WEBVTT 00:00:01.020 --> 00:00:07.060 Sometimes we have functions that look like this. 00:00:07.060 --> 00:00:10.808 Cause of X squared. 00:00:11.770 --> 00:00:18.210 Now there are immediate things that are different about this 00:00:18.210 --> 00:00:24.006 than the straightforward cosine function. It's cause of X 00:00:24.006 --> 00:00:31.734 squared, not just cause of X, and so we call this function 00:00:31.734 --> 00:00:38.174 of a function Y equals a function of another function. 00:00:39.030 --> 00:00:40.650 Now in this case. 00:00:41.250 --> 00:00:48.246 The function F is the cosine function, and the function G is 00:00:48.246 --> 00:00:54.659 the square function. Or we could identify them, perhaps a bit 00:00:54.659 --> 00:01:00.489 more mathematically by saying that F of X is cause 00:01:00.489 --> 00:01:07.485 X&G of X is X squared. Now let's have a look at 00:01:07.485 --> 00:01:09.817 another example of this. 00:01:10.570 --> 00:01:15.536 This time we'll turn it around, so to speak. Let's have a look 00:01:15.536 --> 00:01:17.064 at cause squared X. 00:01:17.760 --> 00:01:23.760 Now, how is this a function of a function? Let's remember what 00:01:23.760 --> 00:01:29.760 cost squared of X means. It means cause X squared cause X 00:01:29.760 --> 00:01:36.260 multiplied by itself. So now if we look at our function of a 00:01:36.260 --> 00:01:42.524 function. Let's see what we can do to identify which is F and 00:01:42.524 --> 00:01:49.116 which is G of X. So here we are squaring, so it's the F is the 00:01:49.116 --> 00:01:53.236 square function, and inside the square function, the thing that 00:01:53.236 --> 00:01:59.416 we are actually squaring is G of X, namely cause X. So here the F 00:01:59.416 --> 00:02:04.772 is the square function and the G is the cosine function. Or write 00:02:04.772 --> 00:02:07.244 it down more mathematically F of 00:02:07.244 --> 00:02:13.820 X. Is X squared? Angie of X is cause X? 00:02:14.580 --> 00:02:18.710 Now how do we differentiate a function like Cos squared 00:02:18.710 --> 00:02:23.666 X or a function like the cause of X squared? What we 00:02:23.666 --> 00:02:27.796 need to do to be able to differentiate something that 00:02:27.796 --> 00:02:30.274 is a function of a function? 00:02:31.620 --> 00:02:38.460 To do that, we need to do two things. One, 00:02:38.460 --> 00:02:41.196 we need to substitute. 00:02:42.870 --> 00:02:50.668 You equals G of X. This would then give us Y equals F of 00:02:50.668 --> 00:02:57.909 you, which of course is much simpler than F of G of X. 00:02:57.909 --> 00:03:05.707 Next we need to use a rule or a formula that's known as the 00:03:05.707 --> 00:03:10.606 chain rule. China rules quite simple, 00:03:10.606 --> 00:03:17.596 says DY by DX is equal to DY by EU 00:03:17.596 --> 00:03:21.091 Times du by The X. 00:03:22.200 --> 00:03:26.607 Notice it looks as though the D use cancel out. If these were 00:03:26.607 --> 00:03:29.997 fractions, which they're not, it looks as though they might 00:03:29.997 --> 00:03:32.031 cancel out. That's a way of 00:03:32.031 --> 00:03:36.292 remembering it. So this is how we're going to approach it. 00:03:36.292 --> 00:03:38.140 Substitute you equals G of X. 00:03:38.660 --> 00:03:43.509 And then apply this rule called the chain rule in order to find 00:03:43.509 --> 00:03:47.985 the derivative. So let's take a number of examples and the first 00:03:47.985 --> 00:03:52.461 one will take is the very first example that we looked at. 00:03:53.190 --> 00:03:59.868 And that was why equals cause of X squared. 00:04:00.380 --> 00:04:07.519 So our first step was to put you equals X squared. 00:04:08.080 --> 00:04:14.648 And then Y is equal to cause you. 00:04:15.840 --> 00:04:23.556 Now the chain rule says DY by the X is equal to 00:04:23.556 --> 00:04:30.629 DY by du times du by the X. So we need. 00:04:31.260 --> 00:04:32.650 Do you buy the X? 00:04:33.330 --> 00:04:37.714 Well, that's 2X. We differentiate this. We multiply 00:04:37.714 --> 00:04:45.386 by the two and take one off the index. That leaves us 2X and 00:04:45.386 --> 00:04:48.674 we need the why by DU. 00:04:49.570 --> 00:04:55.090 And the derivative of Cos you is minus sign you. 00:04:55.990 --> 00:05:02.374 So now we need to put these together DY by the X is equal 00:05:02.374 --> 00:05:09.722 to. Divide by do you minus sign U times do you buy DX 00:05:09.722 --> 00:05:11.826 which is 2 X? 00:05:12.530 --> 00:05:17.834 Now this is all very well, but really we'd like to have 00:05:17.834 --> 00:05:23.580 everything in terms of X. And here we've got you, so we need 00:05:23.580 --> 00:05:29.326 to undo this substitution. If we put you equals X squared, we now 00:05:29.326 --> 00:05:35.514 need to replace EU by X squared, and so we write minus 2X and 00:05:35.514 --> 00:05:40.376 bringing the two X to the front sign of X squared. 00:05:40.890 --> 00:05:46.584 Now they're all done like that. What I'm going to do with all 00:05:46.584 --> 00:05:51.840 the next examples that I do is I'm going to put this 00:05:51.840 --> 00:05:56.658 differentiation of the two bits up here with these two bits. 00:05:58.190 --> 00:06:05.138 Let's have a look at the other one. We had. Y equals 00:06:05.138 --> 00:06:09.770 cause squared X. Let's remember that meant the 00:06:09.770 --> 00:06:12.665 cause of X all squared. 00:06:13.890 --> 00:06:21.702 So this is now the G of X and so we will 00:06:21.702 --> 00:06:25.608 put you equals cause of X. 00:06:26.340 --> 00:06:32.549 Then Y is equal to you squared. 00:06:33.530 --> 00:06:40.460 I can calculate do you buy DX that's minus sign 00:06:40.460 --> 00:06:47.390 X and I can calculate DY by du. That's two 00:06:47.390 --> 00:06:55.180 you. Write down the chain rule the why by DX is 00:06:55.180 --> 00:06:59.026 DY by du times DU by 00:06:59.026 --> 00:07:05.110 The X. And now we can substitute in the bits that we've already 00:07:05.110 --> 00:07:07.700 calculated. The why by do you is 00:07:07.700 --> 00:07:14.434 to you. Times do you buy DX which is minus sign X equals. 00:07:14.434 --> 00:07:21.844 Now again we want this all in terms of X. So what we have to 00:07:21.844 --> 00:07:28.266 do is reverse the substitution we put you equals to cause X and 00:07:28.266 --> 00:07:34.688 now we need to undo that by replacing EU with cause X. So 00:07:34.688 --> 00:07:39.134 bringing the minus sign to the front minus 2. 00:07:39.150 --> 00:07:41.964 Kohl's X 00:07:41.964 --> 00:07:48.450 Sign X. Let's take 00:07:48.450 --> 00:07:52.000 another example. 00:07:53.240 --> 00:08:00.620 Y equals 2X minus five all raised to the power 00:08:00.620 --> 00:08:06.726 10. Now, it might be tempting to say, well surely we could just 00:08:06.726 --> 00:08:11.382 multiply out the brackets, but this is to the power 10 to 00:08:11.382 --> 00:08:15.262 multiply out. Those brackets would take his ages, and there's 00:08:15.262 --> 00:08:20.306 all those mistakes that could be made in doing it. Plus when we 00:08:20.306 --> 00:08:24.574 differentiate it, we may not have the best form for future 00:08:24.574 --> 00:08:29.618 work, so let's use function of a function. So here we will put 00:08:29.618 --> 00:08:31.946 you equals this bit here inside 00:08:31.946 --> 00:08:35.392 the bracket. 2X minus five 00:08:35.392 --> 00:08:42.742 and then. Why is equal to U to the power 00:08:42.742 --> 00:08:43.376 10? 00:08:44.550 --> 00:08:50.400 I can now do the differentiation of the little bits. Do you buy 00:08:50.400 --> 00:08:57.150 the X is just two the derivative of two X just giving us two and 00:08:57.150 --> 00:09:04.350 E? Why by DU is 10 we multiply by the index U to the power 9. 00:09:04.350 --> 00:09:10.200 We take one away from the index. Now we can put this together 00:09:10.200 --> 00:09:15.150 using the chain rule so divided by DX is equal to. 00:09:15.880 --> 00:09:23.390 The why by DU times DU by X, substituting our 00:09:23.390 --> 00:09:30.900 little bits. Here's the why by DU10U to the ninth 00:09:30.900 --> 00:09:34.655 times two do you buy 00:09:34.655 --> 00:09:41.400 DX? 2 * 10 is 20 and I want you to the power 9 and I 00:09:41.400 --> 00:09:47.160 need to get this all in terms of X, so I need to replace the you 00:09:47.160 --> 00:09:53.634 here. By the two X minus five. So that's 2X minus five all to 00:09:53.634 --> 00:09:57.614 the power 9, and that's a compact expression for the 00:09:57.614 --> 00:10:02.390 derivative. Think what it would have been like if I had to 00:10:02.390 --> 00:10:05.176 expand the brackets and differentiate each term. 00:10:06.520 --> 00:10:13.312 I want now to take another trick example and then develope that 00:10:13.312 --> 00:10:19.538 trig example a little bit further to a more general case. 00:10:21.020 --> 00:10:27.242 So we'll take Y equals the 00:10:27.242 --> 00:10:30.353 sign of 5X. 00:10:31.090 --> 00:10:37.954 Very easy here to identify the G of X. It's this bit 00:10:37.954 --> 00:10:43.102 here the 5X so will put you equals 5X. 00:10:43.790 --> 00:10:49.856 And then why will be equal to sign you? 00:10:50.890 --> 00:10:57.810 Differentiating do you buy the X is equal to 5. 00:10:58.540 --> 00:11:04.894 And DY bite U is equal to cause you. 00:11:05.560 --> 00:11:12.640 Put the two bits together with the chain rule DY by the 00:11:12.640 --> 00:11:19.720 X is equal to DY by EU Times DU by The X. 00:11:20.710 --> 00:11:22.950 DIY Bindu is cause you. 00:11:23.460 --> 00:11:29.980 Times by and you by the ex here is 5. 00:11:30.900 --> 00:11:38.112 So let's bring the five to the front 5 cause of, and 00:11:38.112 --> 00:11:43.521 now let's reverse the substitution. You is equal to 00:11:43.521 --> 00:11:47.127 5X, so will replace EU by 00:11:47.127 --> 00:11:52.840 5X. Now, notice how that five here and here is apparently 00:11:52.840 --> 00:11:58.972 appeared there, and it did so because the derivative of 5X was 00:11:58.972 --> 00:12:05.615 five. So the question is, could we do this with any number that 00:12:05.615 --> 00:12:12.769 appeared there in front of the X bit five or six? Or 1/2 or 00:12:12.769 --> 00:12:16.346 not .5? Or for that matter, an 00:12:16.346 --> 00:12:23.374 so? Have a look at Y equal sign NX. 00:12:25.130 --> 00:12:28.770 You equals an X that 00:12:28.770 --> 00:12:35.830 sour substitution. And then Y is equal to sign you we can 00:12:35.830 --> 00:12:41.938 differentiate EU with respect to X and the derivative of NX is 00:12:41.938 --> 00:12:45.501 just N because N is a constant 00:12:45.501 --> 00:12:52.960 and number. And the why by DU is equal to cause you. 00:12:52.960 --> 00:12:56.434 We can now put this back 00:12:56.434 --> 00:13:03.630 together again. My by the X is DY by EU Times 00:13:03.630 --> 00:13:06.310 DU by The X. 00:13:07.110 --> 00:13:13.860 Equals. DIY Bindu is cause you. 00:13:14.470 --> 00:13:17.646 Times by du by DX, which is just 00:13:17.646 --> 00:13:25.395 N. So let's move the end to the front. That's 00:13:25.395 --> 00:13:27.660 N cause NX. 00:13:27.670 --> 00:13:31.438 So the ends of behaved in exactly the same way that the 00:13:31.438 --> 00:13:34.578 fives behaved in the previous question. And of course this 00:13:34.578 --> 00:13:38.660 means. Now we're in a position to be able to do any question 00:13:38.660 --> 00:13:40.544 like this simply by writing down 00:13:40.544 --> 00:13:46.620 the answer. So if we're just going to write down the answer, 00:13:46.620 --> 00:13:53.680 let's take. Why is sign of 6X? 00:13:54.240 --> 00:14:02.220 And then divide by the X is just six cause 6X just by using 00:14:02.220 --> 00:14:05.640 the standard result that we had 00:14:05.640 --> 00:14:12.678 before. Similarly, if we had Y was equal to cause 1/2, X 00:14:12.678 --> 00:14:18.376 will just because it's cause it's not going to be any 00:14:18.376 --> 00:14:24.592 different to sign really and so D. Why by DX is equal 00:14:24.592 --> 00:14:30.290 to 1/2 and the derivative of causes minus sign 1/2 X. 00:14:31.330 --> 00:14:38.315 Now let's take one more example. In this particular style, so 00:14:38.315 --> 00:14:45.935 will take Y equals E to the X cubed. And here this 00:14:45.935 --> 00:14:53.555 X cubed is our G of X. So we will put you 00:14:53.555 --> 00:14:55.460 equals X cubed. 00:14:55.460 --> 00:15:02.474 And then why will be equal 2 E to the power you we can 00:15:02.474 --> 00:15:08.486 differentiate. So do you find the X is equal to three X 00:15:08.486 --> 00:15:14.498 squared? We multiply by the power and take one off it. And 00:15:14.498 --> 00:15:20.510 why by DU is equal to E to the power you we 00:15:20.510 --> 00:15:25.019 differentiate with respect to you. The exponential function is 00:15:25.019 --> 00:15:29.128 it's. Own derivative, and so it stays the same. 00:15:30.130 --> 00:15:35.863 Bringing those two bits together to give us the why by DX through 00:15:35.863 --> 00:15:42.919 the chain rule DY by DX is DY by du times by DU by The X. 00:15:43.520 --> 00:15:50.736 The why Bindu is E to the power you times by du by DX which is 00:15:50.736 --> 00:15:52.089 3 X squared. 00:15:52.890 --> 00:15:56.886 Move the three X squared through to the front. 00:15:57.910 --> 00:16:02.746 E to the power. And now we made the substitution that gave us 00:16:02.746 --> 00:16:07.210 you we put you equals X cubed and now we need to. 00:16:07.760 --> 00:16:14.645 Reverse that substitution and replace U by X cubed. 00:16:16.040 --> 00:16:21.270 So let's have a look at what we've got here. 00:16:21.910 --> 00:16:28.880 E to the X cubed. Now if we think of this as E to EU then E 00:16:28.880 --> 00:16:34.210 to EU is just the derivative of the F with respect to you. 00:16:34.210 --> 00:16:38.310 Because the exponential function is own derivative and here the 00:16:38.310 --> 00:16:43.230 three X squared is simply the derivative of U with respect to 00:16:43.230 --> 00:16:48.560 X. In other words, it's the derivative of G of X. Now let's 00:16:48.560 --> 00:16:53.070 see if we can put that together for a general case. 00:16:54.060 --> 00:17:01.470 So we've got Y equals F of G of X. 00:17:02.730 --> 00:17:09.828 And we're putting you equals G of X and our chain rule tells 00:17:09.828 --> 00:17:15.288 us that the why by the X is equal to. 00:17:15.850 --> 00:17:23.178 DY by EU Times DU by The X. 00:17:23.750 --> 00:17:28.722 Equals let's start here. You remember that in most of the 00:17:28.722 --> 00:17:34.146 examples that I've done, I've been pushing the result of du by 00:17:34.146 --> 00:17:37.310 DX forward to the front of the 00:17:37.310 --> 00:17:43.603 expression. So let's do it to begin with. Instead, at the end 00:17:43.603 --> 00:17:49.159 when we've been simplifying, so do you buy DX? This is the 00:17:49.159 --> 00:17:55.641 derivative of G of X, so that's DG by D of X by DX. 00:17:56.550 --> 00:18:03.401 Times by and this is the why by DU. So that's the derivative 00:18:03.401 --> 00:18:08.671 of F of G of X with respect to you. 00:18:09.850 --> 00:18:16.192 Now can we use this? Let's have a look at an example why equals 00:18:16.192 --> 00:18:22.534 the tan of X squared and so DY by the X is equal to. 00:18:23.170 --> 00:18:28.574 Well, this bit here is the G of X, so we need its derivative 00:18:28.574 --> 00:18:33.206 and we're going to write it down at the front. That's easy, 00:18:33.206 --> 00:18:37.452 that's 2X. And now we need to differentiate tan as though 00:18:37.452 --> 00:18:42.084 this was tamu with respect to you and the derivative of tan 00:18:42.084 --> 00:18:47.874 is just sex squared. And then we need to put in the G of X 00:18:47.874 --> 00:18:49.418 sex squared X squared. 00:18:50.530 --> 00:18:55.174 Notice how short this is. 2 lines. Let's just look back at 00:18:55.174 --> 00:19:00.205 the example that we did before and it took us all of this. 00:19:00.800 --> 00:19:05.430 What we've done is we've contracted this whole process to 00:19:05.430 --> 00:19:08.208 one that takes place in our 00:19:08.208 --> 00:19:14.594 head. Quicker. You can still do it this way if you like, but if 00:19:14.594 --> 00:19:19.118 you can get into the habit of doing this, it's much quicker. 00:19:19.118 --> 00:19:24.019 So let's have a look at some examples and try and keep track 00:19:24.019 --> 00:19:29.297 of what we're doing. So let's begin with Y equals E to the one 00:19:29.297 --> 00:19:30.428 plus X squared. 00:19:30.440 --> 00:19:33.555 Sunday why by DX is equal to. 00:19:34.080 --> 00:19:39.722 Well, we can identify the G of X. It's this up here. The power 00:19:39.722 --> 00:19:44.155 of the exponential function, and so if we differentiate that it's 00:19:44.155 --> 00:19:48.185 2X. And now we want the derivative of the exponential 00:19:48.185 --> 00:19:53.500 function. It is its own derivative, so that's E to the 00:19:53.500 --> 00:19:55.204 one plus X squared. 00:19:56.420 --> 00:19:58.688 Able to write it down straight 00:19:58.688 --> 00:20:06.092 away. Let's take Y equals the sign of X 00:20:06.092 --> 00:20:10.062 Plus E to the X. 00:20:10.100 --> 00:20:13.845 The why by DX is equal to 00:20:13.845 --> 00:20:19.804 well here. We can identify the G of X, so we want the derivative 00:20:19.804 --> 00:20:24.088 of this to write down the derivative of X is one. The 00:20:24.088 --> 00:20:28.372 derivative of E to the X is just E to the X. 00:20:28.890 --> 00:20:32.958 Times by and we want the derivative of this. 00:20:33.860 --> 00:20:39.827 As though this were you and the derivative of sine is cause, and 00:20:39.827 --> 00:20:43.499 then X Plus E to the power X. 00:20:44.030 --> 00:20:49.794 We take Y equals 00:20:49.794 --> 00:20:56.545 the tan. All X squared 00:20:56.545 --> 00:20:59.860 plus sign X. 00:20:59.860 --> 00:21:06.964 Why buy the X is equal to identify the G of X? 00:21:06.964 --> 00:21:12.884 That's this lump here, the X squared plus sign X 00:21:12.884 --> 00:21:15.844 differentiate it 2X plus cause 00:21:15.844 --> 00:21:22.310 X. And now the derivative of tan as though it were tan of 00:21:22.310 --> 00:21:27.040 U. Well that will be sex squared. But instead of EU 00:21:27.040 --> 00:21:33.060 we want you is G of X which is X squared plus sign X. 00:21:36.490 --> 00:21:42.509 Y equals this time will have a look at one that's got brackets 00:21:42.509 --> 00:21:49.454 and powers in it. 2 minus X to the 5th, all raised to the 9th 00:21:49.454 --> 00:21:56.454 power. The why by DX is got to identify the G of X first and 00:21:56.454 --> 00:22:02.844 the G of X is this bit inside the bracket the two minus X to 00:22:02.844 --> 00:22:07.530 the power 5. So we need to differentiate that first, so 00:22:07.530 --> 00:22:11.790 that's going to be minus 5X to the power 4. 00:22:12.650 --> 00:22:16.970 Remember the derivative of two, which is a constant is 0 and 00:22:16.970 --> 00:22:21.650 then multiply by the Five and take one off the index of minus 00:22:21.650 --> 00:22:23.450 5X to the power 4. 00:22:24.080 --> 00:22:29.267 Now we've got to differentiate to the power 9 as though it was 00:22:29.267 --> 00:22:34.853 you to the power 9 will that would be times by 9, and then 00:22:34.853 --> 00:22:36.848 instead of EU we want. 00:22:37.360 --> 00:22:42.834 The two minus X to the 5th and we take one away from the 00:22:42.834 --> 00:22:47.526 power. We get that little bit of tidying up to do here, 00:22:47.526 --> 00:22:51.045 'cause we've got a multiplication that we can do 00:22:51.045 --> 00:22:57.301 minus five times by 9 - 45 X to the power four 2 minus X to 00:22:57.301 --> 00:23:00.038 the fifth, raised to the power 8. 00:23:01.460 --> 00:23:04.760 Take 1 final example. 00:23:05.540 --> 00:23:13.010 Let's take Y equals the log of X plus sign 00:23:13.010 --> 00:23:17.912 X. We can identify this as our G of X. 00:23:19.080 --> 00:23:25.926 So D why by DX is equal to we can write down the derivative 00:23:25.926 --> 00:23:32.283 of this one plus cause X times by and now we want the 00:23:32.283 --> 00:23:33.750 derivative of log. 00:23:34.440 --> 00:23:40.121 Well, the derivative of log of you would be one over you, so 00:23:40.121 --> 00:23:47.550 this is one over and this is EU or the G of X one over X plus 00:23:47.550 --> 00:23:54.105 sign X and it be untidy to leave it like that. So we move that 00:23:54.105 --> 00:23:58.475 numerator one plus cause X. So it's more obviously the 00:23:58.475 --> 00:24:03.719 numerator of the fraction. So what we've seen there is the use 00:24:03.719 --> 00:24:05.467 of the chain rule. 00:24:06.070 --> 00:24:10.318 But then modifying it slightly so we get used to writing down 00:24:10.318 --> 00:24:13.858 the answer more or less straight away. You can still 00:24:13.858 --> 00:24:18.460 use the chain rule. You can still do it at full length, but 00:24:18.460 --> 00:24:23.416 if you can get into the habit of doing it like this, you find 00:24:23.416 --> 00:24:27.310 it quicker and easier to be able to manage and more 00:24:27.310 --> 00:24:31.204 complicated problems will be easier to do if you can short 00:24:31.204 --> 00:24:33.682 cut short circuit some of the processes.