WEBVTT 00:00:05.000 --> 00:00:11.060 Oh, we know how to find the derivative of all polynomials. And why do we know how to 00:00:11.060 --> 00:00:19.797 find the derivative? Or what rules do we use today? Power rule and then some 00:00:19.797 --> 00:00:20.380 different 00:00:20.380 --> 00:00:24.220 It takes care of polynomials. 00:00:25.560 --> 00:00:28.240 And then rational functions. 00:00:31.660 --> 00:00:32.900 Why are we able to do 00:00:32.900 --> 00:00:36.420 rational functions now? Because it's a quotient rule. 00:00:38.220 --> 00:00:39.640 Radical functions. 00:00:42.380 --> 00:00:44.760 Like y equals the square root of 00:00:44.760 --> 00:00:46.660 x squared plus three or something. 00:00:47.660 --> 00:00:57.300 both by the power rule and then because of the chain rule: polynomials, rationals, 00:00:58.140 --> 00:01:10.480 radicals, exponentials, okay we proved it for the function e to the x and then we 00:01:10.480 --> 00:01:14.500 did something the other day because of the chain rule for any exponential 00:01:14.500 --> 00:01:17.740 of function. And of course, what's the derivative of this? 00:01:19.060 --> 00:01:20.900 b to the x. What is the derivative of this? 00:01:23.460 --> 00:01:26.200 B to the x times ln of the base. 00:01:28.460 --> 00:01:29.760 All trig functions. 00:01:31.280 --> 00:01:34.280 We prove for sine and cosine, and then we use 00:01:34.280 --> 00:01:38.720 quotient rule for tangent. C tan, cos c tan, 00:01:38.820 --> 00:01:43.280 c tan. What functions are we still missing? 00:01:43.840 --> 00:01:49.511 that we don't know how to find the derivative of. We kind of did piece-wise, we just 00:01:49.511 --> 00:01:52.180 have to break it up into different pieces. 00:01:53.780 --> 00:02:00.424 We have not done any logarithms. And then the only other type we're going to learn 00:02:00.424 --> 00:02:02.500 as well is inverse trig. 00:02:06.440 --> 00:02:09.000 So today we're talking about the derivative of logarithms. 00:02:14.200 --> 00:02:19.740 So let's talk about this function, our favorite logarithm. 00:02:21.440 --> 00:02:23.480 What is the base of this logarithm? 00:02:26.520 --> 00:02:27.060 No, no. 00:02:27.060 --> 00:02:28.260 e. 00:02:30.080 --> 00:02:30.980 Yes. 00:02:32.860 --> 00:02:37.200 If it's just log, what's the base? 10. 00:02:39.240 --> 00:02:42.820 ln, the natural log, is log base e of x. 00:02:42.820 --> 00:02:44.200 of x. 00:02:46.760 --> 00:02:49.200 If we start with the definition, 00:03:09.260 --> 00:03:13.040 Can we manipulate this at all? 00:03:22.240 --> 00:03:24.000 What can we do with this? 00:03:28.220 --> 00:03:29.820 Uh oh. 00:03:33.220 --> 00:03:35.140 What is the difference of two logarithms? 00:03:41.880 --> 00:03:45.840 It is one logarithm, and inside we divide. 00:03:45.840 --> 00:03:50.160 ln of x plus h over x. 00:03:58.300 --> 00:03:59.960 Is that going to help at all? 00:04:02.400 --> 00:04:09.700 We play in zero for h. We get ln of x over x, which is ln of one. 00:04:09.880 --> 00:04:10.780 What is ln of one? 00:04:14.040 --> 00:04:17.200 0 over 0. That didn't help much. 00:04:20.500 --> 00:04:22.880 We could do this. 00:04:35.420 --> 00:04:39.880 Now, what can I do with the coefficient of a logarithm? 00:04:47.900 --> 00:04:49.720 We can, because it is one over h. 00:04:49.840 --> 00:04:50.740 It's not a constant. 00:04:51.580 --> 00:04:57.600 But it is a coefficient of a logarithm. 00:04:57.600 --> 00:05:23.600 This can become the power of the inside. 00:05:25.760 --> 00:05:27.940 Did this help at all? 00:05:39.140 --> 00:05:40.420 Any ideas now? 00:05:40.960 --> 00:05:42.260 Now what if we plug in zero? 00:05:46.380 --> 00:05:47.340 Wait, is this an x? 00:05:47.640 --> 00:05:51.360 Is this an x? Sorry, is that what you were talking about? Thanks. 00:05:55.160 --> 00:06:02.920 As h goes to 0, this goes to x over x which is 1. 00:06:04.660 --> 00:06:07.600 And what does that go to? 00:06:07.600 --> 00:06:18.582 1 over 0 goes to either infinity or negative infinity. What is 1 to the infinity or 00:06:18.582 --> 00:06:20.520 1 to the 00:06:20.520 --> 00:06:31.793 negative infinity? This is an indeterminate form. Okay, long story short, this is 00:06:31.793 --> 00:06:32.660 not 00:06:32.660 --> 00:06:39.720 gonna work but it was a good review of rules of logarithms. Difference of logs 00:06:39.720 --> 00:06:47.000 is one log with quotient coefficients to become exponents but anyways long search 00:06:47.000 --> 00:07:02.100 yeah it's not gonna work so we gotta try something else let's go back to the 00:07:04.920 --> 00:07:11.062 So, before you learned about logarithms, what did logarithms, when did we come up 00:07:11.062 --> 00:07:11.940 with logarithms? 00:07:11.940 --> 00:07:13.380 Logarithms are inverses. 00:07:15.260 --> 00:07:18.920 We knew any exponential looks like this, right? 00:07:19.980 --> 00:07:21.940 Some would depending on the base. 00:07:27.020 --> 00:07:29.160 This was a one to one function. 00:07:29.420 --> 00:07:30.600 What's a one to one function? 00:07:34.180 --> 00:07:38.980 It's a function. How do we know that there's a function? 00:07:40.620 --> 00:07:44.580 It's a vertical line test. Then it's a one to one 00:07:44.580 --> 00:07:48.060 function if it passed the vertical line test and it passes the horizontal. 00:07:50.400 --> 00:07:57.540 There's only going to be for every input there's only one output and never 00:07:57.540 --> 00:08:07.340 repeats the same y-value. Okay in every inverse function sorry I just gave away 00:08:07.340 --> 00:08:13.300 the answer. Every one to one function is guaranteed to have an inverse. 00:08:17.000 --> 00:08:23.878 If we reflect about the line it's this. This is the inverse of B to the X. And we 00:08:23.878 --> 00:08:24.240 just 00:08:24.240 --> 00:08:33.794 call this function a log with the base, wherever the base B is. Does this ring a 00:08:33.794 --> 00:08:35.480 bell? I don't 00:08:35.480 --> 00:08:38.860 No. What is an inverse functional? 00:08:44.200 --> 00:08:52.480 It is simply if we have some ordered pair in F, what is in F inverse? 00:08:54.680 --> 00:08:57.060 B, A. It's literally just all the ordered pairs. 00:08:57.060 --> 00:09:07.920 way. Okay. And that's why it was a reflection about the line y equals x. Okay 00:09:07.920 --> 00:09:13.780 and then inverses have a very very important property. What's the most 00:09:13.780 --> 00:09:24.980 important property of inverses? You compose a function and its inverse and 00:09:24.980 --> 00:09:31.320 And you end up with x whatever you started with there and vice versa 00:09:37.180 --> 00:09:38.080 Okay 00:09:42.420 --> 00:09:44.880 This is very important if 00:09:45.700 --> 00:09:47.100 We have to inverse something 00:09:48.320 --> 00:09:53.020 Okay, so our function e to the x and 00:09:53.140 --> 00:09:56.620 ln of x are inverse functions. 00:10:02.660 --> 00:10:12.660 That tells us e composed with ln of x is equal to x, 00:10:15.880 --> 00:10:19.460 ln of e to the x is equal to x. 00:10:23.140 --> 00:10:24.300 We agree with that 00:10:26.340 --> 00:10:27.240 Okay 00:10:34.920 --> 00:10:37.060 So let's go back to this 00:10:43.020 --> 00:10:48.420 I'm gonna plug both sides of this equation into our function e to the x 00:10:49.980 --> 00:10:52.560 So the left-hand side becomes e 00:10:53.140 --> 00:10:57.280 to the y and the right hand side is 00:11:01.400 --> 00:11:04.280 e to the ln of x 00:11:08.640 --> 00:11:13.880 And e to the ln of x is x because they're inverse functions 00:11:17.040 --> 00:11:22.927 Okay, sometimes we call this the exponential form and call this the logarithmic 00:11:22.927 --> 00:11:23.677 form. 00:11:25.280 --> 00:11:33.544 Right, it's literally if y equals e to the x, remember this is how we found 00:11:33.544 --> 00:11:34.120 inverses, 00:11:34.120 --> 00:11:40.334 you just switch the ordered pairs, x equals e to the y, which is the same thing 00:11:40.334 --> 00:11:40.700 here. 00:11:40.700 --> 00:11:51.640 Okay, so this is exactly the same thing as this. 00:11:51.660 --> 00:11:53.300 Just written two different ways. 00:11:55.460 --> 00:12:07.799 Okay, now we're going to take the derivative of this using what kind of 00:12:07.799 --> 00:12:08.680 differentiation? 00:12:11.840 --> 00:12:16.973 implicit because now we have functions on both sides of the equation. We're using 00:12:16.973 --> 00:12:17.340 implicit 00:12:17.340 --> 00:12:24.409 differentiation. And we are looking right we're looking for the derivative of this 00:12:24.409 --> 00:12:26.040 which is dy 00:12:26.040 --> 00:12:41.780 dx. So we're going to take the derivative with respect to x here. So implicit with 00:12:41.780 --> 00:12:57.968 respect x. And the derivative of e to the y is e to the y times the derivative of 00:12:57.968 --> 00:12:58.820 our 00:12:58.840 --> 00:13:01.520 inside function y, what is the derivative of y? 00:13:04.800 --> 00:13:06.000 dy dx. 00:13:07.780 --> 00:13:09.900 Okay, that's the derivative of the left side. 00:13:12.080 --> 00:13:19.040 Derivative of x is either dx dx or just the number one. 00:13:24.520 --> 00:13:26.300 And then I solve for dy dx. 00:13:27.320 --> 00:13:40.660 I get 1 over e to the y and what is oops 00:13:40.660 --> 00:14:05.360 What is e to the y equal to? x. dy over dx is 1 over x. 00:14:05.360 --> 00:14:11.000 derivative of the natural log turns out to be this rational function. 00:14:13.800 --> 00:14:19.120 Okay, this is very important we need to memorize it. 00:14:35.760 --> 00:14:39.979 Okay, so now we know the derivative of the natural log. What about for any 00:14:39.979 --> 00:14:40.729 logarithm? 00:14:42.840 --> 00:14:58.400 Let's say y equals log base b of x. We're gonna do the same exact process, except 00:14:58.400 --> 00:15:04.680 without ease. What do we do first? 00:15:13.160 --> 00:15:20.880 Yeah we need to change it from logarithmic form to exponential form. So 00:15:20.880 --> 00:15:25.760 So I'm going to go B to both sides. 00:15:38.700 --> 00:15:43.200 B to the log base B of X is X, 00:15:43.400 --> 00:15:45.040 because they're inverse functions, 00:15:45.300 --> 00:15:46.200 composed through inverse. 00:15:47.060 --> 00:15:49.680 So this is the same function. 00:15:50.880 --> 00:15:55.340 As this so now we're going to differentiate using implicit differentiation 00:16:03.720 --> 00:16:10.580 Derivative of B to the Y is B to the Y times 00:16:13.500 --> 00:16:20.140 Ln of the base I think we talked about that at the very top 00:16:24.100 --> 00:16:34.983 derivative of any exponential function is the same function times ln of the base. 00:16:34.983 --> 00:16:35.760 Then 00:16:35.760 --> 00:16:41.536 we multiply by the chain rule by the derivative of the inside. The derivative of y 00:16:41.536 --> 00:16:42.980 is dy over dx. 00:16:45.460 --> 00:16:47.260 So that's the derivative of the left side. 00:16:47.700 --> 00:16:52.020 The derivative of the right side is 1 or dx over dx. 00:16:53.160 --> 00:16:55.760 And we solve for this. 00:17:10.780 --> 00:17:20.140 And then, B to the Y is simply a... 00:17:30.280 --> 00:17:32.200 And there it is. 00:17:33.540 --> 00:17:50.040 The derivative of any logarithm with any base is simply 1 over x times 00:17:50.040 --> 00:17:59.960 the natural log of the base. Which really this is the coefficient of the x but we 00:17:59.960 --> 00:18:02.840 usually write it like that. 00:18:04.560 --> 00:18:07.480 of any logarithm. 00:18:16.280 --> 00:18:28.600 So what is the derivative of log of x? 00:18:28.600 --> 00:18:40.240 What's the base? There's no base here. 10. It's gonna be 1 over x times the 00:18:40.240 --> 00:18:42.680 natural log of 10. Easy. 00:18:55.380 --> 00:18:56.760 Okay, let's do some more 00:19:18.640 --> 00:19:24.340 Okay, what role do we have these? 00:19:24.600 --> 00:19:30.260 We're going to use lots of the big ideas chain rule. 00:19:30.420 --> 00:19:34.380 We got a function sine of x plus x cubed inside another function log base 4. 00:19:35.460 --> 00:19:37.120 So this is my u. 00:19:41.440 --> 00:19:44.340 What is the derivative of log base 4 of u? 00:19:49.920 --> 00:19:59.120 1 over u times ln of 4, but what is u equal to? 00:20:05.820 --> 00:20:11.200 sine of x plus x cubed times 00:20:13.000 --> 00:20:14.880 the natural logarithm of 4. 00:20:16.440 --> 00:20:19.520 Now we've got to multiply by the derivative of this. 00:20:24.660 --> 00:20:39.000 By the chain rule, the derivative is cosine of x plus 3x squared. 00:20:39.000 --> 00:20:40.540 We write it as one fraction. 00:21:04.160 --> 00:21:05.560 So 00:21:06.720 --> 00:21:13.358 Again, so if there's anything inside the logarithm, that's our U. It's 1 over U 00:21:13.358 --> 00:21:13.800 times 00:21:13.800 --> 00:21:19.687 ln of the base times the derivative of the inside U. By the chain rule. Let's try 00:21:19.687 --> 00:21:20.437 some more. 00:21:44.100 --> 00:21:46.880 So we got a function inside a function. 00:21:48.140 --> 00:21:50.340 What is the derivative of ln of u? 00:21:52.680 --> 00:21:53.920 One over u. 00:21:56.720 --> 00:21:58.400 Times the derivative of u. 00:21:58.400 --> 00:22:05.400 We have x to the 7th is 7x to the 6th. 00:22:09.380 --> 00:22:14.920 And what does that simplify to? 7 over x. 00:22:19.520 --> 00:22:21.420 Sorry, 7 over x. 00:22:33.760 --> 00:22:40.918 We could have done this one other way. How could I have changed this function? An 00:22:40.918 --> 00:22:42.260 exponent inside a 00:22:42.260 --> 00:22:50.407 logarithm becomes a coefficient. Now we don't need a chain rule, the constant 00:22:50.407 --> 00:22:51.660 multiple rule. 00:22:54.960 --> 00:23:03.360 It's 7 times the derivative of ln of x which is 1 over x and you're done. 00:23:13.360 --> 00:23:18.611 Most of the time in this class we're really just going to deal with the natural log 00:23:18.611 --> 00:23:18.920 to 00:23:18.920 --> 00:23:22.980 log base 4 or some other base. But not always. 00:23:24.620 --> 00:23:27.000 And you can always change bases too before you 00:23:27.000 --> 00:23:30.920 want it. So for like this, we could have 00:23:30.920 --> 00:23:32.660 used the change of base formula. 00:23:34.580 --> 00:23:37.840 What would have been the change of base formula? If we want to use the natural log, 00:23:39.320 --> 00:23:40.640 it's ln of the inside 00:23:44.980 --> 00:23:46.020 divided by 00:23:47.380 --> 00:23:48.680 ln of the base. 00:23:51.740 --> 00:23:52.980 And then this is just the coefficient. 00:23:53.360 --> 00:23:58.900 one over ln 4 times ln of sine x plus x cubed. 00:23:59.440 --> 00:24:01.200 Anyways, and then if you differentiate this, 00:24:02.140 --> 00:24:05.940 you end up with this times... 00:24:05.940 --> 00:24:08.040 Oops, that's a 4, right? 00:24:09.640 --> 00:24:12.520 Times 1 over ln of 4, and you get that same thing. 00:24:16.900 --> 00:24:19.520 Okay, let's try another one. 00:24:30.700 --> 00:24:33.780 Okay, you definitely have to use the chain rule. 00:24:36.180 --> 00:24:49.560 Derivative of ln u is 1 over u times the derivative of 5x plus 6. 00:24:49.560 --> 00:24:51.620 Five. 00:25:15.920 --> 00:25:18.380 What kind of function is this? 00:25:20.100 --> 00:25:21.240 It's a piecewise. 00:25:24.300 --> 00:25:28.940 Uh oh. Anything involving absolute value is a piecewise function. 00:25:37.160 --> 00:25:39.740 It's going to be ln of x, if what? 00:25:42.860 --> 00:25:53.900 if X is greater than zero and it's going to be if X is less than zero, Ln of 00:25:53.900 --> 00:26:02.460 negative X, oops, get it right this inside is the definition of the absolute value. 00:26:03.820 --> 00:26:06.820 What is the domain here? 00:26:11.600 --> 00:26:12.840 All real numbers 00:26:12.840 --> 00:26:13.740 except for 00:26:14.480 --> 00:26:17.720 zero. Unlike 00:26:21.020 --> 00:26:22.220 this function 00:26:22.220 --> 00:26:24.820 just ln of x, what is the domain here? 00:26:29.640 --> 00:26:30.540 Uh oh. 00:26:30.800 --> 00:26:32.160 Only positive numbers. 00:26:34.760 --> 00:26:39.240 Okay, right, we talked about the graph of this already. Ln of x looks like that. 00:26:50.580 --> 00:26:52.760 What does the graph of this look like? 00:26:55.780 --> 00:26:58.060 To the right of zero, it is ln of x. 00:27:00.420 --> 00:27:03.260 What does this look like? 00:27:05.140 --> 00:27:07.940 This is like that graph like reflected on the 00:27:09.680 --> 00:27:12.000 y axis like in the classes. That's correct. 00:27:13.140 --> 00:27:14.040 And how'd you know that? 00:27:15.760 --> 00:27:17.400 Okay, that's a good guess. 00:27:19.680 --> 00:27:24.260 A negative in front of an x inside a function reflects about the y axis. 00:27:26.240 --> 00:27:27.320 For example, 00:27:27.320 --> 00:27:38.302 If you plug in negative 1 for x, you get ln of negative negative 1, which is ln of 00:27:38.302 --> 00:27:38.880 1. 00:27:38.880 --> 00:27:39.900 a1, ln of 1. 00:27:39.940 --> 00:27:41.940 So it just basically reflects all these values. 00:27:42.640 --> 00:27:48.200 Anyways, so to find the derivative of a piecewise, what do we have to do? 00:27:52.040 --> 00:27:53.640 Take two separate derivatives. 00:27:53.640 --> 00:28:03.240 So let's start with this one. If x is greater than 0, then f of x which equals 00:28:03.240 --> 00:28:14.380 ln of x equals ln of x. And the derivative would be, derivative ln of x is 1 over x. 00:28:25.580 --> 00:28:35.280 Right now we have f prime of x is 1 over x if x is greater than 0. 00:28:37.420 --> 00:28:48.640 And then if x is less than 0, f of x equals ln the absolute value of x. 00:28:49.600 --> 00:28:55.640 Since x is less than zero, our function is this. 00:29:07.640 --> 00:29:10.420 Now when we take the derivative, we've got to use the chain rule. 00:29:10.420 --> 00:29:23.100 So, the derivative of ln of u is 1 over u, 1 over negative x times the derivative of 00:29:23.100 --> 00:29:30.280 the inside derivative of negative x, negative 1, just 1 over x. 00:29:40.820 --> 00:29:46.480 Since it's the same, we just write it as one thing. 00:29:47.160 --> 00:29:52.840 F' of x is simply 1 over x. 00:29:58.240 --> 00:30:04.484 And we already know that domain, all numbers except for zero, because you can't plug 00:30:04.484 --> 00:30:04.900 in 00:30:04.900 --> 00:30:05.800 zero there. 00:30:11.260 --> 00:30:17.680 So, the derivative of ln x is 1 over x. 00:30:20.520 --> 00:30:28.280 The derivative of ln of the absolute value of x is also equal to 1 over x. 00:30:33.700 --> 00:30:38.680 So what's the only, actually let me write it differently. 00:30:46.100 --> 00:30:47.420 Same story thing. 00:30:59.920 --> 00:31:02.640 So what is the difference between these two then? 00:31:07.600 --> 00:31:09.220 The domain is the difference. 00:31:14.020 --> 00:31:16.760 What is the domain of this? 00:31:19.960 --> 00:31:21.120 From 0 to infinity. 00:31:21.380 --> 00:31:22.500 The only positive numbers. 00:31:25.220 --> 00:31:31.215 That tells us the domain of this. There are no slopes of tangent lines to the left 00:31:31.215 --> 00:31:31.965 of zero. 00:31:34.440 --> 00:31:38.500 The domain here is also zero to infinity. 00:31:41.020 --> 00:32:01.160 This is the graph of ln x and this is the graph of its derivative, 1 over x. 00:32:01.160 --> 00:32:11.009 Here the domain is all numbers except for zero: negative infinity to zero, union 00:32:11.009 --> 00:32:13.120 zero to infinity, 00:32:15.530 --> 00:32:29.710 which is also the domain of this. 00:32:30.150 --> 00:32:31.270 Okay. 00:32:35.270 --> 00:32:36.390 Okay. 00:32:36.950 --> 00:32:39.330 Ln of the absolute value of x. 00:32:41.230 --> 00:32:42.310 Looks like this. 00:32:45.970 --> 00:32:53.414 and this is this which exactly the same on the right hand side over here the only 00:32:53.414 --> 00:32:54.290 difference is 00:32:58.030 --> 00:32:59.710 We have a different domain. 00:33:00.710 --> 00:33:12.713 Okay. So when we're finding derivatives, even though the derivative is 1 over X 00:33:12.713 --> 00:33:13.570 here, 00:33:14.990 --> 00:33:16.370 the domain is restricted. 00:33:16.690 --> 00:33:17.990 It's not the domain of this. 00:33:18.330 --> 00:33:20.650 The domain is only positive numbers because the domain 00:33:20.650 --> 00:33:22.210 of the function is only positive numbers. 00:33:23.490 --> 00:33:29.065 So that is very important. If the domain is not all real numbers for the function, 00:33:29.065 --> 00:33:32.550 the domain is not all real numbers for the derivative. 00:33:32.830 --> 00:33:36.350 It's maximum of whatever the domain of the function is. 00:33:38.990 --> 00:33:47.773 The distinction between these two is going to be very important in several months 00:33:47.773 --> 00:33:50.910 when we're talking about antiderivatives. 00:33:52.450 --> 00:33:59.410 What are antiderivatives? Antiderivatives are you're given a derivative and we want 00:33:59.410 --> 00:34:00.570 to find 00:34:00.570 --> 00:34:08.391 a function whose derivative is that. Okay, and this is gonna be very important 00:34:08.391 --> 00:34:09.141 later. 00:34:09.250 --> 00:34:18.732 But the main point of this is the domain of the function is also the most possible 00:34:18.732 --> 00:34:19.290 domain 00:34:19.290 --> 00:34:28.110 of this right when I say most possible like the square root of X the derivative 00:34:28.110 --> 00:34:43.930 is 1 over 2 square root x. The domain here is zero to infinity. What is the domain 00:34:43.930 --> 00:34:54.330 here? Zero to infinity, not including zero. Okay, so sometimes the numbers in the 00:34:54.330 --> 00:35:07.910 domain go down, right? Now we're missing one number. But anyways, my point was the 00:35:07.910 --> 00:35:12.910 domain of the function is at most the domain of the derivative. I don't know if 00:35:12.910 --> 00:35:19.650 I'm saying that quite right but it can be more restricted like it is here. 00:35:26.010 --> 00:35:28.770 Okay one more thing. 00:35:39.190 --> 00:35:42.590 I told you your function. 00:35:53.350 --> 00:35:58.530 Is this a power function? 00:36:00.470 --> 00:36:03.890 Anyway, if that's the case, we're going to use the power rule. 00:36:06.890 --> 00:36:09.730 What is the definition of a power function? 00:36:12.590 --> 00:36:22.318 X to any power. And the power has to be any number, any real number. So is this a 00:36:22.318 --> 00:36:22.830 power 00:36:22.830 --> 00:36:34.492 function? Here is a variable, it's not a real number. So it's not a power function. 00:36:34.492 --> 00:36:35.950 So we 00:36:35.950 --> 00:36:36.990 We can't use power rule. 00:36:42.670 --> 00:36:43.850 Is it an exponential function? 00:36:50.130 --> 00:36:53.270 So for an exponential function, what does the base have to be? 00:36:56.410 --> 00:37:00.570 Greater than zero and not equal to one. 00:37:03.410 --> 00:37:05.050 Is this an exponential function? 00:37:05.050 --> 00:37:07.370 No. No, why? 00:37:09.890 --> 00:37:12.890 Because the base is not a number. 00:37:14.290 --> 00:37:16.590 So this is neither a 00:37:16.590 --> 00:37:21.090 power function or an exponential function. So we can't use 00:37:21.090 --> 00:37:25.030 the power rule or we can't use the rule for derivative of exponential function. 00:37:26.790 --> 00:37:28.390 So what in the world are we going to do? 00:37:33.990 --> 00:37:35.030 How do you 00:37:35.030 --> 00:37:38.870 Okay, so we can't use any of the rules that we know 00:37:40.610 --> 00:37:42.970 because it's neither of those type of functions. 00:37:43.570 --> 00:37:45.910 So we're gonna first manipulate the function 00:37:45.910 --> 00:37:49.250 or the equation and then take derivatives. 00:37:50.670 --> 00:37:52.990 And as we said, we're gonna start 00:37:52.990 --> 00:37:56.390 by taking the natural log of both sides or any logarithm. 00:37:59.010 --> 00:38:00.870 We always choose the natural log 00:38:00.870 --> 00:38:02.610 because that's the simplest derivative. 00:38:03.830 --> 00:38:07.990 So we take the natural log of both sides. 00:38:17.670 --> 00:38:19.630 And then what can I do with this? 00:38:22.350 --> 00:38:23.390 It becomes a coefficient. 00:38:39.270 --> 00:38:44.276 Now here we just got some function. We don't have to take the derivative of ln now. 00:38:44.276 --> 00:38:47.810 Here we got a product. We're going to use the product rule. 00:38:49.090 --> 00:38:52.712 And we know the derivative of ln of x. So since there's functions on both sides, 00:38:52.712 --> 00:38:54.070 what are we going to use? 00:38:56.990 --> 00:38:58.750 implicit differentiation. 00:39:06.570 --> 00:39:09.050 The derivative of ln y is 00:39:10.970 --> 00:39:17.490 1 over y times the derivative of y, which is 00:39:18.990 --> 00:39:20.270 dy dx. 00:39:22.850 --> 00:39:25.170 Okay, that's just really the left side. 00:39:27.750 --> 00:39:31.910 Now, the derivative of the right side, we need the product rule. 00:39:37.050 --> 00:39:39.830 So, derivative of the 3x is 3. 00:39:45.110 --> 00:39:45.670 The 00:39:45.670 --> 00:39:48.830 derivative of 3x is 3, and the derivative of ln x is 1 over x. 00:40:03.110 --> 00:40:05.590 These happen to simplify, right? Plus 3. 00:40:07.490 --> 00:40:11.390 And we're trying to find dy dx, the derivative. 00:40:12.290 --> 00:40:13.470 So we solve for this. 00:40:23.210 --> 00:40:24.670 Times y. 00:40:27.790 --> 00:40:31.310 Except now we don't want x's and y's. 00:40:33.230 --> 00:40:38.790 What was y equal to? x to the 3x. 00:40:40.370 --> 00:40:41.770 So. 00:40:53.030 --> 00:40:56.430 And there it is. 00:41:08.550 --> 00:41:17.610 Okay, so when are we going to do this? 00:41:17.890 --> 00:41:19.010 First take the natural log. 00:41:25.590 --> 00:41:30.690 Whenever you have a variable both in the base and in the exponent. 00:41:32.090 --> 00:41:34.770 Because it's not a power function, it's not an exponential function. 00:41:35.390 --> 00:41:38.990 So this is called logarithmic 00:41:40.910 --> 00:41:45.470 I spell logarithmic. 00:41:49.970 --> 00:41:52.530 I'm blanking. 00:41:53.150 --> 00:41:54.330 Logarithm. 00:41:57.230 --> 00:41:58.130 There we go. 00:42:00.730 --> 00:42:01.630 Differentiation. 00:42:12.610 --> 00:42:14.630 Use whenever 00:42:17.950 --> 00:42:21.390 there is a variable 00:42:27.530 --> 00:42:36.670 And both the base and the exponent. 00:42:41.270 --> 00:42:49.210 And all you do is start by taking logs of both sides 00:42:57.590 --> 00:43:01.130 and then use implicit differentiation. 00:43:21.690 --> 00:43:26.210 Let's try another one. 00:43:40.270 --> 00:43:42.030 Let's see what this function looks like. 00:43:58.020 --> 00:43:59.180 It is kind of cool. 00:44:02.700 --> 00:44:05.220 Why are we missing a bunch of gaps in it? 00:44:10.380 --> 00:44:16.092 So, there's definitely some asymptote right here, but there is no, the function is 00:44:16.092 --> 00:44:16.500 not 00:44:16.500 --> 00:44:17.800 defined right here. 00:44:18.960 --> 00:44:27.240 It is not defined from pi to 2 pi. 00:44:27.760 --> 00:44:28.660 Why is that? 00:44:34.380 --> 00:44:35.900 What's that? 00:44:38.480 --> 00:44:39.540 It has to do with the 00:44:42.840 --> 00:44:46.300 We cannot have a negative base. 00:44:51.600 --> 00:44:52.500 Oh. 00:44:53.800 --> 00:44:56.080 We can't have negative numbers in a base. 00:44:57.620 --> 00:44:58.280 Like, negative numbers in a base. 00:44:58.280 --> 00:45:02.420 4 to the power of x is not an exponential function. The base has to be positive. 00:45:03.560 --> 00:45:04.580 So long story short, 00:45:07.520 --> 00:45:10.900 whenever sine is a negative number, the function doesn't exist. 00:45:12.160 --> 00:45:13.320 That's why there's a bunch of gaps. 00:45:15.420 --> 00:45:18.240 So, let's find dy by dx. 00:45:24.440 --> 00:45:29.937 The derivative. And we're going to have to use logarithmic differentiation. Why is 00:45:29.937 --> 00:45:30.687 that? 00:45:33.560 --> 00:45:41.897 There's a variable in the base and there's a variable in the exponents. Okay. Can't 00:45:41.897 --> 00:45:44.120 use the power rule. 00:45:44.120 --> 00:46:00.480 can't use the exponential rule. So first step is take the log of both sides and 00:46:00.480 --> 00:46:06.580 what happens it will be do this exponent can become the coefficient 00:46:12.900 --> 00:46:16.640 Okay, so it's the same function as written differently. 00:46:20.860 --> 00:46:25.560 And now we differentiate using implicit. 00:46:29.980 --> 00:46:36.460 The derivative of ln of y is 1 over y times the derivative of the inside, which is 00:46:36.460 --> 00:46:36.820 dy 00:46:36.820 --> 00:46:37.720 dx. 00:46:41.020 --> 00:46:43.280 Then we've got a product rule. 00:46:49.600 --> 00:47:04.120 The derivative of cosine is negative sine, times ln of sine x, plus cosine x times 00:47:07.780 --> 00:47:34.100 the derivative of ln of sine x, which is one over sine x times cosine x. 00:47:50.300 --> 00:47:53.760 And then again, we're solving for dy dx. 00:47:53.760 --> 00:47:55.820 We multiply both sides by y. 00:48:18.560 --> 00:48:21.560 And then we replace Y with where it was, which was what? 00:48:21.560 --> 00:48:36.800 sine x raised to cosine x. There it is. 00:48:36.800 --> 00:48:47.646 functions. We've never even thought about before. And when do we use logarithmic 00:48:47.646 --> 00:48:48.480 differentiation? 00:48:51.720 --> 00:48:57.291 Variable in the base and a variable in the exponent. You have to use it or it's the 00:48:57.291 --> 00:48:57.600 only 00:48:57.600 --> 00:49:06.069 option. Okay you could use logarithmic differentiation for other functions. Okay for 00:49:06.069 --> 00:49:07.480 example, and 00:49:07.490 --> 00:49:19.710 wouldn't do this but let's say we have this sine of x to the fourth power we 00:49:19.710 --> 00:49:24.050 would normally just use the chain rule. 00:49:27.190 --> 00:49:34.750 You get 4 sine cubed x times cosine x. 00:49:34.750 --> 00:49:41.368 It's easier. You could use logarithmic differentiation by first manipulating the 00:49:41.368 --> 00:49:42.118 equation. 00:50:04.750 --> 00:50:14.010 You have to use implicit differentiation, so 1 over y times dy dx is simply 4 times 00:50:14.910 --> 00:50:16.170 derivative of ln assigned. 00:50:17.470 --> 00:50:23.090 For ln of u would be 1 over u times the derivative of sine and cosine. 00:50:31.590 --> 00:50:39.550 over sine of x times y. What was y equal to? 00:50:40.950 --> 00:50:42.850 sine of x to the fourth. 00:50:55.670 --> 00:50:58.030 And then we can simplify this. 00:51:00.670 --> 00:51:01.450 sine of x 00:51:01.450 --> 00:51:04.550 divided by, or sine of x to the fourth, this becomes 00:51:04.550 --> 00:51:06.690 sine of x cubed. 00:51:08.030 --> 00:51:09.070 We have four, 00:51:10.070 --> 00:51:19.533 cosine of x times sine of x cubed, which is this right here. It's a long story 00:51:19.533 --> 00:51:28.997 short. You could use it for other problems we rarely ever do. Once or while it makes 00:51:28.997 --> 00:51:30.110 things easier. 00:51:31.270 --> 00:51:33.930 It definitely did not make it easier here, right? 00:51:36.430 --> 00:51:38.250 But you could use logarithmic 00:51:39.810 --> 00:51:41.870 differentiation. You have to use it 00:51:41.870 --> 00:51:45.690 whenever you've got a variable both in the base and in the exponent. 00:51:47.630 --> 00:51:49.990 Okay, any questions on this?