1 00:00:00,012 --> 00:00:06,082 , In our quest for a function which with it's own derivative, we met e to the x. 2 00:00:06,207 --> 00:00:11,523 Remember, the derivative of e to the x is e to the x. What's the inverse function 3 00:00:11,535 --> 00:00:17,492 for e to the x? What function undoes that sort of exponentiation? Well, we really 4 00:00:17,504 --> 00:00:23,235 don't have a name for that function yet, so we're just going to call it Log. 5 00:00:23,238 --> 00:00:28,486 So in symbols, if e to the x is equal to y, then log y equals x, right? 6 00:00:28,490 --> 00:00:35,685 Log is the inverse function for e to the, the log of something I must raise e to get 7 00:00:35,697 --> 00:00:42,941 back the thing I plugged into log. These logs or logarithms are super important for 8 00:00:42,953 --> 00:00:49,052 a ton of reasons. Take a look at this. Since e to the x plus y is e to the x 9 00:00:49,064 --> 00:00:55,348 times e to y, right? This is the property of exponents, if you like. There's a 10 00:00:55,360 --> 00:01:01,970 corresponding statement about log. Log of a times b is log of a plus log of b. Or, a 11 00:01:01,982 --> 00:01:08,308 shorthand way to say that is that logarithms transform products into sums. 12 00:01:08,314 --> 00:01:14,332 This is a big reason why we care so much about logs. Once we've got this new 13 00:01:14,344 --> 00:01:20,344 function, log, we can ask what's the derivative of log? So, if f of x is e to 14 00:01:20,356 --> 00:01:27,418 the x, the inverse function is log. If I want to now differentiate log, I can 15 00:01:27,430 --> 00:01:35,230 use the inverse function theorem. So, the derivative of the inverse function is 1 16 00:01:35,242 --> 00:01:42,743 over the derivative of the original function evaluated at the inverse function 17 00:01:42,755 --> 00:01:49,026 of x. Now, the neat thing here is that the derivative of e to the x is itself. So, f 18 00:01:49,038 --> 00:01:55,293 prime is just f, and I'm left with f of f inverse of x. It's e to the log of x. But 19 00:01:55,744 --> 00:02:01,949 log of x tells me what I have to plug into e to get out the input, right? f of the 20 00:02:01,961 --> 00:02:06,886 inverse function of f is just, it would be the, the same input again. So, this is 1 21 00:02:06,886 --> 00:02:12,649 over x. So, the derivative of log x is just 1 over x. And you can really see this 22 00:02:12,661 --> 00:02:17,685 fact on the graph. Here's a graph of y equals log x. And I should warn you right 23 00:02:17,697 --> 00:02:22,671 off the bat that the x-axis and the y-axis have totally different scales. The x-axis 24 00:02:22,996 --> 00:02:28,023 goes from 1 to 100. The y-axis in this plot goes from 0 to 5. it's going to make 25 00:02:28,035 --> 00:02:33,322 it not so easy to tell the exact values of the slopes and tangent lines, but you can 26 00:02:33,334 --> 00:02:38,188 see from this graph the important qualitative feature. That the graph is 27 00:02:38,200 --> 00:02:43,836 getting less and less slopey. And if you like, it's flattening out as the input 28 00:02:43,848 --> 00:02:49,665 gets bigger. if I put down a tangent line and I start moving the point that I'm 29 00:02:49,677 --> 00:02:54,335 taking the tangent line at to the right, you can see the tangent line slope is 30 00:02:54,347 --> 00:02:59,095 getting closer and closer to zero. And, of course, that's reflected by knowing the 31 00:02:59,107 --> 00:03:03,915 derivative of log x is 1 over x. So, if x is really big, the tangent line at x is 32 00:03:03,927 --> 00:03:09,617 really close to zero in slope. Think about log of a really big number. For instance, 33 00:03:09,738 --> 00:03:14,973 what's log of a million? A log of a million is about 13.815510. And, of 34 00:03:14,985 --> 00:03:20,690 course, it keeps going. I, it's an irrational number. But, now the derivative 35 00:03:20,702 --> 00:03:26,205 of log, right? Is 1 over its input. So, what does that tell you that you might 36 00:03:26,217 --> 00:03:31,723 think log of a million and 1 is equal to? Well, the derivative tells you how much 37 00:03:31,735 --> 00:03:37,308 wiggling input affects the output. So, if I wiggle the input by 1, you expect the 38 00:03:37,320 --> 00:03:42,972 output to change by about the derivative. And yeah, log of a million and 1 is about 39 00:03:42,984 --> 00:03:48,902 13.815511, right? What's being affected here is in the millionths place after the 40 00:03:48,914 --> 00:03:54,569 decimal point, right? It's the 6th digit after the decimal point because it's being 41 00:03:54,581 --> 00:03:59,145 affected like a change of 1 over a million. All right? I'm changing the 42 00:03:59,157 --> 00:04:04,178 output by about a millionth. At this point, we can also handle logs with other 43 00:04:04,190 --> 00:04:09,092 bases. So, let's suppose I want to differentiate log of x base b, right? This 44 00:04:09,104 --> 00:04:14,153 is the number that I'd raise b to, to get back x. Well, there's a change of base 45 00:04:14,165 --> 00:04:20,034 formula for log. This is the same as the derivative of, say, the natural log of x 46 00:04:20,046 --> 00:04:26,019 over the log of b. But the log of b is a constant, and the derivative of a constant 47 00:04:26,031 --> 00:04:31,938 multiple is just that constant multiple times the derivative. So, this is 1 over 48 00:04:31,950 --> 00:04:37,428 log b times the derivative of, here's a natural log of x. But I know the 49 00:04:37,440 --> 00:04:41,518 derivative of the natural log of x, it's 1 over x. 50 00:04:41,519 --> 00:04:47,948 So, the derivative of log of x base b is 1 over log b times 1 over x. Or maybe 51 00:04:47,960 --> 00:04:54,951 another way to write this would be 1 over x times log b, if you prefer writing it 52 00:04:54,963 --> 00:05:00,778 that way. e to the x is a sort of key that unlocks how to understand the derivative 53 00:05:00,790 --> 00:05:05,991 of a ton of other exponential functions. For example, now that we know how to 54 00:05:06,003 --> 00:05:11,035 differentiate e to the x, we can also differentiate 2 to the x. So, let's 55 00:05:11,047 --> 00:05:16,023 suppose I want to differentiate 2 to the x. Now, you might just memorize some 56 00:05:16,035 --> 00:05:21,677 formula for differentiating this. But it's easier, I think better, to just recreate 57 00:05:21,689 --> 00:05:26,590 this function out of the functions that you already know all the derivatives of. 58 00:05:26,701 --> 00:05:32,082 So, in this case, let's replace 2 by e to the log 2 to the x, right? So, instead of 59 00:05:32,094 --> 00:05:37,294 writing 2 here, I've just written e to the log 2, this is just 2. But I've got e to 60 00:05:37,306 --> 00:05:42,383 the log 2 to the x and that's the same as e to the log 2 times x. You know, this is 61 00:05:42,395 --> 00:05:47,272 a composition of functions that I know how to differentiate. I know how to 62 00:05:47,284 --> 00:05:52,580 differentiate e to the, and I know how to differentiate constant multiple times x. 63 00:05:52,962 --> 00:05:57,789 So, by the chain rule, it's the derivative of the outside function. So, which is 64 00:05:57,801 --> 00:06:02,221 itself, e to the, at the inside function, which is log 2 times x, times the 65 00:06:02,233 --> 00:06:07,423 derivative of the inside function which in this case is log 2 log x. So, I'm just 66 00:06:07,435 --> 00:06:12,635 going to multiply by log 2. Now, I could kind of make this look a little bit nicer, 67 00:06:12,752 --> 00:06:18,195 right? e to the log 2 times x, well, that's just 2 to the x times, again log 2. 68 00:06:18,312 --> 00:06:23,330 So, the derivative of 2 to the x is 2 to the x times log 2. And, of course, 2 69 00:06:23,342 --> 00:06:29,170 didn't play any significant role here. I could have replaced 2 by any other number 70 00:06:29,422 --> 00:06:34,695 and I'd get the same kind of formula. What I hope you're seeing is that all of the 71 00:06:34,707 --> 00:06:39,879 derivative laws are connected. With practice, you'll be able to differentiate 72 00:06:39,891 --> 00:06:45,088 any function that you build by combining our standard library of functions and 73 00:06:45,100 --> 00:06:47,112 operations on those functions.