1 00:00:00,012 --> 00:00:03,953 [music]. Let's combine the change rule and the 2 00:00:03,953 --> 00:00:10,681 derivative of sign to differentiate a slightly more complicated function. 3 00:00:10,681 --> 00:00:14,706 Let's try to differentiate sine of x squared. 4 00:00:14,706 --> 00:00:20,506 We can realize this function as the composition of two functions. 5 00:00:20,506 --> 00:00:26,428 I can write sine of x squared as the composition of f and g, where f is the 6 00:00:26,428 --> 00:00:30,365 sine function and g is the squaring function. 7 00:00:30,365 --> 00:00:35,015 Now, how do I differentiate a composition of two functions? 8 00:00:35,015 --> 00:00:39,940 I use the chain rule. So I differentiate f and the derivative of 9 00:00:39,940 --> 00:00:44,165 f is cosine x, where the derivative of sine is cosine. 10 00:00:44,165 --> 00:00:52,021 And the derivative of g is just 2x. So now I want to differentiate the 11 00:00:52,021 --> 00:01:01,109 composition of f and g and that's by the chain rule f prime of g of x times g prime 12 00:01:01,109 --> 00:01:05,240 of x. In this case, f prime is cosine, so it's 13 00:01:05,240 --> 00:01:12,273 cosine of just g of x, which is x squared times the derivative of g, which is 2 x. 14 00:01:12,273 --> 00:01:19,555 So, the derivative of sine x squared with respect to x, is cosine of x squared times 15 00:01:19,555 --> 00:01:23,629 2x. Honestly, this is a pretty neat example. 16 00:01:23,629 --> 00:01:28,957 In magnitude, this function, sine of x squared, is no bigger than 1. 17 00:01:28,957 --> 00:01:33,471 And yet, what do we know about this function's derivative? 18 00:01:33,471 --> 00:01:39,056 Well, the derivative of sine of x squared is cosine of x squared times 2x. 19 00:01:39,056 --> 00:01:42,513 And that function can be as large as you like. 20 00:01:42,513 --> 00:01:48,564 You can make cosine of x squared times 2x as big as you want, as long as you choose 21 00:01:48,564 --> 00:01:52,870 x appropriately. So what we have here is a function which 22 00:01:52,870 --> 00:01:57,052 isn't very big. The function's value is no bigger than 1 23 00:01:57,052 --> 00:02:01,656 in magnitude, but the function's derivative is very large. 24 00:02:01,656 --> 00:02:06,080 And you can see that on the graph. The values of this function really aren't 25 00:02:06,080 --> 00:02:08,692 that large. The values are all hugging zero. 26 00:02:08,692 --> 00:02:12,345 But the derivative, the slope of the tangent line is enormous. 27 00:02:12,345 --> 00:02:15,284 Look over here. If you imagine a tangent line, that 28 00:02:15,284 --> 00:02:17,837 tangent line is going to have enormous slope. 29 00:02:17,837 --> 00:02:20,718 The derivative over here is going to be very large. 30 00:02:20,719 --> 00:02:25,932 In spite of the fact, that the actual values of the function really aren't that 31 00:02:25,932 --> 00:02:29,096 large. This actually provides another lesson. 32 00:02:29,096 --> 00:02:34,553 Just because 2 functions are nearby in value, doesn't mean that their derivatives 33 00:02:34,553 --> 00:02:39,750 are anything close to each other. For instance, here is the graph of the 34 00:02:39,750 --> 00:02:44,484 cosine function. And here is the graph of a function sine 35 00:02:44,484 --> 00:02:51,200 of x square over 10 plus cosine of x. Since sine of x squared is between minus 1 36 00:02:51,200 --> 00:02:56,462 and 1, this differs from the cosine by no more than a tenth. 37 00:02:56,462 --> 00:03:02,230 And, yeah, you can see the graph is really close to the graph of cosine and yet this 38 00:03:02,230 --> 00:03:07,353 graph is way more wriggly. Let's zoom in and we can see the same sort 39 00:03:07,353 --> 00:03:10,799 of thing. Here's a zoomed in copy of just a cosine 40 00:03:10,799 --> 00:03:14,161 curve. And here's what happens if you zoom in on 41 00:03:14,161 --> 00:03:18,284 this other function. And in terms of the value, this other 42 00:03:18,284 --> 00:03:22,380 function really isn't different from cosine very much. 43 00:03:22,380 --> 00:03:27,658 But in terms of derivative, this function is totally different than cosine. 44 00:03:27,658 --> 00:03:33,208 This function is super wiggly, so the derivative of this function is enormous, 45 00:03:33,208 --> 00:03:38,173 even though the derivative of cosine is no bigger than 1 in magnitude. 46 00:03:38,173 --> 00:03:42,499 If you think this is kind of an interesting example, it's worth trying to 47 00:03:42,499 --> 00:03:47,396 cook up an even more elaborate example. Here's a very specific challenge for you. 48 00:03:47,396 --> 00:03:52,154 Can you find a function that, just make one up, so that your functions values and 49 00:03:52,154 --> 00:03:56,906 magnitude are less than c, and your functions derivative and magnitude is less 50 00:03:56,906 --> 00:03:59,922 than c? So I want a specific number c, so that no 51 00:03:59,922 --> 00:04:04,947 matter what value of x you plug in, the function's value there and the function's 52 00:04:04,947 --> 00:04:07,869 derivative there is less than c in magnitude. 53 00:04:07,869 --> 00:04:12,685 But, I want that function to have second derivative which can't be bounded by a 54 00:04:12,685 --> 00:04:17,104 constant. I want you to cook up a function so that, 55 00:04:17,104 --> 00:04:22,495 yeah, the values of the first derivative are bounded by c. 56 00:04:22,495 --> 00:04:29,781 But the second derivative can be as big or as negative as I'd like, by choosing x 57 00:04:29,781 --> 00:04:35,934 appropriately. Can you find a function like that?