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These are the user uploaded subtitles that are being translated: 1 00:00:00,300 --> 00:00:05,880 It we're going to talk about how to find f of x even the function second derivative to complete this 2 00:00:05,880 --> 00:00:10,910 problem we'll work backwards and take the integral of the second derivative to find the first derivative 3 00:00:11,400 --> 00:00:15,090 and then the integral of the first Serota to find the original function. 4 00:00:15,090 --> 00:00:19,290 In this particular problem we've been asked to find f of x f f l prime. 5 00:00:19,350 --> 00:00:24,500 The second derivative of x is equal to two thirds X to the two thirds. 6 00:00:24,870 --> 00:00:32,970 So in order to find f of x the original function from prime of X or the second derivative we need to 7 00:00:32,970 --> 00:00:34,220 work backwards. 8 00:00:34,410 --> 00:00:41,010 What we'll do is take the integral of prime of X to find the first tentative prime of X and then take 9 00:00:41,010 --> 00:00:43,590 the integral of that to find other backs. 10 00:00:43,650 --> 00:00:54,570 If we have the second derivative f double prime of X equal to two thirds X to the two thirds and to 11 00:00:54,570 --> 00:01:04,610 find the first derivative of prime of X will take the integral of the second derivative. 12 00:01:04,740 --> 00:01:07,570 Two thirds X to the two thirds. 13 00:01:07,830 --> 00:01:12,680 And when we take the integral we have to have the corresponding DX notation here. 14 00:01:12,690 --> 00:01:18,840 So you take the integral we'll just use simple power rule and because we have X to the two thirds here 15 00:01:19,290 --> 00:01:21,150 we'll add 1 to the exponent. 16 00:01:21,160 --> 00:01:29,550 So two thirds plus one or we can say two thirds plus three thirds will give us 5 thirds so we'll get 17 00:01:30,000 --> 00:01:38,880 X to the 5 thirds and then we want to take the coefficient two thirds and divide it by the new exponent 18 00:01:38,960 --> 00:01:40,080 5 thirds. 19 00:01:40,110 --> 00:01:42,110 So two thirds divided by five thirds. 20 00:01:42,300 --> 00:01:48,870 But instead of picking a fraction divided by a fraction we can say two thirds and instead of dividing 21 00:01:48,900 --> 00:01:57,160 by five thirds that's the same thing as multiplying by the inverse of bibe thirds which is three fifths. 22 00:01:57,180 --> 00:02:00,920 So that is the integral of the second derivative. 23 00:02:00,960 --> 00:02:07,650 But of course whenever we take an integral we have to add c so we get plus C here and all we need to 24 00:02:07,650 --> 00:02:09,650 do is simplify. 25 00:02:09,660 --> 00:02:14,560 Notice obviously that the three in the numerator and denominator here will cancel. 26 00:02:14,730 --> 00:02:26,030 So we'll get to Firth's X to the five thirds plus C and that is f prime of x are first derivative function. 27 00:02:26,040 --> 00:02:30,120 Now we need to take the integral of that to get back to at the max. 28 00:02:30,150 --> 00:02:36,720 So ever x will be equal to the integral of our first derivative function. 29 00:02:36,720 --> 00:02:47,140 So to fix X to the 5 third's plus C and of course as always our dx notation. 30 00:02:47,160 --> 00:02:50,930 Now in order to take this in a roll do the same thing we did last time. 31 00:02:51,000 --> 00:02:54,410 Add 1 to the exponent so x to the 530. 32 00:02:54,410 --> 00:03:00,460 Here we add 1 to 5 thirds 5 thirds plus one will just give us 8 thirds. 33 00:03:00,490 --> 00:03:03,920 So we'll get X to the 8 thirds. 34 00:03:04,080 --> 00:03:09,080 And now we want to take two fifths the coefficient and divide it by the new exponent. 35 00:03:09,150 --> 00:03:17,190 So two fifths divided by eight thirds is the same thing as two fifths times three eighths So multiply 36 00:03:17,190 --> 00:03:17,700 them. 37 00:03:18,210 --> 00:03:22,670 And now see we can treat as a constant. 38 00:03:22,680 --> 00:03:30,100 This is basically the same thing as saying See times x to the zero power X to zero or anything to the 39 00:03:30,120 --> 00:03:32,930 zero power for that matter is equal to 1. 40 00:03:33,210 --> 00:03:35,900 So this is the same as saying See times 1. 41 00:03:35,900 --> 00:03:36,720 Or just see. 42 00:03:36,810 --> 00:03:41,770 So by multiplying by X to zero we haven't changed anything in it. 43 00:03:41,790 --> 00:03:45,940 Now allows to treat this the same way that we did this X of the five third's here. 44 00:03:46,110 --> 00:03:51,840 We'll just add one to the exponent so we'll get zero plus one just gives us one and then we'll divide 45 00:03:51,840 --> 00:03:56,660 C by the new exponent so see you divided by 1 is what we'll get. 46 00:03:56,880 --> 00:04:02,870 And because we took another integral we need to account for the constant of integration and add C again. 47 00:04:03,060 --> 00:04:05,880 We want to distinguish it because we've already added C.. 48 00:04:06,090 --> 00:04:10,250 So instead of using C we'll use the next variable and call it D. 49 00:04:10,380 --> 00:04:14,180 So now we just need to simplify this as much as we can to get our final answer. 50 00:04:14,280 --> 00:04:24,030 Well get f of x is equal to two fifths times 8 will give us 6 over 40 which is the same thing as 3 over 51 00:04:24,450 --> 00:04:33,560 20 x to the 8 thirds plus see X plus the. 52 00:04:33,750 --> 00:04:34,460 And that's it. 53 00:04:34,470 --> 00:04:37,100 That is our original function f of x. 54 00:04:37,140 --> 00:04:42,560 If the second derivative f double prime of x is equal to two thirds X to the two thirds. 5975