1 00:00:01,440 --> 00:00:06,156 In this video, we're going to be looking at how we might 2 00:00:06,156 --> 00:00:11,038 differentiate. Functions of Y with respect to X. 3 00:00:11,800 --> 00:00:17,389 Let's begin by looking at an equation like this. 4 00:00:18,060 --> 00:00:23,702 X squared Plus Y 5 00:00:23,702 --> 00:00:27,005 squared. Minus 6 00:00:27,005 --> 00:00:33,930 4X. +5 Y minus 8 equals 7 00:00:33,930 --> 00:00:40,109 0. Now here the X is and wiser all tangled 8 00:00:40,109 --> 00:00:43,175 together and the wise Y squared 9 00:00:43,175 --> 00:00:48,506 and Y. It will be quite difficult to rearrange this, so 10 00:00:48,506 --> 00:00:51,194 it said Y equals a function of 11 00:00:51,194 --> 00:00:58,594 X. We could perhaps given values of X workout what values of why 12 00:00:58,594 --> 00:01:01,204 were and thereby draw graph. 13 00:01:01,810 --> 00:01:05,370 But nevertheless, these things are intimately connected, and 14 00:01:05,370 --> 00:01:09,820 differentiating something like this is going to be much harder, 15 00:01:09,820 --> 00:01:16,050 so it would seem than if it just said why equals some function of 16 00:01:16,050 --> 00:01:21,195 X. But that's what we're going to have a look at in this video. 17 00:01:21,195 --> 00:01:24,045 How can we differentiate a function and equation that looks 18 00:01:24,045 --> 00:01:27,180 like this where the X and wiser all tangled up together? 19 00:01:27,770 --> 00:01:31,554 We're going to use something known as the chain rule or 20 00:01:31,554 --> 00:01:35,682 function of a function. It's got those two names. Chain rule an 21 00:01:35,682 --> 00:01:37,058 function of a function. 22 00:01:37,760 --> 00:01:41,218 I'm going to try and stick with using the name chain rule, but 23 00:01:41,218 --> 00:01:43,346 you may know it as function of a 24 00:01:43,346 --> 00:01:48,938 function. There is a video which covers this particular rule 25 00:01:48,938 --> 00:01:51,800 explicitly. So I'm just going to 26 00:01:51,800 --> 00:01:58,994 revise it. So let's take an example such as Y 27 00:01:58,994 --> 00:02:06,316 equals. 5 + 2 X to the 10th power. And 28 00:02:06,316 --> 00:02:11,932 let's say I want to differentiate this. Then the 29 00:02:11,932 --> 00:02:15,676 chain rule says that if I 30 00:02:15,676 --> 00:02:21,493 port. You equals 5 31 00:02:21,493 --> 00:02:24,575 + 2 32 00:02:24,575 --> 00:02:27,860 X. Then 33 00:02:29,150 --> 00:02:34,330 Why will be equal to you to the power 10? 34 00:02:34,830 --> 00:02:41,254 And we can form the why by DX 35 00:02:41,254 --> 00:02:47,678 by doing DY by du times DU by 36 00:02:47,678 --> 00:02:50,890 DX and it's this. 37 00:02:51,560 --> 00:02:54,170 That is the chain rule. 38 00:02:54,700 --> 00:02:58,429 Or some of you may know it function of a function. 39 00:02:59,330 --> 00:03:04,290 So this is what we've got to do. We've got to work out the why by 40 00:03:04,290 --> 00:03:08,630 DU and workout. Do you buy the X? So I'm going to turn the 41 00:03:08,630 --> 00:03:12,350 page. I'm going to virtually write this out again in order to 42 00:03:12,350 --> 00:03:16,070 make sure that I've got the space in which to do it. 43 00:03:17,060 --> 00:03:20,804 So we begin with Y equals 44 00:03:20,804 --> 00:03:24,422 5. Plus 2X to the 45 00:03:24,422 --> 00:03:31,260 power 10. We put you equals 5 + 46 00:03:31,260 --> 00:03:38,060 2 X and then Y is equal to 47 00:03:38,060 --> 00:03:41,460 U to the power 48 00:03:41,460 --> 00:03:48,860 10. DY by the X is given by the Y by 49 00:03:48,860 --> 00:03:55,856 EU times. Do you buy the X and we can workout each 50 00:03:55,856 --> 00:04:02,852 of these two divided by DU&U by DX. So let's have a 51 00:04:02,852 --> 00:04:09,848 look at that why by EU is the derivative of U to 52 00:04:09,848 --> 00:04:13,346 the power 10 with respect to 53 00:04:13,346 --> 00:04:19,478 you. Which is 10 U to the power 9. Remember we multiply by the 54 00:04:19,478 --> 00:04:25,442 index and take one of the index to give us nine and so that's 55 00:04:25,442 --> 00:04:32,890 10. 5 + 2 X to the Power 9 replacing EU 56 00:04:32,890 --> 00:04:36,070 by 5 + 2 X. 57 00:04:36,970 --> 00:04:41,690 And we can calculate you by The X. 58 00:04:42,800 --> 00:04:48,960 That's the derivative of 5 + 2 X. 59 00:04:49,300 --> 00:04:54,217 With respect to X, and that's just two because the derivative 60 00:04:54,217 --> 00:05:00,475 of five 5 is a constant stats zero, the derivative of two X is 61 00:05:00,475 --> 00:05:07,078 just two. So now we know what divided by du is. It's this and 62 00:05:07,078 --> 00:05:12,655 we know what do you buy the access. It's this so we can 63 00:05:12,655 --> 00:05:13,942 write those in. 64 00:05:14,850 --> 00:05:22,131 10 Times 5 + 2 X to the 65 00:05:22,131 --> 00:05:29,407 9th. Times by two and of course, the two times by 10 66 00:05:29,407 --> 00:05:30,988 gives us 20. 67 00:05:31,570 --> 00:05:38,162 So we've used our chain rule in order 68 00:05:38,162 --> 00:05:43,930 to be able to differentiate this function. 69 00:05:44,730 --> 00:05:51,154 So let me just write our chain rule 70 00:05:51,154 --> 00:05:57,578 down here DY by X is equal to 71 00:05:57,578 --> 00:06:03,199 Y by DU times du by X. 72 00:06:05,080 --> 00:06:08,840 Now. Let's suppose that we 73 00:06:08,840 --> 00:06:14,557 had zed. And said was a function of Why? 74 00:06:15,360 --> 00:06:22,120 Then D zed by the X would be equal to 75 00:06:22,120 --> 00:06:24,980 D zed. Bye. 76 00:06:25,500 --> 00:06:29,424 DY times DY 77 00:06:29,424 --> 00:06:35,580 by X. Using our chain rule again. 78 00:06:36,410 --> 00:06:42,053 Let's take an example. Let's say that zed is 79 00:06:42,053 --> 00:06:44,561 equal to Y squared. 80 00:06:45,650 --> 00:06:52,420 Then these Ed by the X would be equal to. 81 00:06:53,240 --> 00:06:59,252 The derivative of Y squared with respect to Y. 82 00:07:00,050 --> 00:07:06,920 Times DY by the X and the derivative of Y 83 00:07:06,920 --> 00:07:13,790 squared with respect to Y is equal to two Y 84 00:07:13,790 --> 00:07:16,538 times DY by X. 85 00:07:17,540 --> 00:07:23,260 Notice what we seem to have done is just differentiated. Why 86 00:07:23,260 --> 00:07:30,020 square root respect to Y and multiplied by DY by the X? And 87 00:07:30,020 --> 00:07:35,740 I'll draw attention to that again later on. For now, let's 88 00:07:35,740 --> 00:07:41,980 have a look at some examples. Begin with this one Y squared. 89 00:07:42,570 --> 00:07:45,930 Plus X cubed. 90 00:07:46,780 --> 00:07:50,150 Minus Y cubed plus 6. 91 00:07:50,700 --> 00:07:55,800 Equals 3 Y. We want to differentiate this with respect 92 00:07:55,800 --> 00:08:01,920 to X, so we're looking for the derivative of Y squared with 93 00:08:01,920 --> 00:08:03,450 respect to X. 94 00:08:03,980 --> 00:08:11,068 Derivative of X cubed with respect to X. 95 00:08:11,070 --> 00:08:15,366 Derivative of Y cubed with respect to X. 96 00:08:16,530 --> 00:08:22,558 Derivative of the six with respect to X and the derivative 97 00:08:22,558 --> 00:08:29,682 of three Y with respect to X. Not remember how we said we 98 00:08:29,682 --> 00:08:36,806 would do this? We said we will take the derivative of Y squared 99 00:08:36,806 --> 00:08:38,998 with respect to Y. 100 00:08:39,750 --> 00:08:45,426 And multiply by DY by the X that was our chain rule. 101 00:08:46,070 --> 00:08:50,410 Plus this is straightforward, don't need to worry about this. 102 00:08:50,410 --> 00:08:56,052 This is the derivative of X cubed with respect to X. We know 103 00:08:56,052 --> 00:09:01,694 that is 3 X squared multiplied by the index and take one away 104 00:09:01,694 --> 00:09:07,982 minus. And here again, we've got to apply our chain rule. So 105 00:09:07,982 --> 00:09:13,174 we've got the derivative of Y cubed with respect to Y. 106 00:09:13,880 --> 00:09:17,324 Times by DY. By 107 00:09:17,324 --> 00:09:21,210 The X. Plus 108 00:09:22,020 --> 00:09:25,972 Now the derivative of six well six is just a constant, so we 109 00:09:25,972 --> 00:09:27,492 know its derivative is 0. 110 00:09:28,690 --> 00:09:34,282 Equals and again we want the derivative of three Y, so our 111 00:09:34,282 --> 00:09:39,874 chain rule tells us this is the derivative of three Y with 112 00:09:39,874 --> 00:09:43,136 respect to Y times DY by DX. 113 00:09:43,770 --> 00:09:48,593 Now we can go back and work out each of these derivatives with 114 00:09:48,593 --> 00:09:51,190 respect to Y, so that will give 115 00:09:51,190 --> 00:09:57,140 us 2Y. DYIDX plus three 116 00:09:57,140 --> 00:10:04,052 X squared. Minus the derivative of Y Cube 117 00:10:04,052 --> 00:10:11,228 with respect to Y is 3 Y squared divided by DX. We can 118 00:10:11,228 --> 00:10:12,884 ignore the zero. 119 00:10:13,460 --> 00:10:20,324 And the derivative of three Y with respect to Y is 3 times 120 00:10:20,324 --> 00:10:21,908 divided by DX. 121 00:10:22,930 --> 00:10:29,235 Now. If we get together all the terms that have a DY by the X in 122 00:10:29,235 --> 00:10:35,660 them. Then, having done that, we can sort out what divided by DX 123 00:10:35,660 --> 00:10:40,940 actually is, so I'm going to gather together all the terms in 124 00:10:40,940 --> 00:10:48,420 do I buy DX over on this side of the equation so I can keep .3 X 125 00:10:48,420 --> 00:10:53,700 squared there on its own. So I have three X squared equals. 126 00:10:54,600 --> 00:11:01,739 Now here on this side I've got 3D Y by X. 127 00:11:02,700 --> 00:11:08,524 I'm going to take this away from each side, so that's minus two 128 00:11:08,524 --> 00:11:15,558 Y. DY by X and I'm going to add this term to both sides 129 00:11:15,558 --> 00:11:22,248 plus three Y squared DY by The X. what I can see here is that 130 00:11:22,248 --> 00:11:28,938 I've got a common factor of the why by DX that I can take out, 131 00:11:28,938 --> 00:11:33,844 so that's what we're going to do next, so will have. 132 00:11:34,970 --> 00:11:38,410 Three X squared equals. 133 00:11:39,920 --> 00:11:45,951 Bracket. And taking out that common factor of the why 134 00:11:45,951 --> 00:11:51,117 by DX. So let's just go back and have a lot. What do I 135 00:11:51,117 --> 00:11:54,069 buy? DX was multiplying. It was multiplying A3. 136 00:11:55,150 --> 00:12:02,248 It was multiplying a minus two Y and it was multiplying A plus 137 00:12:02,248 --> 00:12:09,346 three Y squared, so it was multiplying a 3 - 2 Y and 138 00:12:09,346 --> 00:12:17,062 a plus. 3 Y squared. So now we can get divided by DX on 139 00:12:17,062 --> 00:12:23,494 its own if we divide throughout by this expression sode, why by 140 00:12:23,494 --> 00:12:29,390 DX is equal to three X squared and dividing throughout dividing 141 00:12:29,390 --> 00:12:35,286 both sides by 3 - 2 Y plus three Y squared. 142 00:12:36,260 --> 00:12:41,130 And then we've got our expression for DY by DX. 143 00:12:41,820 --> 00:12:44,816 That was a reasonably straightforward example. The 144 00:12:44,816 --> 00:12:49,096 work that many complications and it followed very directly from 145 00:12:49,096 --> 00:12:54,660 our first look at this. So now let's look at a slightly more 146 00:12:54,660 --> 00:12:58,512 complicated example, one where in fact we've got other 147 00:12:58,512 --> 00:13:04,574 functions. Of X&Y. So in this case will start up with a sign 148 00:13:04,574 --> 00:13:09,161 Y where we've got the Axis and the wise actually combined 149 00:13:09,161 --> 00:13:13,331 together, so we've got X squared times by Y cubed. 150 00:13:15,040 --> 00:13:22,540 Minus calls X and let's say equals 2 Yi. Want to be 151 00:13:22,540 --> 00:13:29,415 able to differentiate this with respect to X, so that's the 152 00:13:29,415 --> 00:13:36,290 derivative of sine Y with respect to X plus the derivative 153 00:13:36,290 --> 00:13:39,415 of X squared Y cubed. 154 00:13:40,010 --> 00:13:47,990 With respect to X minus the derivative of Cos X with respect 155 00:13:47,990 --> 00:13:55,970 to X equals the derivative of two Y with respect to X. 156 00:13:56,970 --> 00:14:02,510 Now let's remember what our function of a function rule 157 00:14:02,510 --> 00:14:09,158 tells us that this is done as the derivative of sine Y 158 00:14:09,158 --> 00:14:14,144 with respect to Y times by DY by DX. 159 00:14:16,340 --> 00:14:21,870 This one. Bit of a problem 'cause This is X squared times 160 00:14:21,870 --> 00:14:24,768 by Y cubed. So it's a product. 161 00:14:25,370 --> 00:14:31,498 It's a U times by AV, so let me just write a little U over the 162 00:14:31,498 --> 00:14:33,796 top and a little V there. 163 00:14:34,670 --> 00:14:41,276 You is X squared and V is Y cubed. 164 00:14:41,810 --> 00:14:43,518 So this is plus. 165 00:14:44,330 --> 00:14:49,136 Let's remember how we differentiate a product. We take 166 00:14:49,136 --> 00:14:55,847 you. And we multiply it by the derivative of V. That's the 167 00:14:55,847 --> 00:14:59,903 derivative of Y cubed with respect to X. 168 00:15:01,110 --> 00:15:08,190 Plus the an we multiply that by the derivative of U, which 169 00:15:08,190 --> 00:15:13,500 in this case is the derivative of X squared. 170 00:15:14,100 --> 00:15:21,211 Minus now we can do this one. The derivative of Cos X with 171 00:15:21,211 --> 00:15:27,775 respect to X. The derivative of causes minus sign, so a minus 172 00:15:27,775 --> 00:15:34,339 and minus makes a plus sign. X equals the derivative of and 173 00:15:34,339 --> 00:15:40,903 again my chain rule tells me that this is the derivative of 174 00:15:40,903 --> 00:15:44,732 two Y with respect to Y times 175 00:15:44,732 --> 00:15:48,508 by. Divide by The X. 176 00:15:50,150 --> 00:15:54,605 So this has been much more complicated, but notice how it 177 00:15:54,605 --> 00:15:58,250 follows the standard rules that we've already got for 178 00:15:58,250 --> 00:16:01,895 differentiation. So now the derivative of sine wired with 179 00:16:01,895 --> 00:16:04,730 respect to Y is just cause why. 180 00:16:05,420 --> 00:16:08,549 DY by X. 181 00:16:09,340 --> 00:16:13,580 Plus X squared 182 00:16:14,840 --> 00:16:20,660 Times by now this will be the derivative of Y cubed with 183 00:16:20,660 --> 00:16:22,115 respect to Y. 184 00:16:22,660 --> 00:16:26,698 Times by DY. By The X. 185 00:16:28,100 --> 00:16:35,250 Plus this one Y cubed times by now the derivative of X squared 186 00:16:35,250 --> 00:16:39,100 with respect to X is just 2X. 187 00:16:39,820 --> 00:16:42,118 Plus sign X. 188 00:16:42,880 --> 00:16:47,430 We've already done that one equals and hear the derivative 189 00:16:47,430 --> 00:16:53,800 of two Y with respect to Y. These two times DY by The X. 190 00:16:55,080 --> 00:17:02,140 Almost done now we still got a little bit of 191 00:17:02,140 --> 00:17:09,200 differentiation in here to do so. Let's do that cause 192 00:17:09,200 --> 00:17:16,260 YDY by DX plus I think I can fairly safely 193 00:17:16,260 --> 00:17:23,320 remove these brackets now. X squared times 3 Y squared 194 00:17:23,320 --> 00:17:28,968 DY by X +2 XY cubed plus sign. 195 00:17:28,990 --> 00:17:34,625 X equals 2 DY by The X. 196 00:17:35,240 --> 00:17:39,465 Where at the same stage as we were last time we've got the 197 00:17:39,465 --> 00:17:43,040 differentiation Dom and it's this thing. The why by DX that 198 00:17:43,040 --> 00:17:47,265 we want. So what we gotta do is get all those terms that 199 00:17:47,265 --> 00:17:51,490 involved why by DX on one side of the equation and the other 200 00:17:51,490 --> 00:17:55,390 terms on the other side. Now these are the two terms that 201 00:17:55,390 --> 00:17:59,940 don't have a divided by DX in them, so I'm going to keep them 202 00:17:59,940 --> 00:18:03,515 at this side. So it's two XY cubed plus sign X. 203 00:18:05,780 --> 00:18:11,128 Two, XY cubed plus 204 00:18:11,128 --> 00:18:13,802 sign X 205 00:18:13,802 --> 00:18:20,312 equals. Let's just go back and see what we've got. We've got a 206 00:18:20,312 --> 00:18:22,242 2 divided by DX here. 207 00:18:22,280 --> 00:18:28,370 2. DY by The X. 208 00:18:29,090 --> 00:18:34,576 And we're going to bring these two terms over to this side by 209 00:18:34,576 --> 00:18:39,218 taking them away from both sides. So we're going to take 210 00:18:39,218 --> 00:18:42,594 away 'cause why do I buy the X? 211 00:18:43,330 --> 00:18:49,921 On both sides, minus cause YDY by the X, and then we're going 212 00:18:49,921 --> 00:18:56,512 to take away the other term. This is the X squared times by 213 00:18:56,512 --> 00:19:02,089 three Y squared divided by DX, right that a little bit 214 00:19:02,089 --> 00:19:08,680 differently. When I do it, so we have minus three X squared Y 215 00:19:08,680 --> 00:19:10,708 squared divided by X. 216 00:19:11,990 --> 00:19:17,464 Again, we see we've got a set of terms here, each with divided by 217 00:19:17,464 --> 00:19:19,419 DX as a common factor. 218 00:19:19,920 --> 00:19:26,526 Two, XY cubed plus sign X is equal to. 219 00:19:27,500 --> 00:19:33,704 Let's take out this common factor of DY by The X. 220 00:19:34,300 --> 00:19:39,085 Well, it's multiplying two, so we've got a two there. It's 221 00:19:39,085 --> 00:19:43,435 multiplying minus cause Y, so we've gotta minus cause why 222 00:19:43,435 --> 00:19:47,785 there? And it's multiplying minus three X squared Y squared 223 00:19:47,785 --> 00:19:53,005 minus three X squared Y squared. And now finally we can get 224 00:19:53,005 --> 00:19:57,790 divided by DX on its own, because we can divide throughout 225 00:19:57,790 --> 00:20:02,140 divide both sides of the equation by what's in this 226 00:20:02,140 --> 00:20:05,185 bracket. So we have two XY cubed 227 00:20:05,185 --> 00:20:12,623 plus sign. X all over 2 minus cause Y minus three 228 00:20:12,623 --> 00:20:15,531 X squared Y squared. 229 00:20:16,090 --> 00:20:19,058 And there's our divide by the eggs. 230 00:20:20,290 --> 00:20:22,649 Now let's just look back at this 231 00:20:22,649 --> 00:20:28,390 one. Look at all this complicated differentiation that 232 00:20:28,390 --> 00:20:30,490 we had here. 233 00:20:31,120 --> 00:20:35,476 Now I went through it slowly and carefully, but when we're doing 234 00:20:35,476 --> 00:20:39,469 calculations on our own, we might make slips. It would be 235 00:20:39,469 --> 00:20:43,462 helpful if we could automate some of this process so it 236 00:20:43,462 --> 00:20:47,092 instead of writing down the derivative of sine wave with 237 00:20:47,092 --> 00:20:50,722 respect to X is and going through the chain rule. 238 00:20:51,430 --> 00:20:56,302 We went automatically to all we need to do is differentiate sign 239 00:20:56,302 --> 00:21:01,174 wired with respect to Y. That's cause Y an multiplied by divided 240 00:21:01,174 --> 00:21:06,452 by DX. So we automate we would miss out at least these two 241 00:21:06,452 --> 00:21:10,512 lines and go direct from there to there. Similarly, the 242 00:21:10,512 --> 00:21:15,790 derivative of two Y with respect to X would be the derivative of 243 00:21:15,790 --> 00:21:21,474 two Y with respect to Y two times divided by DX. And so we 244 00:21:21,474 --> 00:21:23,098 go direct from there. 245 00:21:23,120 --> 00:21:28,688 To there, so let's have a look at that in another example. 246 00:21:29,460 --> 00:21:37,122 So we'll take Y squared plus 247 00:21:37,122 --> 00:21:44,784 X cubed minus XY plus cause 248 00:21:44,784 --> 00:21:50,780 Y. Equals note this time we're going to go. 249 00:21:51,310 --> 00:21:55,534 Direct to the differentiation, we're not going to go by the 250 00:21:55,534 --> 00:22:00,142 chain rule. We're going to use it, of course, but we aren't 251 00:22:00,142 --> 00:22:04,750 going to write it down, so we want to differentiate this with 252 00:22:04,750 --> 00:22:10,510 respect to X. So the first term is Y squared, so we know that to 253 00:22:10,510 --> 00:22:13,966 differentiate Y squared with respect to X, we differentiate 254 00:22:13,966 --> 00:22:16,270 this with respect to why that's 255 00:22:16,270 --> 00:22:22,105 too why. And we multiply by DY by DX. 256 00:22:22,690 --> 00:22:28,722 Now we want the derivative of X cubed with respect to X, so 257 00:22:28,722 --> 00:22:31,140 that's 3X. Squared 258 00:22:31,920 --> 00:22:33,950 Minus. 259 00:22:35,060 --> 00:22:39,948 Now I am going to write this down in full because it's the 260 00:22:39,948 --> 00:22:44,836 derivative of XY with respect to X. It's a product again, it's X 261 00:22:44,836 --> 00:22:45,964 times by Y. 262 00:22:46,960 --> 00:22:52,680 Plus, the derivative of Cos Y with respect to X, which we do 263 00:22:52,680 --> 00:22:58,400 as a derivative of cause Y with respect to Y that's minus Sign 264 00:22:58,400 --> 00:23:05,760 Y. Times DY by DX, the derivative of 0 is 265 00:23:05,760 --> 00:23:11,560 just zero. So let's get these two terms together because 266 00:23:11,560 --> 00:23:18,085 they both got the why by DX. So I have two Y minus Sign Y. 267 00:23:18,180 --> 00:23:24,668 Times Ty by DX plus three X squared. 268 00:23:25,400 --> 00:23:32,860 Minus. Now this is a product. It is a U times 269 00:23:32,860 --> 00:23:36,764 by AV. So we 270 00:23:36,764 --> 00:23:43,880 want you. Times the derivative of Y with respect to X, which 271 00:23:43,880 --> 00:23:47,527 is just the wise by The X. 272 00:23:48,050 --> 00:23:55,670 Plus V, which is Y Times the derivative of X, which is 273 00:23:55,670 --> 00:23:59,720 just one. Equals 0. 274 00:24:00,940 --> 00:24:06,040 So we see here that we've got another term now involving the 275 00:24:06,040 --> 00:24:12,415 why by DX it's minus X, so we can put that in the bracket so 276 00:24:12,415 --> 00:24:15,815 we can have two Y minus sign Y 277 00:24:15,815 --> 00:24:22,996 minus X. DY by the X plus three X squared 278 00:24:22,996 --> 00:24:30,216 and then minus this term here minus Y equals 0. 279 00:24:31,690 --> 00:24:37,670 If we take this over to the other side, In other words, we 280 00:24:37,670 --> 00:24:43,650 take this away from both sides and add that to both sides. You 281 00:24:43,650 --> 00:24:50,550 have two Y minus sign Y minus X times DY by X is equal to. 282 00:24:50,550 --> 00:24:56,070 Adding this one to both sides. Why taking this one away from 283 00:24:56,070 --> 00:24:58,370 both sides, we get that. 284 00:24:58,960 --> 00:25:03,952 Now it's clear how we would finish this off. We will take 285 00:25:03,952 --> 00:25:07,696 this factor here and divide both sides by it. 286 00:25:08,400 --> 00:25:15,540 So we get DY by the X was equal 2. 287 00:25:16,280 --> 00:25:23,624 Just go back. We've got Y minus three X squared to go 288 00:25:23,624 --> 00:25:27,296 in the numerator on the top. 289 00:25:28,420 --> 00:25:34,660 And this factor to go on the bottom in the denominator two Y 290 00:25:34,660 --> 00:25:36,100 minus Sign Y. 291 00:25:36,610 --> 00:25:40,600 Minus. 292 00:25:40,600 --> 00:25:44,500 X. And then 293 00:25:44,500 --> 00:25:51,552 we have. Our derivative, Let's just take one 294 00:25:51,552 --> 00:25:55,995 more example. Y 295 00:25:55,995 --> 00:26:01,405 cubed minus 296 00:26:01,405 --> 00:26:04,860 X. Sign 297 00:26:04,860 --> 00:26:08,786 Y. Plus Y squared 298 00:26:08,786 --> 00:26:13,170 over X. Equals 8. 299 00:26:14,520 --> 00:26:19,976 Again, all the axes and Wise bundled up together if possible. 300 00:26:19,976 --> 00:26:27,416 We want to be able to do this directly. We want to be able to 301 00:26:27,416 --> 00:26:31,880 differentiate it straight away without going through the chain 302 00:26:31,880 --> 00:26:38,328 rule. So the derivative of Y cubed with respect to Y times by 303 00:26:38,328 --> 00:26:44,776 DY by DX. So that's three Y squared times DY by X minus. 304 00:26:44,840 --> 00:26:51,171 We want the derivative of this. This is a product so let's see 305 00:26:51,171 --> 00:26:58,476 if we can do it all in one go. Again you want X Times the 306 00:26:58,476 --> 00:27:03,833 derivative of sine Y with respect to X. That's X times 307 00:27:03,833 --> 00:27:05,781 cause YDY by X. 308 00:27:06,310 --> 00:27:12,790 And now we want sign Y times the derivative of X so that sign Y 309 00:27:12,790 --> 00:27:15,814 and the derivative of X is just 310 00:27:15,814 --> 00:27:20,832 one. This one is a bit trickier. This is a quotient Y squared 311 00:27:20,832 --> 00:27:25,785 over X, so we want plus. Now let's remember what we do with 312 00:27:25,785 --> 00:27:32,920 the quotient. It's V which is on the bottom that's X Times the 313 00:27:32,920 --> 00:27:39,280 derivative of what's on the top. The derivative of Y squared with 314 00:27:39,280 --> 00:27:45,110 respect to X, which is the derivative of Y squared with 315 00:27:45,110 --> 00:27:52,000 respect to Y2Y times DY by the X minus Y squared times the 316 00:27:52,000 --> 00:27:57,300 derivative of V, which in this case is just X. 317 00:27:57,300 --> 00:28:03,470 All over V squared, which is X squared equals 0. 318 00:28:04,430 --> 00:28:06,418 Now this needs a little bit of 319 00:28:06,418 --> 00:28:10,530 tidying up. We've got a denominator here that we can 320 00:28:10,530 --> 00:28:13,410 probably multiply out by, so let's do that. 321 00:28:13,990 --> 00:28:21,116 And do some tidying on the way, so this will be three X squared 322 00:28:21,116 --> 00:28:24,170 Y squared DY by The X. 323 00:28:25,070 --> 00:28:32,966 Minus X cubed cause YDY by the X 324 00:28:32,966 --> 00:28:36,914 minus X squared Sign 325 00:28:36,914 --> 00:28:44,182 Y. +2 XYDY by X minus Y squared equals 0, 326 00:28:44,182 --> 00:28:50,760 so I've multiplied throughout by this X squared so the X 327 00:28:50,760 --> 00:28:55,544 squared is appeared there, multiplying that it's appeared 328 00:28:55,544 --> 00:29:00,926 there inside that X cubed, multiplying that it's appeared 329 00:29:00,926 --> 00:29:07,504 there, multiplying the sign Y, and it's gone. From here, 'cause 330 00:29:07,504 --> 00:29:09,896 we've multiplied by so. 331 00:29:09,930 --> 00:29:15,110 Multiplying and dividing by, in effect, leaving this on changed. 332 00:29:15,110 --> 00:29:21,844 Now let's get together all the terms in DY by DX, so we 333 00:29:21,844 --> 00:29:28,578 have the why by DX times this term, three X squared Y squared. 334 00:29:29,100 --> 00:29:33,000 This term minus X cubed 335 00:29:33,000 --> 00:29:39,958 cause why? This term +2 XY. 336 00:29:41,630 --> 00:29:46,306 And here I've got minus X squared sign Y or I think I want 337 00:29:46,306 --> 00:29:50,982 to add that to the other side. So that's plus X squared sign Y, 338 00:29:50,982 --> 00:29:54,990 and here I've got minus Y squared. Again, I think I want 339 00:29:54,990 --> 00:29:59,666 to add that to both sides, so I get plus Y squared over there. 340 00:30:00,340 --> 00:30:08,060 Now why by X is equal to X squared sign 341 00:30:08,060 --> 00:30:15,780 Y plus Y squared in the numerator and dividing by 342 00:30:15,780 --> 00:30:22,728 this expression as the denominator. Three X squared Y 343 00:30:22,728 --> 00:30:28,904 squared minus X cubed cause Y, +2 XY. 344 00:30:28,940 --> 00:30:31,706 Notice how much shorter automating that 345 00:30:31,706 --> 00:30:35,394 Differentiation's made? What's quite a complicated problem, and 346 00:30:35,394 --> 00:30:40,465 that's something you want to work at. Trying to automate your 347 00:30:40,465 --> 00:30:45,536 differentiation so you don't have to go through the rules and 348 00:30:45,536 --> 00:30:47,841 write them down every time.