We're going to derive the MacLaurin series representation of the function F. Of x equal cubic root of X plus eight. Then we will compute at least the first three coefficients and show and explain all steps. Then we have the function cubic root of X plus eight can be written as X plus eight to the one third. Tom written this way is easier to calculate the derivative of F. And what we want to calculate now is is Michael O'Ryan series representation of dysfunction. And that's a Taylor series around zero. We write that here, are you serious? He's taylor serious. And syria they love develop around zero. Mm. And so we need the derivatives of this function at zero. First we calculate the derivatives at any point. So we have the first derivative Is equal to 1 3rd times X plus eight To the 1 3rd -1 times the derivative respect to X of X plus eight. Applying the sheen rule is equal to one third times X plus eight to the genetic disease. It's Times one because the derivative of expose it is equal to one and this is equal to 1/3 times cubic root of X plus eight square. So this is a preservative can be viewed at the at this formula or as this formula. Now the second there is a tv sequel Two and we are going to use this formula to calculate the second derivative is 1/3 times negative two thirds times X plus eight To the negative 2/3 -1 times the derivative of the base explicit. And from now on, we notice derivative is one. We can't put it any more. So this is negative two over three square Times X-plus eight to the negative. Uh huh. Five thirds. And we can rewrite this as to over three square cubic root of X-plus eight to the fifth. Yeah, the third derivative of F is And we are going to use this formula here is negative 2/3 square times. And at the 5/3 Times X-plus eight To the negative 5/3 when it's one times the narrative of the base which is one and this is equal to Now, the result of the coefficient here is positive. So it's two times five over three cube times explicit to the negative eight thirds. And we can rewrite is at two times five over three cube times cubic root of X plus eight to the eighth. Okay, you're going to do one more derivative, fourth derivative in order to generalize the result, then we have two times five times So here is over three Cube in that times. Okay, Foreign -8/3 Times X-plus eight To the negative 8/3 -1 Times the derivative of the base, which is one. We have now a negative result here two times 5 times eight Over 3 to the 4th Times X-plus eight To the -11/3. That is negative. Two times five times 8 Over 3 to the 4th kevin Garrett of X-plus 8 to the 11 and now we can generalize this formula here And we can say that the K- three relative of F at X is equal to. So the first thing we note here is that the signs are alternate. So the first relative here, mhm has positive coefficient. One third, second derivative Has a negative coefficient negative to over three square. The third derivative is positive, the coefficient and four through the TV is negative and so on. So we have alter main science and the coefficient is fraction. In the numerator we have the multiplication of the previous coefficients or the previous exponents if you want. So in this case, for example, This multiplication two times 5 is the product of the previous numerator is in the exponents. If you look at this following multiplication here is these eight times this five year times this to here. So it's a product of the previous new operators of these opponents without negative sign it. Of course. And in the denominator of that fraction we have the power of three. And the exponents of that power is the same as the order of the derivative Here we have three to the fourth and we have here the 4th derivative. 3, 3, 2 deserve here, third derivative, three square second derivative and so on. And the second part is Expose eight race to a negative power. The denominator result was three but the numerator is a number which is changing here. And that number is in fact the number will correspond to the last uh turned or factor in the multiplication of the next term. So we can use all these observations to conclude that the case derivative Campbell written this way. So we have a numeric fraction. And we're going to see that this numeric fraction is the product. As we have seen here. The Maria Teresa product From G equals zero To K -1. That is one less than the order of the derivative. All the factors of the Form 1 -3. J. Why that because we know that the results Of the formula of the derivative is always subtracting one. It means that we are always striking one. Um the # three. One more time than the previous relative. That is the first derivative is we have expanded n numerator one or in the function. And the secondary body we have one minus three. And if we do this way, we have the sign already, I get to When we do this, say this next relative. We are going to subtract one. And it means in the narrator we subtract from this 2, 3, 3. So we get 1 -3 -3 is 1 -6. It's nearly five. It's following his opponent here And so on. So we have a subtraction of 1 -93 times a day. Okay, A coefficient representing the derivative. So that is why we have the factor of this form changing the coefficient J from zero. Where we we are here to one less than the order to derivative. Okay. Mhm. In the denominator we have uh huh Power of 3% before three to the order of the derivative. Here we have three to the fourth and it's the fourth derivative. He had three to the third. It's his third derivative and so on. So this is numeric fraction and it's important to notice that in the numerator we already have to sign the alternative sign. Let's see when keiko's wanda is way we are at the first derivative here We have a positive sinus 1/3. And in fact when we have the product from J equals 0 to 0 because for k equals one. The Over bar is zero of one month, three day we have only one factor which is 11 of three times zero. She's one. We did the following derivative. The 2nd 1 From Gable Syria to one. Because Putin gave was to here We will have an Upper Limited one. We have two factors, 1 -3 times zero times 1 -3 times one. That is one times -2. And we have -2. So the first Coefficient was positive positive and the second one was negative. This negative two. We found here. Yeah, I did too. Mhm And if we do one more, we are going to convince ourselves that With one more factor that is for Keiko three we give us an upper bound to we had one times -2 times in the last one is 1 -1 six is 95. And we have two negative numbers. The black means there's all these positive. So we have one Times two Times 5. That is to thank five. Which is this coefficient with plus sign or positive side. So that sign is already the alternative sign is already included in the result of the product of these factors here. So we don't need to to add another term to have that alternative sign. Yeah. And now we have the power of X-plus eight. And as we said before, we have the next Factor we should have in the numerator here. So we have one 3K which will be the next term of this form over three. And for example for cables one, we will have That is the first derivative who have 1 -3/3, that is negative 2/3. Which is in fact this component of X plus eight in the first derivative from second derivative. That is for cables to we'll have in spawning of one minus six Over three. It is 95/3 in the 3rd derivative. And that is in the second derivative. Sorry, that is is here and so on. So this is our general expression for any K Career than or equal to one. We cannot include they function itself in this formula because we will have a product here with an upper limit, negative upper limit. Maybe doing some uh considerations additional considerations, we could include it. We're going to let that is separate. So this is our general formula of the derivative of the function at any point. But now we want that at zero. So the K three relative of F zero is the product From Jake will 0 to K -1 of the terms. 1 -3 day over 3 to the K. S. power time eight As we put here zero. And we get eight to the Yeah 1 -3. K. Over three. Okay, and we can say that is equal to the product from she equal zero, Gay -1 of 1 -3. J Over 3 to the Cave Power Times 2 to the third time is raised to the 1 -3. K. Over three. And we are remembering that the power of a power is a new power with the multiplication of exponents. So we're going to cancel out this three here with this spoon in here and we'll get finally Okay, The product from Jacob 0 to K -1 Of 1 -3. J Over 3 to the cave power times 2 to the 1 -30 K. We can't even do some more here. He sees The product from J Cole 0 to K -1 of one minute 3. J Over three to the cave power. And this here is a product of 2-1 Time stood to the 93K. So we have a factor to here. And to to them three. Key To the -3 K. This can be put in the denominator already that is. We write it around here and again we remember that. This can be written us to the third And that to the case power. That is This will end up being two times product from J equals zero two. Gay -1 is 1 -3. J. over. Okay. Three to the cave power times eight to the cave power. And Mhm. This become uh this comes from the fact that two to the three K. Is equal to two to the third and that to the kids power. And Mhm. And now we have a proud of powers with the same exponents. So we know this equal finally to two times the product from J. Well zero to k minus juan Of 1 -3 J. Over. Uh huh. 24. There is three times 8 to the case power. So we have a very good beautiful formula for The case derivative of F0 and dad for K. Very vulnerable to our So we can say that team a chlorine serious representation. Okay. Of if Yeah. Yes. Yeah. So we know the general formulas. We're going to put it in compressed form But it is a serious from Keiko 0 to Infinity of the case derivative of F at zero over K. Factorial times eggs. To the cave power. Yeah. And we can separate this as F at zero, which is the first term here Because for cable zero get After the theory theory of derivative is the function itself at zero. Zero factorial is one and K two, the theory extra zero is one. So the first term as this series is the functional zero. And then we have the series from K goes one. So we have separate the value of the first term of the series because we don't have uh we cannot use the general formula formula K. To remedy for that. In some cases it can be done. But here it's not convenient to do that. So we have this And always substitute the values effort zero is cubic root zero plus eight plus is serious from cake was want to infinity. And now we right is term which we found here. Uh huh. So we will have to Product from Jacob 0 to K -1 Of 1. 1, 3. J over 24 to the case power, That's derivative. The derivative at zero. And that divided by K. Victoria's. So we got to put here. Kay factorial and that times X. To the cave power. Let's say this is serious and uh having this. Yeah. Uh we know here that we have cubic root of eight because eight is two to the third. We have to hear and we can take this two out of the series. And we get finally this expression here, 2-plus 2 times a series from chemicals want infinity of The product from J0 To K -1 Of 1 -3. J over 24 to the case power times K. Factorial In that Times two eggs to the cave power. And this is it. And now we have taken this to out but we got to remember that each coefficient has he's factor too. It's very important that So now we can calculate the three, the first three coefficients of the series. This will be a result, general result of the McLaren, serious of the function F cubic group of experts. Eight. And now we calculate the first three coefficients. And so the first one is f at zero which is to found here. Now we have 2nd proficient will be two times. So sorry here I'm going to write it properly because it's not going to talk about if a zero, you can see it without this case. First efficient If just effort zero. This way is better. The second coefficient will be in this case two times. And we got to use remember his expression here without the eggs to the cave power. Because this isn't this expression here is the numerical coefficient of the series. So it's two which we put out. It's a common factor times the product from Jacob zero. In this case for k equals one. We get up to zero of 1 -3 days. So we will be only one term over 24 to the one times one victoria fine pictorial, That's equal to two times The only term here is one minus three times 0/24 because one victoria's one, so we get zero here. So we get to over 24 is 1/12, it's always our second coefficient and a third efficient is two times a product from Jacob 0- 1. 1.3 day over 24 square times two factorial And this is equal to two times And here we'll have two factors one minus three times zero times one minus three times one over 24 times 24 which is 24 square times 22 factorial is too so we cancel out these factors to here and we will get one. This factory is one, this factor is negative 21123 over 24 Times 24. We simplify this to with one Those 24th here And this is equal to negative because one times negative two is negative number. We will have one because we simplify this to within 24 we get 12 times 24 And then this is secret to Nettie one over and this give us um something like 288. This is the 3rd coefficient and we can say finally that the fourth proficient if we won't calculate that Is two times broader From Jacob 0 to 2 of 1. 1 of three, J over 24 Cube Times three Factorial. And this is equal to two times one times negative. Two times 85 over 24 times 24 times 24 times three times two. This is three factory and this is 24 Q. And this is equal to we simplified it to hear and one is two with 24 and we get There. So it's positive because we are -2 times 75. So we get five over 12 times 24 times 24 times three. And now we do all this multiplication in the denominator and we get five over. Okay, mm hmm. 20,000 736. So this is our fourth coefficient. And from then from that term a lot on that is from the fourth coefficient on the fractions will have denominators which are big numbers. So uh it is, we show a little bit how it goes. But the important thing here is that we have a general formula for the MacLaurin series of the function killed a group of experts aid at any point X.