find-the-nth-derivative-of-each-function-by-calculating-the-first-few-derivatives-and-observing-the-

Question

Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

(a) $f(x)=x^{n}$                   (b) $f(x)=1 / x$

Clarissa Noh · Accepted Answer

Hi, this is Claire. So instruction 3.3. Number 50. Ito&#x27;s too. It&#x27;s about calculus book. You&#x27;re gonna find the interpretive of each function with party we&#x27;re going to for the first few d&#x27;oeuvres it it&#x27;s so the first derivative moves in Times X on Linus Ling. This is just a power formula. So if you do this with a second or a bit of then time, Linus long next ones, too. They&#x27;re driven sometimes a new awareness planning. I still x to the n minus three. And here we have a clear, um you pattern So are conservative equal to n factorial. Moving on to part B. We&#x27;re going to do the same. The same approach. You could have one over X square, second derivative wounds. I could have one square two over x cute. In our third derivative, I get a one you mystery over next to the flaws. And here we have another pattern. Sonar, and they&#x27;re a bit of ends. They could have one to the end Pictorial over. Next on dde waas One books for answer and we&#x27;re done

Clarissa Noh · Answer

he It&#x27;s clear. So in new Marine here. So we&#x27;re gonna find the first few derivatives you have ever X is equal to X to the end power when we have our first derivative, which is X and times X to the n minus one power. We have our second derivative, which is n times and minus one X to the N minus two. And we see that, uh, to the then power derivative is equal to and times and minus one times and minus two all the way. Tow n minus and close one times X to the n minus. And and this becomes equal to in factorial For part B, we&#x27;re gonna do the same thing. Just write a couple of derivatives to see the pattern. X Square becomes next for the second er, bit of negative one negative too. Next to the negative three years power. And then we have, uh, then derivative, which is equal to negative one times Negative too, comes negative three all the way to negative and times X to the negative and plus one power which is equal to negative one to the end. Power factorial over X to the end, plus one

Ernest Castorena · Answer

a function of X is equal to two over X plus one. And then let&#x27;s see, we take our derivative Oh God, negative two over X plus one squared. Well, let&#x27;s take our second derivative now. No end up with Let&#x27;s see two times negative one squared times two over X plus one cubed. Now let&#x27;s take our third derivatives on the of the Rexes Equity two times Negative one times two times three times, one over explosive, one to the court. Let&#x27;s take our fourth derivative. You can kind of see the pattern already round up two times negative one to the pork two times, three times four times, one over X plus one to the fifth. And we can guess our formula for the answer. A bit of X. Now that&#x27;s just two times negative. One Teoh end part times and factorial all of that over X plus one to the end plus month. Part

Linh Vu · Answer

Thanks. Wrinkled your agent eggs. And we know that the riveting off exponents show each of the next it was stays dissenting and even the secondary with Dio It was also the 70. And the theory would dio Could you need you to eggs and even fonder f and derivative. I&#x27;m sorry, Could you digital? Thanks.

Ernest Castorena · Answer

for this problem. They want us to take driven us until we find a formula for the end derivative. So let&#x27;s take our first derivative. Let&#x27;s see in general Uh huh. Two X plus one to the negative one, huh? And under a bit of the inside times drinks. Now see, second derivative on the boy is seen. I know who to elect is equal to you. Currency. Negative one over too quick. This is a negative one. So the let&#x27;s take you. Yeah, that&#x27;s good. And then we&#x27;ll have to X plus one to the negative 3/2 parts and then we&#x27;ll multiplying by. Let&#x27;s see, it&#x27;s your ex on. We don&#x27;t want to experience the derivative of two experts. One on the inside is just too, so we just have a tear in front of their. It&#x27;s the race that properly, that&#x27;s just a to so over here, we&#x27;re going to multiply line under two. We have a T Square. Let&#x27;s take our their individual. We&#x27;ll end up with negative one to the let&#x27;s see four, then 1/2 one time. See over to Cuba two X plus one to the negative 5/2 times Que? Let&#x27;s see the Force group. That report thanks is equal to you. Negative one to the power and one times the times over to to the poor two x plus one to the negative. Seven. Overtime. You hear the word? Okay, Now it&#x27;s not necessary here. The tune of the fourth is what&#x27;s not really necessary. My if we had to guess are interpreted, it would be seen negative one to the emplacement. And then we have one. I&#x27;m saying times seen to and and then all of this will be over and see, this is for an is equal to one on later over. Is it? If I for one, let&#x27;s see Yeah, it does cause we have negative one to the two Firing Mel, etc. We were over two acts, plus one to the Let&#x27;s see, that&#x27;s a seven. So why don&#x27;t they dio to end minus one over? Yeah, that looks good

Matthew Lee · Answer

we need to find a pattern or I&#x27;m gonna function. Isn&#x27;t given a name to call you back physical function. They wanted two different J extra and want to find a formal effort the interview of G&#x27;s dysfunction. They do that. Let&#x27;s start by finding the front of it is the second derivative. And then, um, tryingto find a pattern. Cheap paperbacks. We need to get the popular here. That&#x27;s going to be X time derivative of the second function. Plus the second home. We need to find a pattern, or I&#x27;m gonna function. Option back on a derivative of the birth. You have one. Okay, Tim, apart. That gun X f from a box epidemic. All right, here&#x27;s the first year that it kind of hard to be a pattern right now. Find the second Derby, didn&#x27;t you? The pattern Devil crime of X or the year after year, The practical again. So again, we&#x27;ll get X. I&#x27;m second function, which is now a crime course. A crime. But they didn&#x27;t give a name. McCall chair backs. What is your function? They wanted two different J extra and want to find a formal effort. The interview of G&#x27;s function do that. Let&#x27;s start by finding the structure of it is second derivative. And then, um, tryingto find a pattern. Keep coming back. We need you need a factual here. That&#x27;s going to be X time derivative of the second function. Plus the second part of it of sorry, you&#x27;re the derivative of X. I&#x27;m double parked. What? Second function? Which prime of X sign. First function derivative functions. One. What derivative? This function here Weapon do also after that time and we can simplify this thing. To accept double kind of X doesn&#x27;t have probably except one another crime about two primavera. Okay, let&#x27;s do one more derivative. I&#x27;m just make sure we have the paranoid We need to rev attention back on a derivative of the birth. Where do you have one? Okay, so far. This gun, except from a box Productive. All right, here&#x27;s the first year that it&#x27;s kind of hard to be a pattern right now. Find the second Derby, didn&#x27;t you? To the pattern devil prime of X, or that you&#x27;re actually the practical again. So again we&#x27;ll get next. I&#x27;m second function, which is now a crime course a crime. But the derivative dysfunction Here. Practice again. X times the derivative of the function of a triple front back. Plus that second function after the crime time one, You&#x27;re going back. What? Your own It is functioning the Tuesday there. But then we have a second derivative. No. Okay. Come to find her. The actual crime last even after prime here, too. After the crime that makes the reactor prime. Sorry. You&#x27;re the derivative Excitement back double parked. The second function. Which prime of X sign. First function derivative functions. One. What? Be derivative. This function here. Weapon do all after that time, Enrique in temple by this thing to accept double kind of X doesn&#x27;t have proper back someone. You&#x27;re another crime about two. Prime of X. Okay, let&#x27;s do one more derivative. I&#x27;m going to make sure we have the paranoid. We need a derivative. Okay, think round of information to write a formula Org the derivative of the function. Well, first of all, we always have a next and then the power. The derivative. It&#x27;s the same derivative that we&#x27;re taking the whole function. So the interpretive effort. Well, plus, I have a number here. That number coefficient matches the number of derivative that we&#x27;re doing. You hear the thunder emanated a breather, and then I can drive it. About two. There back. You&#x27;re the first driven and dysfunction. You&#x27;re a practical again. Next time, the derivative of the function of a triple run back. Plus that second function after the crime time. One back. What you wrote it is punching the Tuesday there. But then we have a second derivative. No. Okay. Come to find her the actual crime. Unless you have an active crime here, too. After the crime that makes the reactor will prime. Oh, that, uh, one day you could even think of fish and being a hero derivative. And I have zero there. So that means that this and time, um, s and then here, the derivative for after one arrest. Then it is for over here. So here we have a second. Everybody clear the third. You&#x27;re That means that there should be and money one and minus one

Problem

For what values of $x$ does the graph of $f(x)=x+…

Problem 58 Hard Difficulty

Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

(a) $f(x)=x^{n}$ (b) $f(x)=1 / x$

Answer

(a) $f^{(n)}(x)=n !$
(b) $f^{(n)}(x)=(-1)^{n} \frac{n !}{x^{n+1}}$

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James Stewart

Biocalculus Calculus for the Life Sciences

Grace H.

Anna Marie V.

Michael J.

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Precalculus Review - Intro

Functions on the Real Line - Overview

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(a) $f^{(n)}(x)=n !$(b) $f^{(n)}(x)=(-1)^{n} \frac{n !}{x^{n+1}}$

Biocalculus Calculus for the Life Sciences

Grace H.

Anna Marie V.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Anna Marie V.

Michael J.

Joseph L.

(a) $f^{(n)}(x)=n !$
(b) $f^{(n)}(x)=(-1)^{n} \frac{n !}{x^{n+1}}$

James Stewart

Biocalculus Calculus for the Life Sciences