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These are the user uploaded subtitles that are being translated: 1 00:00:00,390 --> 00:00:05,220 Today we're going to talk about how to find f of x even the functions third derivative. 2 00:00:05,530 --> 00:00:06,760 Complete this problem. 3 00:00:06,810 --> 00:00:11,200 We'll work backwards taking the integral of the third derivative to get the second derivative. 4 00:00:11,460 --> 00:00:15,870 The integral of the second derivative they get the first derivative and then the integral of the first 5 00:00:15,870 --> 00:00:18,570 derivative to find the original function. 6 00:00:18,570 --> 00:00:24,270 In this particular problem we've been asked to find f of x if f triple prime or the third derivative 7 00:00:24,270 --> 00:00:28,460 of x is equal to x minus the square root of x. 8 00:00:28,830 --> 00:00:35,700 So in order to find our way from F triple prime or the third derivative X back to f of x we're going 9 00:00:35,700 --> 00:00:37,620 to have to work our way backwards. 10 00:00:37,650 --> 00:00:45,380 So we'll start with the third derivative of X and we'll we'll try to find the second derivative X on 11 00:00:45,480 --> 00:00:50,160 we found it will find the first derivative and then use the first derivative to find the original function. 12 00:00:50,400 --> 00:00:56,780 So we'll start with the third derivative of X and in order to find the second derivative of x f double 13 00:00:56,790 --> 00:01:03,430 prime we'll need to take the integral or the antiderivative of the third derivative. 14 00:01:03,480 --> 00:01:09,360 So X minus the square root of x and whenever you have integral notation like this you're going to need 15 00:01:09,480 --> 00:01:13,220 a DX corresponding notation to go with it. 16 00:01:13,530 --> 00:01:17,980 So let's go ahead and make a change in this integral to make it easier on ourselves. 17 00:01:17,980 --> 00:01:20,050 We're going to be taking the integral of x. 18 00:01:20,130 --> 00:01:26,160 I want to convert the square root of x here to x to the one half which is the same thing because now 19 00:01:26,160 --> 00:01:29,920 we can use power rule to easily take integral of this term. 20 00:01:29,940 --> 00:01:35,850 So to take the integral or remember that we're going to use simple power rule here. 21 00:01:35,850 --> 00:01:41,580 We have X to the first power x to the one that take the integral we'll add 1 to the exponent. 22 00:01:41,670 --> 00:01:48,770 We'll get one plus one equals two and we'll divide the coefficient which right now is just 1 divided 23 00:01:48,780 --> 00:01:52,220 by the new exponent which we already found to be 2. 24 00:01:52,530 --> 00:01:57,740 So that's how we use power rule to take the integral of simple polynomial terms like this. 25 00:01:57,750 --> 00:02:02,850 So now we need to take the integral of negative x to the one half in order to do that. 26 00:02:02,910 --> 00:02:04,220 Add 1 to the exponent. 27 00:02:04,260 --> 00:02:12,390 So one half plus one will give us X to the three halves and then we'll divide the coefficient which 28 00:02:12,390 --> 00:02:18,000 is currently 1 by the new exponent so divided by three halves. 29 00:02:18,000 --> 00:02:24,790 When we simplify this we'll get 1 1/2 x squared minus two thirds. 30 00:02:25,810 --> 00:02:32,130 X to the three halves but then keep in mind that when ever we take an integral we always have to add 31 00:02:32,170 --> 00:02:34,370 C. or the constant integration. 32 00:02:34,570 --> 00:02:38,020 So we need to add c to account for the constant of integration here. 33 00:02:38,110 --> 00:02:41,250 And of course that would carry down into our simplification. 34 00:02:41,320 --> 00:02:45,580 So what we solve for here is the second derivative of f of x. 35 00:02:45,580 --> 00:02:47,920 Now we need to go ahead and find the first derivative. 36 00:02:47,920 --> 00:02:56,580 So the first derivative will be f prime of X and that will be equal to the integral of the second derivative. 37 00:02:56,710 --> 00:03:08,930 So 1 1/2 x squared minus two thirds X to the three halves class C D. 38 00:03:09,190 --> 00:03:14,080 So again we'll use power rule to take the integral of this term by term. 39 00:03:14,080 --> 00:03:15,410 We have x squared year. 40 00:03:15,460 --> 00:03:23,090 Add one to the exponent two plus one gives us three and then we divide the coefficient one half by three. 41 00:03:23,150 --> 00:03:26,770 One half divided by three is one sixth. 42 00:03:26,860 --> 00:03:28,680 So we get one 6 there. 43 00:03:28,750 --> 00:03:34,420 Now we have negative two thirds X to the three halves that take the integral will add 1 to the exponent 44 00:03:34,860 --> 00:03:38,040 three halves plus one will give us five halves. 45 00:03:38,080 --> 00:03:44,040 So X to the five halves and then we'll divide negative two thirds by five halves. 46 00:03:44,050 --> 00:03:52,120 The new exponent but negative two thirds divided by five halves is the same thing as negative two thirds 47 00:03:52,330 --> 00:03:54,600 times two fifths. 48 00:03:54,760 --> 00:03:55,060 Right. 49 00:03:55,060 --> 00:04:00,850 Remember instead of dividing by a fraction we can multiply by its inverse So this is negative two thirds 50 00:04:00,940 --> 00:04:02,580 times two fifths. 51 00:04:02,850 --> 00:04:10,320 And then when we take the integral of C we'll just get C X because remember that C is just a constant. 52 00:04:10,570 --> 00:04:17,860 And this is basically c times x to the zero power x to the zero and anything raised to the zero is 1. 53 00:04:17,890 --> 00:04:24,370 So essentially we haven't changed this at all because we're just saying See times 1 which is see this 54 00:04:24,370 --> 00:04:31,180 now follows though the same rules as the other two terms so since we have X to the 0 will add 1 to the 55 00:04:31,180 --> 00:04:37,750 exponent and get X to the first power and then divide the coefficients C at a new exponent 1. 56 00:04:38,020 --> 00:04:43,430 But of course because we've integrated We also want to add another concept since we've already used 57 00:04:43,470 --> 00:04:48,030 SI to account for a constant we add D instead. 58 00:04:48,170 --> 00:04:55,300 Now we just need to simplify this as much as we can so we'll get one sixth X to the third we have negative 59 00:04:55,300 --> 00:04:56,910 two thirds times two fifths. 60 00:04:56,950 --> 00:05:09,420 So we'll get minus four over 15 x to the five halves plus C X plus D. 61 00:05:09,650 --> 00:05:12,250 And that is our first derivative. 62 00:05:12,290 --> 00:05:20,360 Now in order to finally get back to our original function f of x we will take the integral of our answer 63 00:05:20,360 --> 00:05:31,580 there so our backs will be the integral of one sixth next to the third minus for over 15 x to the five 64 00:05:31,580 --> 00:05:35,830 halves plus X plus D. 65 00:05:36,950 --> 00:05:42,740 And as always we'll have our the X here to take the integral we'll do the same thing that we've been 66 00:05:42,740 --> 00:05:47,700 doing well add 1 to the exponent to x to the third becomes X to the fourth. 67 00:05:47,870 --> 00:05:56,330 And now we have 1 6 divided by 24 which will be 1 24th taking the integral of this X to the five halves 68 00:05:56,390 --> 00:06:00,760 will add 1 to the exponent 5 habes plus 1 gives us seven halves. 69 00:06:00,980 --> 00:06:07,000 So X to the seven hands will get negative for over 15. 70 00:06:07,040 --> 00:06:16,580 Instead of dividing by seven halves will multiply by the inverse to sevens and we get Plus we add 1 71 00:06:16,580 --> 00:06:17,260 to the exponent. 72 00:06:17,260 --> 00:06:21,500 This is essentially X to the first so get x squared. 73 00:06:21,650 --> 00:06:27,810 And we'll divide the coefficients C by the new exponent which is two and then same thing here. 74 00:06:27,800 --> 00:06:30,750 The this is essentially the times x to the 0. 75 00:06:30,800 --> 00:06:36,500 We add one to the explant and we get access to the first power. 76 00:06:36,500 --> 00:06:38,480 We divide the by the new exponent. 77 00:06:38,480 --> 00:06:40,330 So we get the over 1. 78 00:06:40,420 --> 00:06:46,190 But because we're integrating Of course we have to add another constant of integration this time we'll 79 00:06:46,190 --> 00:06:47,560 call it e. 80 00:06:47,870 --> 00:06:56,210 So when we simplify this we'll get our final answer for f of x which is just going to be 1 over 24 x 81 00:06:56,210 --> 00:07:10,400 to the fourth negative for 15 times to sevenths will be negative 8 over 1 0 5 x 2 the 7 hands plus C 82 00:07:10,460 --> 00:07:17,980 over to x squared plus the X plus E. 9283

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