08:52,542 We couldn't have produced that number e without using a limit. 125 00:08:52,542 --> 00:08:56,165 E^x is the function that only calculus could provide us with.