WEBVTT 00:00:02.640 --> 00:00:07.671 In this video, we're going to be looking at how we can use 00:00:07.671 --> 00:00:10.767 logarithms to help simplify certain functions before we 00:00:10.767 --> 00:00:11.541 differentiate them. 00:00:12.720 --> 00:00:17.160 To start off with, let's just remind ourselves 00:00:17.160 --> 00:00:21.045 about logarithms themselves. So if we have 00:00:21.045 --> 00:00:27.150 Y equals the log of X, then this means log to 00:00:27.150 --> 00:00:32.145 base E, and if we differentiate that why by 00:00:32.145 --> 00:00:36.030 DX is equal to one over X? 00:00:37.380 --> 00:00:41.678 But of course. Inside this logarithm here this might 00:00:41.678 --> 00:00:47.880 not just be X, it might be a function of X, so it might 00:00:47.880 --> 00:00:52.310 be why equals the log of a function of X. 00:00:53.550 --> 00:00:57.834 So what happens when we differentiate this? Well, we 00:00:57.834 --> 00:01:03.546 have a rule that tells us why by DX is equal to. 00:01:04.400 --> 00:01:09.611 The derivative of F of X. That's F dash 00:01:09.611 --> 00:01:12.506 decks over F of X. 00:01:14.290 --> 00:01:18.110 Now. We're going to be making use these results. 00:01:18.910 --> 00:01:24.470 But also the properties of logarithms themselves so. 00:01:27.090 --> 00:01:31.320 Let's take our first example, why? 00:01:32.700 --> 00:01:34.480 Equals the natural log. 00:01:36.440 --> 00:01:42.533 3X to the 4th plus Seven all raised to 00:01:42.533 --> 00:01:44.564 the fifth power. 00:01:45.850 --> 00:01:48.104 And we want to be able to 00:01:48.104 --> 00:01:50.030 differentiate this. Well. 00:01:50.590 --> 00:01:55.320 It looks very complicated in here, but here we are raising. 00:01:56.020 --> 00:02:01.684 3X to the power 4 + 7 to the power five and one of the things 00:02:01.684 --> 00:02:05.932 we do know about logarithms is that if we are raising what's 00:02:05.932 --> 00:02:06.994 inside the log. 00:02:07.870 --> 00:02:12.690 To the power five, that's exactly the same as multiplying 00:02:12.690 --> 00:02:19.920 the log by 5. So In other words, we can rewrite this as Y equals 00:02:19.920 --> 00:02:27.632 5 times the log of three X to the 4th plus 7 using our laws of 00:02:27.632 --> 00:02:32.452 logarithms. And now this is now much easier to differentiate 00:02:32.452 --> 00:02:36.790 because this inside the log is as very straightforward 00:02:36.790 --> 00:02:42.860 function. So we can now differentiate the why by DX is 00:02:42.860 --> 00:02:45.470 equal to. 5. 00:02:46.280 --> 00:02:51.296 Times by now, let's remember what we do. This is the 00:02:51.296 --> 00:02:54.944 function inside the log, so we differentiate that. 00:02:56.500 --> 00:03:04.420 So that gives us 12X cubed and the derivative of the 70s 00:03:04.420 --> 00:03:11.020 zero over the function itself 3X to the 4th +7. 00:03:11.790 --> 00:03:17.740 And if we just do a little bit of simplifying five times by 12 00:03:17.740 --> 00:03:23.690 gives us 60 X cubed over 3X to the 4th plus Seven and what 00:03:23.690 --> 00:03:27.940 looked like up here quieter fearsome derivative that we were 00:03:27.940 --> 00:03:33.040 going to have to do here, it's turned out quite simply. So 00:03:33.040 --> 00:03:35.590 let's have a look at another 00:03:35.590 --> 00:03:41.210 example. Supposing we've got Y equals the log of. 00:03:42.300 --> 00:03:49.748 We've got 1 - 3 X divided by 1 + 2 X. Now what 00:03:49.748 --> 00:03:56.664 we've got here is the log of a quotient, and there is a 00:03:56.664 --> 00:03:59.324 formula for differentiating a quotient. 00:04:00.500 --> 00:04:05.373 That's going to lead to a very very complicated expression, but 00:04:05.373 --> 00:04:11.132 what we can do is make use of the laws of logarithms because 00:04:11.132 --> 00:04:16.891 the laws of logarithms tell us that when we're doing the log of 00:04:16.891 --> 00:04:22.207 a quotient than that is simply the log of the numerator minus 00:04:22.207 --> 00:04:26.637 the log of the denominator. Because in logarithms we do 00:04:26.637 --> 00:04:30.624 division by subtracting the logs of the respective parts. 00:04:31.050 --> 00:04:35.660 And now these two are both very easy to differentiate. 00:04:36.500 --> 00:04:39.836 So we can have the why by DX 00:04:39.836 --> 00:04:45.927 equals. And this is a function of X 1 - 3 X, so we 00:04:45.927 --> 00:04:47.403 differentiate it that's minus 00:04:47.403 --> 00:04:53.230 three. Over the function itself, 1 - 3 X minus that 00:04:53.230 --> 00:04:58.477 minus sign, and now we do the same for this logarithm. 00:04:58.477 --> 00:05:02.770 Here's the function of X which we differentiate, so 00:05:02.770 --> 00:05:08.971 the derivative of 1 + 2 X is 2 over the function itself. 00:05:08.971 --> 00:05:10.879 1 + 2 X. 00:05:12.070 --> 00:05:17.790 Now that is a penalty to pay for getting away with such a 00:05:17.790 --> 00:05:19.990 straightforward differentiation. What we have 00:05:19.990 --> 00:05:25.270 to do now is put these two together. We have to bring 00:05:25.270 --> 00:05:29.670 them together all over the same denominator and this is 00:05:29.670 --> 00:05:32.750 our denominator. The product of these two. 00:05:34.930 --> 00:05:39.767 So that's 1 - 3 X times 00:05:39.767 --> 00:05:43.222 by 1 + 2 X. 00:05:44.830 --> 00:05:49.087 So we ask ourselves, how many times does this 00:05:49.087 --> 00:05:55.236 divide into this? And the answer is 1 + 2 X, so we 00:05:55.236 --> 00:06:01.385 have minus 3 * 1 + 2 X minus two and then how 00:06:01.385 --> 00:06:06.115 many times does this go into this? And the answer 00:06:06.115 --> 00:06:12.264 is 1 - 3 X so it's 2 * 1 - 3 X. 00:06:13.420 --> 00:06:20.349 Now we need to simplify this so we can keep the denominator as 00:06:20.349 --> 00:06:27.811 it is 1 - 3 X times by 1 + 2 X and let's 00:06:27.811 --> 00:06:32.608 multiply out this bracket minus three times everything inside 00:06:32.608 --> 00:06:40.070 the bracket, so it's minus 3 - 6 X and then minus two times 00:06:40.070 --> 00:06:43.801 everything inside the bracket, so that's minus. 00:06:43.890 --> 00:06:46.730 2 + 6 X. 00:06:47.950 --> 00:06:54.082 Now we need to simplify this bit on top, and we've minus 6X plus 00:06:54.082 --> 00:07:01.090 6X, so that's no axis minus 3 - 2 - 5. So we have minus 5. 00:07:01.910 --> 00:07:08.890 Over 1 - 3 X times by 1 + 2 00:07:08.890 --> 00:07:12.030 X. But so long as we're good 00:07:12.030 --> 00:07:16.240 with the algebra. This differentiation is well 00:07:16.240 --> 00:07:22.590 worth the simplifying as we have done here, so let's 00:07:22.590 --> 00:07:24.495 take another example. 00:07:26.500 --> 00:07:34.192 This time let's take Y equals X to the power sign X. 00:07:35.080 --> 00:07:35.490 Now. 00:07:37.240 --> 00:07:42.311 Real problems having the unknown the variable appear yet again in 00:07:42.311 --> 00:07:48.765 the power in the index, so this sign X is the problem. If it 00:07:48.765 --> 00:07:54.297 wasn't in the index, we might be able to differentiate it, but 00:07:54.297 --> 00:08:01.212 one way of getting it down, so to speak out of the index is to 00:08:01.212 --> 00:08:07.205 take logs of both sides so we have log Y equals the logs. 00:08:07.620 --> 00:08:13.820 X to the power sign X and if we remember. 00:08:14.430 --> 00:08:16.750 By using our log laws. 00:08:17.330 --> 00:08:23.630 When we raise the function inside the log to a power, then 00:08:23.630 --> 00:08:28.880 that's just the same as multiplying the log of the 00:08:28.880 --> 00:08:30.980 function by the power. 00:08:31.830 --> 00:08:35.726 And now this is a straightforward product. It's 00:08:35.726 --> 00:08:38.648 sign X times by Log X. 00:08:39.240 --> 00:08:43.096 We know that we can differentiate this. What 00:08:43.096 --> 00:08:48.398 about this at the left hand side log Y? Well, if 00:08:48.398 --> 00:08:52.254 we're differentiating with respect to X, we can 00:08:52.254 --> 00:08:56.110 differentiate this log Y implicitly, and so the 00:08:56.110 --> 00:09:02.376 derivative of the log of Y is one over Y times DY by 00:09:02.376 --> 00:09:03.822 the X equals. 00:09:05.540 --> 00:09:11.884 Now we need to differentiate this sign X times by Log X. It's 00:09:11.884 --> 00:09:18.228 a product you times by V and we know that the derivative of 00:09:18.228 --> 00:09:26.036 product is U times DV by DX Plus V Times du by DX. So let's write 00:09:26.036 --> 00:09:32.868 that down you which we said was sign X times divided by DX and 00:09:32.868 --> 00:09:36.284 the derivative of log X is one. 00:09:36.450 --> 00:09:37.830 Over X. 00:09:39.050 --> 00:09:42.540 Close the which is log 00:09:42.540 --> 00:09:47.338 X. Times the derivative of U and 00:09:47.338 --> 00:09:51.454 you re sign X its derivative is 00:09:51.454 --> 00:09:53.218 therefore cause X. 00:09:56.360 --> 00:09:59.790 Now we can do a little bit of simplifying. Here we can put it 00:09:59.790 --> 00:10:05.180 all. Over this denominator of X, and so we have sine X. 00:10:07.020 --> 00:10:11.673 Plus and if we're putting everything over this denominator 00:10:11.673 --> 00:10:17.360 of X, we need to multiply anything that's not over the 00:10:17.360 --> 00:10:24.081 denominator of X already by X, so that's X Times Log X times 00:10:24.081 --> 00:10:26.666 cause X all over X. 00:10:27.700 --> 00:10:33.017 But this side with one over Y from what we're really after is 00:10:33.017 --> 00:10:34.653 DY by The X. 00:10:35.830 --> 00:10:41.905 And so we need to multiply up by why, but we know what? Why is 00:10:41.905 --> 00:10:44.740 its X to the power sign X? 00:10:46.390 --> 00:10:53.174 So that's sine XX log X cause X. 00:10:54.030 --> 00:11:00.764 OX and we're going to multiply up by Y, and that's X sign 00:11:00.764 --> 00:11:06.462 X to the power sign X, and so we finished the 00:11:06.462 --> 00:11:10.088 differentiation and again, something looked impossible at 00:11:10.088 --> 00:11:16.822 the beginning by making use of the laws of logs, we have been 00:11:16.822 --> 00:11:18.376 able to differentiate. 00:11:21.520 --> 00:11:28.637 So let's take a look another example, this time a more 00:11:28.637 --> 00:11:33.166 complicated one. Quotient one really is quite. 00:11:34.050 --> 00:11:35.120 Complex. 00:11:38.670 --> 00:11:44.214 Now again, we could do this as a quotient you over V. 00:11:44.800 --> 00:11:49.545 And use the formula which if we remember is VDU by DX minus 00:11:49.545 --> 00:11:53.560 UDV by the X all over V squared. But simply having 00:11:53.560 --> 00:11:57.210 said it, that's quite frightening. So if we have a 00:11:57.210 --> 00:12:02.685 look at this one of the ways we can do it is to take logs 00:12:02.685 --> 00:12:03.415 both sides. 00:12:10.580 --> 00:12:14.396 And then having taken logs of 00:12:14.396 --> 00:12:20.340 both sides. We can make use of our laws of logarithms to 00:12:20.340 --> 00:12:21.549 simplify this side. 00:12:22.850 --> 00:12:27.050 So let's begin by remembering that if we have a quotient when 00:12:27.050 --> 00:12:30.900 we do the logarithm, then that's the log of the numerator. 00:12:33.980 --> 00:12:38.948 Minus the log of the denominator. 00:12:42.960 --> 00:12:46.710 But we can simplify this one even further because we're 00:12:46.710 --> 00:12:52.335 raising 1 - 2 X to the power three, and in terms of our laws 00:12:52.335 --> 00:12:57.585 of logs, that's exactly the same as multiplying the log of 1 - 2 00:12:57.585 --> 00:12:58.710 X by three. 00:13:00.200 --> 00:13:05.260 Minus now here we've got a square root and the square 00:13:05.260 --> 00:13:10.320 root is the same as raising something to the power 1/2 00:13:10.320 --> 00:13:16.300 so that this is exactly the same as taking a half of the 00:13:16.300 --> 00:13:19.060 log of one plus X squared. 00:13:20.950 --> 00:13:27.060 Now what we see is that this that we've got here is much 00:13:27.060 --> 00:13:28.470 simpler to differentiate. 00:13:29.350 --> 00:13:36.266 So we can do that. The derivative of the log of Y doing 00:13:36.266 --> 00:13:43.714 it implicitly is one over YDY by X equals 3 Times Now we want 00:13:43.714 --> 00:13:50.630 to now differentiate the log of 1 - 2 X. So the function 00:13:50.630 --> 00:13:56.482 is 1 - 2 X. So we differentiate that it's minus 00:13:56.482 --> 00:13:59.674 two over 1 - 2 X. 00:14:00.500 --> 00:14:02.130 Minus 1/2. 00:14:04.080 --> 00:14:08.535 Differentiating the log of one plus X squared, the 00:14:08.535 --> 00:14:13.485 function is one plus X squared, so its derivative is 00:14:13.485 --> 00:14:16.455 2X over one plus X squared. 00:14:17.640 --> 00:14:22.860 Again, there's a penalty to be paid. We really ought to put 00:14:22.860 --> 00:14:26.340 these back together again, so it's all one. 00:14:27.010 --> 00:14:31.930 So let's do that and let's just simplify a little bit here. 00:14:31.930 --> 00:14:35.620 There's a two will cancel there with a 2. 00:14:36.250 --> 00:14:36.740 So. 00:14:38.740 --> 00:14:41.740 We've got one over Y. 00:14:42.540 --> 00:14:47.034 DY by the X is equal to. 00:14:49.760 --> 00:14:55.808 We've got 3 times by minus two over 1 - 2 X, so that's going to 00:14:55.808 --> 00:14:56.942 give us minus. 00:14:57.500 --> 00:14:58.530 6. 00:14:59.610 --> 00:15:03.756 All over 1 - 2 X. 00:15:06.410 --> 00:15:10.793 We've got minus X over one plus X squared. 00:15:18.200 --> 00:15:23.708 Now we need to bring these two terms together so we look 00:15:23.708 --> 00:15:28.757 for a common denominator that must be the product of the 00:15:28.757 --> 00:15:29.675 two denominators. 00:15:32.320 --> 00:15:39.250 And 1 - 2 X goes into this one plus X squared times minus 00:15:39.250 --> 00:15:46.675 six times by one plus X squared minus X times by well one plus X 00:15:46.675 --> 00:15:53.605 squared goes into the whole of this 1 - 2 X times. So we 00:15:53.605 --> 00:15:58.555 needed multiply the minus X by 1 - 2 X. 00:15:59.890 --> 00:16:05.500 Multiply out the bracket minus 6 - 6 X squared. 00:16:06.090 --> 00:16:09.914 Minus X +2 X 00:16:09.914 --> 00:16:17.037 squared. All over 1 - 2 X one plus X 00:16:17.037 --> 00:16:17.650 squared. 00:16:20.840 --> 00:16:22.670 We can simplify the top. 00:16:23.400 --> 00:16:30.848 Minus 6 minus X minus four X squared all over 1 - 2 X 00:16:30.848 --> 00:16:37.764 one plus X squared. And now finally we want DY by the X 00:16:37.764 --> 00:16:44.148 equals, but it's going to be this. Let's take that minus sign 00:16:44.148 --> 00:16:46.808 out so we have minus. 00:16:47.550 --> 00:16:53.605 6 plus X plus four X squared. 00:16:54.530 --> 00:17:02.390 All over 1 - 2 X times by one plus X squared. 00:17:03.680 --> 00:17:10.050 And we need to multiply both sides by Y and let's just recall 00:17:10.050 --> 00:17:15.440 what why was. Why was 1 - 2 X or cubed? 00:17:16.300 --> 00:17:23.250 So that's 1 - 2 X or cubed all over 00:17:23.250 --> 00:17:26.725 one plus X squared square 00:17:26.725 --> 00:17:31.690 root off. And clearly is a little bit more simplified, can 00:17:31.690 --> 00:17:37.270 be done here, because we've got a 1 - 2 X here and a 1 - 2 X 00:17:37.270 --> 00:17:40.680 cubed there so we can cancel that with one of those. 00:17:41.430 --> 00:17:45.668 Here we've got a one plus X squared times by the square root 00:17:45.668 --> 00:17:49.906 of 1 plus X squared, and if we follow our laws of indices, 00:17:49.906 --> 00:17:54.470 that's to the power one. And this is to the power half. So we 00:17:54.470 --> 00:17:58.382 add them together. That gives us to the power three over 2. 00:17:59.800 --> 00:18:06.019 So do you? Why by DX finally will be. 00:18:09.850 --> 00:18:11.500 Let's take this top line. 00:18:13.740 --> 00:18:20.476 Minus 6 plus X plus four X squared. 00:18:23.650 --> 00:18:24.940 Times by. 00:18:26.320 --> 00:18:28.850 1 - 2 X squared. 00:18:30.410 --> 00:18:35.946 1 - 2 X or squared all over. 00:18:37.210 --> 00:18:41.890 Just check on that denominator one plus X squared to the power 00:18:41.890 --> 00:18:47.740 one times by one plus X squared to the power a half and the one 00:18:47.740 --> 00:18:53.980 and add it to the half. Gives us 1 1/2 or three over 2, so that's 00:18:53.980 --> 00:18:58.270 one plus X squared raised to the power three over 2. 00:19:00.940 --> 00:19:05.321 So what we've seen in this video is that we can use the 00:19:05.321 --> 00:19:08.354 laws of logarithms to help us simplify certain functions 00:19:08.354 --> 00:19:10.039 before we come to differentiating. 00:19:11.070 --> 00:19:14.254 That can make the whole process of differentiation 00:19:14.254 --> 00:19:15.448 so much simpler.