1 00:00:00,520 --> 00:00:05,135 In this tutorial, we're going to look at differentiating x to the 2 00:00:05,135 --> 00:00:07,380 power n from first principles. 3 00:00:07,380 --> 00:00:12,242 Now n could be a positive integer, n could be a fraction. 4 00:00:12,242 --> 00:00:15,539 It could be negative, or it could be 0. 5 00:00:15,539 --> 00:00:20,242 So we're going to start by taking the case where n is a positive integer. 6 00:00:20,242 --> 00:00:25,130 So we'll be looking at things like x squared, x to the power 7. 7 00:00:25,130 --> 00:00:27,891 Even x to the power 1. 8 00:00:27,891 --> 00:00:33,250 So we have y equals x to the power n. 9 00:00:33,870 --> 00:00:40,482 Our definition of our derivative function is dy by dx equals 10 00:00:40,482 --> 00:00:47,515 the limit as delta x approaches 0, of f of x + delta x, 11 00:00:47,515 --> 00:00:53,538 minus our function of x, divided by delta x. 12 00:00:54,231 --> 00:01:01,450 So let's just look at this part first of all. Our f of x + delta x 13 00:01:01,450 --> 00:01:08,058 is going to equal x + delta x, all to the power n. 14 00:01:08,860 --> 00:01:10,400 And this is a binomial. 15 00:01:10,940 --> 00:01:15,935 So what we're going to start by doing is actually just expanding 16 00:01:16,851 --> 00:01:20,307 a + b to the power n. 17 00:01:22,040 --> 00:01:24,560 And that is a to the power n, 18 00:01:25,070 --> 00:01:32,154 plus n, times a to the power of n - 1, multiplied by b, 19 00:01:32,154 --> 00:01:39,307 plus and there are lots of other terms in between both containing powers of a and b 20 00:01:39,307 --> 00:01:43,882 along to our final term, which is b to the power n. 21 00:01:44,550 --> 00:01:47,404 Well, now let's look at what we've got. 22 00:01:47,404 --> 00:01:52,317 Instead of our a, we've got x, and instead of the b, we've got delta x. 23 00:01:52,317 --> 00:01:57,371 So, we've got x + delta x to the power n. 24 00:01:57,371 --> 00:02:02,426 So our a, we've got x to the power n, 25 00:02:02,426 --> 00:02:09,838 plus n x to the power n - 1 and our b is delta x plus, and 26 00:02:09,838 --> 00:02:15,506 again, all these terms which will be in terms of x and delta x 27 00:02:15,506 --> 00:02:20,818 Up to our last term, which is delta x to the power n. 28 00:02:21,650 --> 00:02:27,276 Now let's substitute this in into our derivative here. 29 00:02:27,475 --> 00:02:34,554 So our dy by dx. is the limit as delta x approaches 0, 30 00:02:35,510 --> 00:02:38,358 of our function of x + delta x. 31 00:02:39,370 --> 00:02:45,850 Which is x to the power n, plus, n x to the n - 1, delta x, plus, 32 00:02:45,850 --> 00:02:52,330 and again all these terms in both x and delta x, plus delta x to the power n. 33 00:02:52,330 --> 00:02:58,904 Minus our function of x, which is x to the power n. 34 00:02:59,590 --> 00:03:03,060 All divided by delta x. 35 00:03:03,790 --> 00:03:08,821 Oh, here we have x to the power n takeaway x to the power n. 36 00:03:09,450 --> 00:03:15,445 So our derivative is the limit as delta x approaches 0. 37 00:03:16,530 --> 00:03:23,642 Of n x to the n - 1, delta x plus all the terms in x and delta x. 38 00:03:23,642 --> 00:03:31,533 Plus delta x to the power n. All divided by delta x. 39 00:03:34,068 --> 00:03:37,900 Now all these terms, have delta x in them. 40 00:03:38,800 --> 00:03:45,290 And we're dividing by delta x, so we can actually cancel the delta x. 41 00:03:45,290 --> 00:03:49,600 So that we have the limit as delta x approaches 0. 42 00:03:50,620 --> 00:03:57,576 Of n x to the power of n -1, plus, now all these terms here 43 00:03:57,576 --> 00:04:03,808 have delta x squared, delta x cubed and higher powers all the way up 44 00:04:03,808 --> 00:04:08,582 to delta x to the power n. So when we've divided by delta x, 45 00:04:08,582 --> 00:04:13,832 all of these have still got a delta x in them up to our 46 00:04:13,832 --> 00:04:18,165 final term of delta x to the power of n -1. 47 00:04:18,470 --> 00:04:23,462 So when we actually take the limit of delta x approaches 0. 48 00:04:24,150 --> 00:04:29,070 All of these terms are going to approach 0. 49 00:04:29,070 --> 00:04:32,879 They've all got delta x in them so they'll all be 0. 50 00:04:32,879 --> 00:04:40,487 So our derivative is just n times x to the power of n - 1. 51 00:04:41,628 --> 00:04:44,449 So at derivative of x to the power n, 52 00:04:44,760 --> 00:04:50,194 is n times x to the power of n - 1. 53 00:04:50,200 --> 00:04:53,590 Let's have a look at some examples. 54 00:04:53,590 --> 00:04:56,450 Let's say y equals x squared. 55 00:04:56,450 --> 00:05:02,424 So our dy by dx equals 56 00:05:02,424 --> 00:05:05,774 Well, the power comes down in front, so it's 57 00:05:05,774 --> 00:05:11,337 2 times x to the power of 2 take away 1 which gives us 58 00:05:11,337 --> 00:05:16,445 2 times x to the power 1, which is just 2x. 59 00:05:18,636 --> 00:05:20,888 y equals x to the power 7. 60 00:05:23,278 --> 00:05:27,841 dy by dx equals... the power comes down 61 00:05:27,841 --> 00:05:36,410 7 times x to the power of 7 - 1, so we get 7x to the power of 6. 62 00:05:39,460 --> 00:05:41,550 Now. 63 00:05:42,170 --> 00:05:46,427 What happens when we have y equals just x? 64 00:05:46,427 --> 00:05:48,690 x to the power of 1? 65 00:05:48,690 --> 00:05:52,220 Well, we already know the derivative of this, but let's 66 00:05:52,220 --> 00:05:54,691 see how it fits with the rule. 67 00:05:55,880 --> 00:06:02,846 Bring down the power 1 times x to the power of 1 take away 1, 68 00:06:02,846 --> 00:06:06,748 so we have 1 times x to the power of 0. 69 00:06:07,930 --> 00:06:13,060 Well, x to the power of 0 is 1, so it's 1 times 1, so 70 00:06:13,060 --> 00:06:15,831 we end up with the derivative of 1, which is 71 00:06:15,831 --> 00:06:18,360 exactly what we expected because we know the 72 00:06:18,360 --> 00:06:22,587 gradient function of y equals x is 1. 73 00:06:26,020 --> 00:06:32,883 So that was when x was a positive integer. What happens when x is 0? 74 00:06:32,883 --> 00:06:37,958 Well, let's have a look: y equals x to the power 0. 75 00:06:39,090 --> 00:06:45,223 Well, we just saw here that x to the power 0 is actually 1. 76 00:06:47,590 --> 00:06:54,745 And if we find the derivative of y equals 1. Well, y equals 1 is a 77 00:06:54,745 --> 00:06:58,084 horizontal line, so the derivative is 0. 78 00:06:59,700 --> 00:07:06,243 So y equals x to the power 0 when n is 0. The derivative is 0. 79 00:07:11,113 --> 00:07:14,774 Let's have a look at some more complicated examples. 80 00:07:16,457 --> 00:07:25,696 Let's try y equals 6 x cubed, minus 12 x to the power 4, plus 5. 81 00:07:27,988 --> 00:07:32,282 dy by dx is equal to... 82 00:07:32,282 --> 00:07:37,718 If that's 6 multiplied by 3 as we bring the power down, 83 00:07:37,718 --> 00:07:41,421 x to the power, take one from the power, 84 00:07:42,416 --> 00:07:47,870 minus 12 times, bring the power down, 4, 85 00:07:48,530 --> 00:07:54,863 x to power of 4 take away 1 and our derivative of 5 is 0. 86 00:07:56,854 --> 00:07:59,210 (I'll put the plus 0 there.) 87 00:07:59,210 --> 00:08:05,447 So three 6s... 18 x to the power 3 - 1 is 2 88 00:08:06,160 --> 00:08:14,733 minus four 12s... 48 x to the power 4 - 1 is 3. 89 00:08:15,520 --> 00:08:18,348 So there's our derivative. 90 00:08:19,515 --> 00:08:20,780 Let's try another one. 91 00:08:20,780 --> 00:08:31,378 y equals x minus 5x to the power 5 + 6 x to the power 7 + 25. 92 00:08:32,460 --> 00:08:38,952 So our derivative dy by dx equals... Now this is x to the power 1. 93 00:08:40,577 --> 00:08:45,038 So, 1 times x to the power 1 - 1, 94 00:08:46,290 --> 00:08:53,466 minus 5 times, bring the power down, 5 times x to the power of 5 - 1, 95 00:08:56,460 --> 00:09:03,720 plus 6 times, bring the power down, 7 multiplied by x to power 7 - 1, 96 00:09:03,720 --> 00:09:10,078 plus the derivative of 25, again 0. 97 00:09:11,000 --> 00:09:18,090 So we have 1 times x to the power of 0, which is just 1, 98 00:09:18,890 --> 00:09:25,689 minus, five 5s are 25, times x to the power 5 - 1 is 4, 99 00:09:26,390 --> 00:09:33,830 plus six 7s, 42 times x to the power 7 - 1 which is 6. 100 00:09:37,454 --> 00:09:41,743 Now we've proved this result when n is a positive integer, 101 00:09:41,743 --> 00:09:45,299 but it actually works also when n is a fraction or when it's a 102 00:09:45,299 --> 00:09:49,375 negative number. Now we're not going to do the proof of this 103 00:09:49,375 --> 00:09:52,930 because it requires a more complicated version of the 104 00:09:52,930 --> 00:09:56,880 binomial expansion, but we're still going to use the result, 105 00:09:56,880 --> 00:10:02,015 so let's have a look at y equals x to the power a half. 106 00:10:02,910 --> 00:10:10,078 Our dy by dx is going to be a half, as we bring down the power, 107 00:10:10,078 --> 00:10:16,222 x to the power of a half take away 1 which equals 108 00:10:16,222 --> 00:10:21,854 a half times x, and a half take away 1, is minus a half. 109 00:10:22,710 --> 00:10:27,262 Let's have a look now, when n is a negative number, so 110 00:10:27,262 --> 00:10:35,760 y equals x to the power of minus 1. So dy by dx equals, let's bring the power down, 111 00:10:35,760 --> 00:10:43,645 minus 1 times x to the power of minus 1 take away 1, so we have... 112 00:10:43,645 --> 00:10:49,476 Minus x to the power -1 - 1 is -2. 113 00:10:54,120 --> 00:10:56,910 Now, before we finish, let's look at two more complicated 114 00:10:56,910 --> 00:11:00,280 examples where we need to do a little bit of rewriting in 115 00:11:00,280 --> 00:11:04,285 index notation before we can carry out the differentiation. 116 00:11:05,610 --> 00:11:12,461 So let's have a look at y equals 1 over x plus 6x 117 00:11:13,340 --> 00:11:17,680 minus 4x to the power of 3 over 2 plus 8. 118 00:11:19,960 --> 00:11:25,154 Now we need to rewrite this in index notation so that we can 119 00:11:25,154 --> 00:11:30,628 easily differentiate it. So that's x to the power of minus 1 120 00:11:30,628 --> 00:11:36,684 plus 6x minus 4x to the power 3 over 2 plus 8. 121 00:11:37,270 --> 00:11:41,645 So let's differentiate dy by dx equals... 122 00:11:44,448 --> 00:11:50,004 Bring the power down, minus 1 x to the power of minus 1 take away 1, 123 00:11:51,479 --> 00:12:02,234 plus derivative of 6x to the power 1, that's 6 times x to the power 1 - 1, 124 00:12:03,390 --> 00:12:09,417 minus... now 4 times 3 over 2, 125 00:12:10,220 --> 00:12:14,624 multiplied by x to the power of 3 over 2 take away 1, 126 00:12:15,760 --> 00:12:19,360 plus, the derivative of 8, which is 0. 127 00:12:19,870 --> 00:12:26,604 So here we have minus x, -1 - 1 is -2. 128 00:12:27,980 --> 00:12:35,855 Plus 6 x to the power 1 - 1 is 0 which is 1, so it's just plus 6. 129 00:12:36,870 --> 00:12:43,861 Minus... three 4s are 12 divided by 2 gives us 6, 130 00:12:43,861 --> 00:12:47,750 x to the power of 3/2 minus 1... 131 00:12:47,750 --> 00:12:52,486 So that's 1 and a half minus 1, so we end up with x to the power a half. 132 00:12:53,610 --> 00:12:55,228 And there's our derivative. 133 00:12:56,330 --> 00:13:05,400 OK, one more example, y equals 4x to the power of one third, 134 00:13:05,400 --> 00:13:11,947 minus 5x plus 6 divided by x cubed. 135 00:13:12,840 --> 00:13:17,540 Now again, we've got a mixture of notations and to 136 00:13:17,540 --> 00:13:22,710 differentiate it we need to write it all in index notation 137 00:13:22,710 --> 00:13:28,820 rather than having the division. So this will be 6x to the power of -3, 138 00:13:30,250 --> 00:13:34,756 So, our dy by dx equals... 139 00:13:34,756 --> 00:13:39,732 4 multiplied by the power, which is a third, 140 00:13:40,510 --> 00:13:45,966 x to the power of 1/3 take away 1, 141 00:13:45,966 --> 00:13:52,491 minus, this is 5 x to the power 1, so it's 5 times 1, 142 00:13:52,491 --> 00:13:57,080 multiplied by x to the power of 1 - 1, 143 00:13:57,100 --> 00:14:08,149 plus 6 multiplied by minus 3 times x to the power of -3 - 1. 144 00:14:09,080 --> 00:14:12,071 So let's tidy it all up. 145 00:14:12,071 --> 00:14:20,258 We get 4 thirds x to the power 1/3 take away 1 is minus 2/3, 146 00:14:20,258 --> 00:14:23,781 so it's x to the power of minus 2/3, 147 00:14:25,390 --> 00:14:31,422 minus, now here x the power of 1 - 1 is x to the power 0, which is 1, 148 00:14:31,422 --> 00:14:33,105 so it's just minus 5, 149 00:14:34,658 --> 00:14:43,828 Six times -3 is -18 and x to the power -3 - 1 is -4. 150 00:14:43,828 --> 00:14:46,605 So, there's our derivative.