find-fprimex-fprime-primex-and-fprime-prime-primex-fxfracxx2

Question

Find $f^{\prime}(x), f^{\prime \prime}(x),$ and $f^{\prime \prime \prime}(x)$
$$f(x)=\frac{x}{x+2}$$

Amy Jiang · Accepted Answer

Okay, so let&#x27;s rewrites or quotient products sort of exercise exports to probably have one. And I was take a derivative. So we get to vote over facts. That&#x27;s one times, like supposed to through a park later. One plus x times derivative of this term. That&#x27;s the negative exposed to the power of native to okay. And now let&#x27;s take our second derivative. That&#x27;s equal to derivative of this term. That&#x27;s negative. One exposed to to have new to plus we use our product will hear sort of a X that&#x27;s just one. So that&#x27;s thinking of the next closed due to our negative, too. And we have plus the derivative list. ERM, that&#x27;s negative, too. And then we&#x27;re changing negative excellent triples X and then exposed to the power of three threat people to native to exports to our neighbor to and in plus two X experts to part of the debris. And I&#x27;m taking or through a derivative, we use our product there. Okay, that&#x27;s not before exposed to the power of negative three, plus a good group of ex That&#x27;s 1 to 2 x close to who have they got? Three. So you find that to be six of what we have here and in time to develop Liston. And then we have times two x so that&#x27;s moving down and three. So that&#x27;s like three times, too. It&#x27;s not your six x and extras to Power of Nick before. Okay, so we get our three year evidence here.

Alex Roush · Answer

Okay, so our task is to find the first and second derivatives of this function. So the first derivative is clearly, ah, an example of the coercion rule because we have the portion of two functions. So f prime of axe is gonna be the bottom function times the derivative of the top function, which is just one. Since this is linear minus the top function times the derivative of the bottom function, which is also linear the slope of one, all divided by the bottom functions squared. Okay, this is X minus X in the X&#x27;s cancel on. We&#x27;re just left with the to. So the first derivative is thus two divided by X plus two quantity squared. Now finding the second derivative, I&#x27;m assuming we don&#x27;t know that Shane rule, which means we actually can&#x27;t just differentiate this function without expanding it. So I&#x27;m gonna rewrite this. You go do it over here. I&#x27;m gonna rewrite this as X squared, plus twice the product of these guys for X plus four so that we can now differentiate um, a second time. Okay, So the second derivative is the derivative of the first derivative. So we&#x27;re differentiating this function now and again, we have a Kocian of two functions. So we&#x27;re gonna take the bottom function times the derivative of the top function. The top function is a constant and the derivative of a constant zero minus the top function times the derivative of the bottom function. So this is our, um I&#x27;m sorry, this is our bottom function. Using the power rule a couple of times, we&#x27;ll get to acts. Plus for okay. Now we divide it by the denominators squared. Well, the denominator factored is X plus two squared. So way have X plus two squared square toe. Because this is zero, this goes away. You know, we have C. I can factor out another two and I have negative four times X plus two. That&#x27;s gonna be important. Just a second. So now X plus two squared squared. OK, Whenever you&#x27;re raising an exponents to the power of something else, you multiply them two times two is four. And to simplify, I can cancel out one of these. So my answer for the second derivative is negative for divided by X plus two raised to the three parts. That&#x27;s the second derivative. And, um, the first riveted is two over X plus two squared

Nicholas Barvinok · Answer

we want to find the derogative of the function X over X squared minus one. To do this, we&#x27;re going to use the quotient rule for derivatives which states that the derivative of and the quotient f over G for two functions F and G is the bottom times the derivative of the top when it&#x27;s the top times the derivative of the bottom all over the bottom square. So in our case, we have the bottom is X squared minus one the top. The derivative of the top is the drift of the ex, which is just one. We don&#x27;t have to write it minus the top x times, the derivative of the bottom. Now the bottom is X squared minus one. The derivative of X squared is to x and then the door of 10 so minus zero we also don&#x27;t have to write And then the bottom squared is X squared minus one square. Now, while we&#x27;re done, if we expand everything the numerator, it just may so happen that we can simplify a bit. So let&#x27;s see, We have X squared minus one minus two x squared all over X squared, minus one squared and this equals expert minus two X squared. We get minus X squared, minus one over X squared minus one squared and that is it.

Dale Sanford · Answer

Let&#x27;s find the 1st 3 derivatives of this function f of X. To start out, we&#x27;re actually going to simplify f of X that is distributing out this fact squared will give us two X squared plus one over X or X to the negative first power. This is because one multiplying these terms, we would subtract their exponents. That is, we have two minus three so x the negative first, and this is much easier to find The derivative of Let&#x27;s get Started with F Prime of X, the first derivative This will be equal to two times two is four x two minus one is one and then plus well rather minus X to the negative second power or four X minus one over X squared. Next up is half double prime of X, the second derivative. So for this we just get derivative of four X is four and then the negatives will cancel to give us plus X to make the negative third power plus two x the negative third power. I&#x27;m sorry. Ah, that is We carry the two down and the negative. And now, finally, we move on to F triple prime of X the third derivative, the derivative of four is zero and then we have minus two times three is six x to the negative fourth power. Thus, these are the 1st 3 derivatives of our function f of X.

Prakash Hampole · Answer

we&#x27;re looking for F Private X, which is the derivative and that means of prime of X is equal to the derivative of um X squared plus one times Ah, seeking X now notice we have a function multiplied by another function. So we will use product law which tell us that this is equal to the derivative of ah x squared plus one times seeking to x um, plus that it plus the first function X squared plus one. I was a derivative of the second function seeking Tex. Now, in the next line will take those derivatives derivative of X squared plus one is two x two x time seek and X then, ah, we will continue. We have X squared plus one and then the German of Sikh Index is seeking next annex. So you have that. And this is our final answer to X speaking Tex plus X squared plus one time seeking x Tanox

Clarissa Noh · Answer

he It&#x27;s clear. So when you raid here, So we&#x27;re gonna take the derivative. Do you ever d x for X square over one plus eat Vex. You were gonna ply the quotient rule, which is equal to D over DX X square one plus B to the X minus d over dx one plus b to the x Times X square all over one plus e to the x square We simplify and we get to x times one plus eat the x minus Eat the X Times X square all over one plus eat the X square for the second derivative. We&#x27;re also gonna ply the quotient rule Let me get tea over d x for two x times one plus eats the X minus eats the x x square comes one plus Eat the X Square dynasty over d x one plus e to the x square times two x comes one plus heat the x minus Eat the X Times X square This is all over one plus. Eat the X square and square that again, and when we simplify we get is this equal to eat? Two X times X square minus Eat the X X Square minus four e to the two x times X minus four e to the x Times acts plus four e to the X Plus two e two the two x plus two all over one plus eat the X cubed.

Problem

Find $f^{\prime}(x), f^{\prime \prime}(x),$ and $…

Problem 64 Hard Difficulty

Find $f^{\prime}(x), f^{\prime \prime}(x),$ and $f^{\prime \prime \prime}(x)$
$$f(x)=\frac{x}{x+2}$$

Answer

$f^{\prime \prime \prime}(x)=6(x+2)^{-3}+-6 x(x+2)^{-4}$

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William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kristen K.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$f^{\prime \prime \prime}(x)=6(x+2)^{-3}+-6 x(x+2)^{-4}$

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kristen K.

Samuel H.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

William Briggs, Lyle Cochran, Bernard Gillett Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kristen K.

Samuel H.

Michael J.

William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental