1 00:00:01,030 --> 00:00:06,233 We're now going to extend our table of derivatives by looking 2 00:00:06,233 --> 00:00:12,382 at functions such as tanks sex, arcsine X, or sign minus one X&A 3 00:00:12,382 --> 00:00:16,166 to the power X, where a is a 4 00:00:16,166 --> 00:00:23,406 positive constant. Let's start with the table with our 5 00:00:23,406 --> 00:00:25,230 function. F of X. 6 00:00:25,990 --> 00:00:27,919 And our derivative. 7 00:00:30,260 --> 00:00:37,124 Either referred to as DF, DX, or F dashed of X. 8 00:00:38,080 --> 00:00:44,136 And let's start with 9 00:00:44,136 --> 00:00:45,650 tonics. 10 00:00:46,970 --> 00:00:49,340 Sex. 11 00:00:50,380 --> 00:00:56,618 Call Tex. And Cosec X. 12 00:00:56,690 --> 00:01:01,450 And the first thing they want to want to do is to actually write 13 00:01:01,450 --> 00:01:03,490 these in terms of sine and 14 00:01:03,490 --> 00:01:10,506 cosine. So tonics we can write a sign X divided by Cos X. 15 00:01:11,720 --> 00:01:15,605 Sex is 1 divided by Cos X. 16 00:01:16,200 --> 00:01:24,084 Call Tex is Arcos X divided by sine X and cosec, X 17 00:01:24,084 --> 00:01:28,026 is 1 divided by sine X. 18 00:01:28,850 --> 00:01:33,470 So let's have a look now at finding the derivatives of some 19 00:01:33,470 --> 00:01:38,750 of these. Let's start with tonics 20 00:01:38,750 --> 00:01:42,550 Y equals TAN X. 21 00:01:45,850 --> 00:01:51,922 Now we've already written tonics in terms of sine and cosine, so 22 00:01:51,922 --> 00:01:56,982 let's write it a sign X divided by cosine X. 23 00:01:57,510 --> 00:02:04,790 And this is as if we had U&V where you is a function of 24 00:02:04,790 --> 00:02:07,910 X&V is a function of X. 25 00:02:08,760 --> 00:02:10,650 So we have a quotient. 26 00:02:11,200 --> 00:02:16,260 So we're going to use our rule for differentiating quotients. 27 00:02:16,260 --> 00:02:24,084 Do Y by ZX equals V times? Do you buy the X 28 00:02:24,084 --> 00:02:27,996 minus use times DV Pi DX? 29 00:02:28,560 --> 00:02:31,220 All divided by V squared. 30 00:02:32,280 --> 00:02:36,000 So in this case, IQ equals 31 00:02:36,000 --> 00:02:39,910 sign X. So do you buy the 32 00:02:39,910 --> 00:02:43,400 X? It was cool sex. 33 00:02:44,230 --> 00:02:48,076 RV. It's called 34 00:02:48,076 --> 00:02:52,070 sex. So 2 feet by TX. 35 00:02:52,740 --> 00:02:55,180 Equals minus sign X. 36 00:02:56,250 --> 00:02:58,128 Let's substitute now. 37 00:02:58,950 --> 00:03:02,646 Into our derivative do I buy DX? 38 00:03:03,280 --> 00:03:07,222 Is it cool to V which is Cos X? 39 00:03:08,240 --> 00:03:13,270 Multiplied by do you buy DX, which is cause X? 40 00:03:14,650 --> 00:03:15,560 Minus. 41 00:03:16,820 --> 00:03:19,328 You which is synex. 42 00:03:19,870 --> 00:03:27,280 Multiplied by TV by DX, which is minus sign X. 43 00:03:27,280 --> 00:03:30,658 All divided by. 44 00:03:31,360 --> 00:03:37,446 V. Squared. Via skull sex so squared is called squared X. 45 00:03:38,420 --> 00:03:41,318 So we have cost squared X. 46 00:03:42,070 --> 00:03:49,153 Minus times minus gives us positive sign squared X. 47 00:03:49,940 --> 00:03:53,310 Divided by Cos squared X. 48 00:03:54,250 --> 00:03:57,970 Now one of our trigonometric identity's is called squared X 49 00:03:57,970 --> 00:04:00,202 plus sign. Squared X equals 1. 50 00:04:00,710 --> 00:04:07,710 So we can substitute 1 divided by Cos squared X. 51 00:04:08,720 --> 00:04:12,278 Which is the same as sex 52 00:04:12,278 --> 00:04:14,770 squared X. So our derivative of 53 00:04:14,770 --> 00:04:17,670 Tan X. Is sex squared X? 54 00:04:19,230 --> 00:04:26,010 Let's have a look at another example. This time Y 55 00:04:26,010 --> 00:04:32,830 equals sex. And again will write sex in terms of sine and 56 00:04:32,830 --> 00:04:38,386 cosine, which is one over call sex. And again we have our 57 00:04:38,386 --> 00:04:40,238 quotient you over V. 58 00:04:40,940 --> 00:04:47,968 So I see Y by ZX equals V times. Do you buy the X? 59 00:04:48,590 --> 00:04:52,538 Minus U times TV by ZX? 60 00:04:53,160 --> 00:04:55,540 Or divided by 3 squared. 61 00:04:56,600 --> 00:05:03,912 So in this case I you equals 1. So I do you buy DX is 0. 62 00:05:04,810 --> 00:05:08,110 RV equals 63 00:05:08,110 --> 00:05:14,990 Cossacks. So I do feet by DX equals minus sign X. 64 00:05:16,020 --> 00:05:23,720 So our derivative do why by DX is equal to V, so it's cause 65 00:05:23,720 --> 00:05:29,607 X. Multiplied by do you buy DX which is 0? 66 00:05:30,780 --> 00:05:34,040 Minus you choose one. 67 00:05:34,740 --> 00:05:39,790 Times DV by The X which is minus sign X. 68 00:05:40,440 --> 00:05:45,750 All divided by V squared, which is called squared X. 69 00:05:46,880 --> 00:05:49,648 So we have 0. 70 00:05:50,960 --> 00:05:55,649 Minus minus that becomes positive. One lot of Cynex 71 00:05:55,649 --> 00:06:00,338 so have sign X divided by Cos squared X. 72 00:06:01,500 --> 00:06:03,820 And we can write this. 73 00:06:05,850 --> 00:06:13,389 As. One over Cos X multiplied by sign X over 74 00:06:13,389 --> 00:06:17,055 call sex. Which 75 00:06:17,055 --> 00:06:23,760 is. Sex tonics, so a derivative of 76 00:06:23,760 --> 00:06:27,020 sex is sex tonics. 77 00:06:28,250 --> 00:06:30,938 Let's put those in our table. 78 00:06:31,870 --> 00:06:34,258 So our derivative of Tan X. 79 00:06:35,010 --> 00:06:36,870 Sex squared X. 80 00:06:37,950 --> 00:06:40,030 A derivative of sex. 81 00:06:40,600 --> 00:06:44,320 Is sex tonics? 82 00:06:45,080 --> 00:06:50,998 And in a similar way are derivative of Cortex will be 83 00:06:50,998 --> 00:06:53,150 minus cosec squared X. 84 00:06:53,680 --> 00:06:59,746 And the derivative of cosec X is minus Cosec 85 00:06:59,746 --> 00:07:01,094 X Cotex. 86 00:07:03,290 --> 00:07:09,194 Now we going to extend this further to look at what happens 87 00:07:09,194 --> 00:07:12,146 when we have time of MX. 88 00:07:12,930 --> 00:07:15,420 And the sack of MX. 89 00:07:16,240 --> 00:07:21,300 The cast of air Max and the cosec of MX. 90 00:07:22,010 --> 00:07:25,736 Let's have a look at the 91 00:07:25,736 --> 00:07:32,752 calculation. So this time we have Y equals the 92 00:07:32,752 --> 00:07:35,149 time of MX. 93 00:07:36,190 --> 00:07:39,658 And what we're going to do here is to use a substitution. 94 00:07:40,270 --> 00:07:47,134 So you were going to substitute instead of MX, so RY is 95 00:07:47,134 --> 00:07:49,422 equal to tan you. 96 00:07:50,530 --> 00:07:53,729 So if we calculate, do you buy 97 00:07:53,729 --> 00:07:57,479 DX? We get M. 98 00:07:57,990 --> 00:08:03,645 And if we find the derivative of Y with respect to you while 99 00:08:03,645 --> 00:08:08,865 we've just learned how to do that, the derivative of tan you 100 00:08:08,865 --> 00:08:10,605 is sex squared you. 101 00:08:10,770 --> 00:08:13,686 So I do why by DX. 102 00:08:14,220 --> 00:08:21,732 Is equal to do Y fi CU multiplied by to you by 103 00:08:21,732 --> 00:08:26,704 DMX. The Wi-Fi do you? Is sex squared you? 104 00:08:28,040 --> 00:08:33,188 Multiplied by do you buy DX, which is RM. Let's put the 105 00:08:33,188 --> 00:08:37,478 constant first M times this X squared. You now we've 106 00:08:37,478 --> 00:08:43,055 introduced EU and we don't want the you were trying to find the 107 00:08:43,055 --> 00:08:49,490 wide by DX we wanted in terms of X. So what we do is we 108 00:08:49,490 --> 00:08:55,067 substitute for the you so we have M times 6 squared of MX. 109 00:08:56,640 --> 00:09:00,670 So what we get when we differentiate ton of MX? 110 00:09:01,320 --> 00:09:04,362 Is the sex squared MX multiplied 111 00:09:04,362 --> 00:09:10,159 by M? Let's have a look at another example. 112 00:09:10,160 --> 00:09:18,128 This time, let's have a look at Y equals cosec of MX. 113 00:09:19,300 --> 00:09:25,053 Again, we're going to let you equal MX as our substitution. 114 00:09:25,650 --> 00:09:28,332 And therefore Y is equal to 115 00:09:28,332 --> 00:09:30,310 Kosek. Of you. 116 00:09:31,110 --> 00:09:35,100 Do you buy the X is equal 117 00:09:35,100 --> 00:09:39,078 to end? And our DY bye see you. 118 00:09:39,870 --> 00:09:42,490 The derivative of cosec you. 119 00:09:43,690 --> 00:09:47,450 Is minus cosec you 120 00:09:47,450 --> 00:09:54,308 cut you? We know that divide by DX. 121 00:09:54,940 --> 00:10:00,808 It cause divide by do you multiplied by do you buy DX? 122 00:10:00,850 --> 00:10:04,098 RDY by do you? 123 00:10:04,710 --> 00:10:08,590 Is minus cosec you caught 124 00:10:08,590 --> 00:10:13,708 you? Multiplied by do you buy DX which is M? 125 00:10:14,250 --> 00:10:17,020 So we have minus N. 126 00:10:17,650 --> 00:10:20,770 Cosec you caught you. 127 00:10:21,580 --> 00:10:23,578 Now we need to substitute for 128 00:10:23,578 --> 00:10:30,714 these use. So we have minus M, Cosec 129 00:10:30,714 --> 00:10:33,636 MX cult MX. 130 00:10:34,610 --> 00:10:40,550 So the derivative of cosec MX is minus M. 131 00:10:40,550 --> 00:10:43,190 Cosec MX caught MX. 132 00:10:44,460 --> 00:10:48,563 Now let's go back to our table and put those in. 133 00:10:49,150 --> 00:10:56,570 So we found that the derivative of tan of MX. 134 00:10:57,700 --> 00:11:02,240 Is M6 squared MX? 135 00:11:03,680 --> 00:11:06,490 The derivative of SAC MX. 136 00:11:07,430 --> 00:11:14,690 Is M times the sack of MX times a ton of MX? 137 00:11:15,710 --> 00:11:22,046 And in a similar way we would find that the koptev 138 00:11:22,046 --> 00:11:27,230 MX is minus M Times the cosec squared MX. 139 00:11:28,310 --> 00:11:34,846 And cosec MX. The derivative is minus M 140 00:11:34,846 --> 00:11:38,931 times cosec MX cult MX. 141 00:11:40,780 --> 00:11:44,146 OK, we're going to extend our 142 00:11:44,146 --> 00:11:48,730 table further. Find a clean page. 143 00:11:50,810 --> 00:11:57,280 Let's just rewrite our headings are function F of X. 144 00:11:57,800 --> 00:12:00,820 That derivative 145 00:12:01,580 --> 00:12:08,050 DFDX Or F Dash Devex. 146 00:12:09,160 --> 00:12:15,145 And we're going to look now at. 147 00:12:15,850 --> 00:12:19,140 The inverse. Sign 148 00:12:19,140 --> 00:12:22,990 minus 1X. It's 149 00:12:22,990 --> 00:12:26,520 minus 1X. And tonics. 150 00:12:27,190 --> 00:12:30,110 Minus 1 - 1 X. 151 00:12:31,550 --> 00:12:36,973 And then finally will look at A to the power X. 152 00:12:38,190 --> 00:12:43,734 Let's do the calculation now for sign minus 1X. 153 00:12:43,740 --> 00:12:51,076 So this time Y equals sine minus 1X. 154 00:12:53,080 --> 00:13:00,080 What we're going to do here is right this as sign Y equals X. 155 00:13:00,760 --> 00:13:06,520 Now we want to find the derivative of Y with respect to 156 00:13:06,520 --> 00:13:09,400 X. So we want to differentiate. 157 00:13:09,410 --> 00:13:11,828 Side Y with respect to X. 158 00:13:12,470 --> 00:13:19,190 And differentiate RX with respect to X. 159 00:13:19,470 --> 00:13:23,630 His right hand side is very easy because the derivative of X is 160 00:13:23,630 --> 00:13:25,550 just one with respect to X. 161 00:13:26,090 --> 00:13:29,716 This side we're going to differentiate implicitly. 162 00:13:30,740 --> 00:13:38,105 And when we do that, we get the derivative of sine. Y is cause Y 163 00:13:38,105 --> 00:13:40,560 multiplied by divided by DX. 164 00:13:41,790 --> 00:13:49,230 So if we now divide both sides by cause why we get DY by DX 165 00:13:49,230 --> 00:13:52,206 equals 1 divided by cause Y? 166 00:13:53,170 --> 00:13:57,574 Now that's fine, except for the fact that we actually wanted in 167 00:13:57,574 --> 00:13:58,675 terms of X. 168 00:13:59,810 --> 00:14:02,626 So we need to use one of our 169 00:14:02,626 --> 00:14:08,002 trigonometric identity's. I'm looking to use call squared Y 170 00:14:08,002 --> 00:14:11,218 plus sign squared Y equals 1. 171 00:14:12,040 --> 00:14:17,215 And if we rearrange this, we get cause Y. 172 00:14:17,740 --> 00:14:22,966 Equals 1 minus sign squared Y and then we want 'cause why not 173 00:14:22,966 --> 00:14:26,986 cause squared Y? So we're going to square root it. 174 00:14:28,100 --> 00:14:31,928 Now we introduce slight problem here because when we square root 175 00:14:31,928 --> 00:14:35,408 something we could have a positive answer. Or we could 176 00:14:35,408 --> 00:14:40,280 have a negative answer. So we need to just look at that for a 177 00:14:40,280 --> 00:14:42,716 moment. Now if we look at the 178 00:14:42,716 --> 00:14:48,778 diagram. Of Y equals sine minus One X, so here's a graph of Y 179 00:14:48,778 --> 00:14:54,322 equals sine minus One X will see that there are many values of Y. 180 00:14:54,850 --> 00:14:59,650 So our values of X. So for one value of X we could have lots of 181 00:14:59,650 --> 00:15:00,850 different values of Y. 182 00:15:01,620 --> 00:15:06,625 So that's not a function, and for it to be a function, we 183 00:15:06,625 --> 00:15:12,400 just want one set of values of Y for each set of values of X. 184 00:15:12,400 --> 00:15:16,635 So what we need to do is actually restrict the section 185 00:15:16,635 --> 00:15:21,255 that we're looking at, and if we restrict it to between minus 186 00:15:21,255 --> 00:15:25,875 π by two and plus five by two, just this section here. 187 00:15:27,210 --> 00:15:31,542 Then we do have a function because we have just one value 188 00:15:31,542 --> 00:15:34,069 of Y for each value of X. 189 00:15:35,350 --> 00:15:39,718 And if we look at this section, we can see that the gradient. 190 00:15:40,680 --> 00:15:45,234 Is increasing and it's positive, so we're actually going to take 191 00:15:45,234 --> 00:15:49,374 the positive square root when we take the square root. 192 00:15:50,440 --> 00:15:57,070 So if we continue down here one over cause Y, we can actually 193 00:15:57,070 --> 00:16:03,700 write as one over the square root of 1 minus sign squared Y. 194 00:16:04,390 --> 00:16:06,861 Now we still wanted in terms of 195 00:16:06,861 --> 00:16:12,868 X. But if we look here, we know that sign Y is equal to 196 00:16:12,868 --> 00:16:17,500 X, so we can substitute in for sine squared Y here and 197 00:16:17,500 --> 00:16:22,518 put X squared. So we get one divided by the square root of 198 00:16:22,518 --> 00:16:24,062 1 minus X squared. 199 00:16:26,350 --> 00:16:32,095 White was cause minus One X can be done in a similar way and is 200 00:16:32,095 --> 00:16:37,074 left as an exercise. But note there that when you look at the 201 00:16:37,074 --> 00:16:41,287 square root should have to take a negative square root rather 202 00:16:41,287 --> 00:16:42,819 than a positive one. 203 00:16:43,490 --> 00:16:49,358 Let's have a look now at Y equals 10 - 1 X. 204 00:16:49,360 --> 00:16:55,444 In the same way, we're going 205 00:16:55,444 --> 00:17:01,528 to write Tan, Y is equal 206 00:17:01,528 --> 00:17:08,394 to X. And again, take the derivative of 207 00:17:08,394 --> 00:17:11,939 each side with respect to 208 00:17:11,939 --> 00:17:18,763 X. I can right hand side is easy, it's just 209 00:17:18,763 --> 00:17:24,550 one. And we differentiate this side implicitly, which gives us. 210 00:17:25,330 --> 00:17:32,727 The derivative of tan Y is sex squared Y multiplied by DY by 211 00:17:32,727 --> 00:17:39,306 ZX. Sophie now divide both sides by sex squared 212 00:17:39,306 --> 00:17:44,362 Y we get 1 / 6 squared Y. 213 00:17:45,510 --> 00:17:50,346 Now, in a similar way to before we need to use a trigonometric 214 00:17:50,346 --> 00:17:53,154 identity. And we need to use the 215 00:17:53,154 --> 00:17:58,675 one. What sex squared Y equals 1 plus? 216 00:17:59,230 --> 00:18:00,619 10 squared, Y. 217 00:18:02,210 --> 00:18:05,220 So if we substitute that. 218 00:18:05,730 --> 00:18:12,132 In here we have 1 / 1 + 10 squared Y. 219 00:18:12,870 --> 00:18:20,100 And since 10 Y is X, then that's just one over 1 plus X squared. 220 00:18:20,880 --> 00:18:27,306 So the derivative of tan minus One X is 1 / 1 plus X 221 00:18:27,306 --> 00:18:32,430 squared. Let's go back and put those in our table now. 222 00:18:32,430 --> 00:18:39,684 So the derivative of sine minus One X is one over the square 223 00:18:39,684 --> 00:18:43,032 root of 1 minus X squared. 224 00:18:44,020 --> 00:18:50,351 A derivative of Cos minus One X is minus one over the square 225 00:18:50,351 --> 00:18:56,682 root of 1 minus X squared and our derivative of tan minus One 226 00:18:56,682 --> 00:19:00,578 X is one over 1 plus X squared. 227 00:19:01,300 --> 00:19:06,513 Let's have a look now at Y equals 8 to the power X. 228 00:19:07,160 --> 00:19:12,718 Y equals A to the power X and this time we're going to start 229 00:19:12,718 --> 00:19:17,879 by taking natural logarithms of each side. So log to the base E 230 00:19:17,879 --> 00:19:24,231 of Y is equal to log to the base E of A to the power X. 231 00:19:24,830 --> 00:19:28,210 Now we're going to use one of the laws of logarithms to bring 232 00:19:28,210 --> 00:19:33,736 down the power. So on the right hand side we have X times log to 233 00:19:33,736 --> 00:19:35,401 the base E of A. 234 00:19:36,430 --> 00:19:41,662 Now we want to find the derivative of this, so we want 235 00:19:41,662 --> 00:19:48,202 to differentiate log to the base E of Y with respect to X and we 236 00:19:48,202 --> 00:19:53,870 want to differentiate X times log to the base E of A with 237 00:19:53,870 --> 00:19:55,178 respect to X. 238 00:19:55,840 --> 00:20:01,376 Now if we look at the right hand side, first log to the base E of 239 00:20:01,376 --> 00:20:04,836 a is a constant, so it's a constant times X. 240 00:20:05,450 --> 00:20:09,746 So when we differentiate with respect to X, we just get the 241 00:20:09,746 --> 00:20:12,610 constant log to the base E of A. 242 00:20:13,260 --> 00:20:18,832 If we differentiate log to the base E of Y with respect to X. 243 00:20:19,400 --> 00:20:25,080 We get one over Y multiplied by DY by DX. 244 00:20:27,280 --> 00:20:30,928 So do why by DX is going to 245 00:20:30,928 --> 00:20:37,226 be equal. To now, to get this side is being divided by Y, so 246 00:20:37,226 --> 00:20:42,465 want to multiply both sides by Y. So we've got Y multiplied by 247 00:20:42,465 --> 00:20:45,286 log to the base E of A. 248 00:20:46,550 --> 00:20:49,230 But we wanted in terms of X, not 249 00:20:49,230 --> 00:20:54,719 Y. But we know that Y equals A to the power X, 250 00:20:54,719 --> 00:20:59,627 so I do why by DX becomes A to the power X 251 00:20:59,627 --> 00:21:03,308 multiplied by log to the base E of A. 252 00:21:04,810 --> 00:21:09,970 Now it's interesting to note here that actually if the 253 00:21:09,970 --> 00:21:12,550 constant a is equal to. 254 00:21:13,080 --> 00:21:18,800 2.7 one and so on. That's the value of E. 255 00:21:19,640 --> 00:21:26,808 Then what we have is Y equals E to the power X and out 256 00:21:26,808 --> 00:21:30,392 the why by DX is equal to. 257 00:21:31,460 --> 00:21:37,130 A to the power X? Well, that's E. To the power X multiplied by 258 00:21:37,130 --> 00:21:44,015 log to the base E of a, which in this case is E and log to the 259 00:21:44,015 --> 00:21:47,660 base E of E is one, so we get. 260 00:21:48,210 --> 00:21:52,591 Our derivative is E to the power X, which is what we expected. 261 00:21:53,880 --> 00:21:59,232 Let's go and complete our table now, then with the derivative of 262 00:21:59,232 --> 00:22:05,922 A to the X as A to the power X multiplied by log to the 263 00:22:05,922 --> 00:22:07,706 base E of A.