find-fa-fx-x-2

Question

Find $ f'(a) $.

$ f(x) = x^{-2} $

Carson Merrill · Accepted Answer

So for this problem, we&#x27;re going to be taking the derivative of ffx and then we want to evaluate it at a specific value at this point for this problem is just gonna be the value of a So what we have is ffx is equal to X to the negative, too. Now what we know first is we&#x27;ll take the derivative of F with respect to X, and what we do here is we immediately take the exponents and move it out in front. When we do that, we get a negative too Times X and then we get a new exponents. We take the old exponents and subtract one, so it&#x27;s gonna give us a negative three, which means that we end up getting negative two x to the negative three. Now, another thing is we want to evaluate it at a so f prime of a is just going to be a negative to a to the negative three. And this will be our final answer for this problem.

Carson Merrill · Answer

So here for this problem were given F double prime of X to be equal to one over X squared. And I&#x27;m gonna write that as X to the negative too. Then I&#x27;ll make it simpler for us in the long run, that tells us that F prime of X is going to be X to the negative one over a negative one, which is just negative one over Acts plus C. That tells us that F of X is going to be equal to negative the anti derivative of one over X is actually going to be the natural log of the absolute value of X. So then based on that, what we end up having is um plus C. X. Class D. Um And the reason why we have plus six is because this is some constant. So it&#x27;s anti derivative of B plus C X. And then we have plus D. For the final general constant.

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

Clarissa Noh · Answer

it&#x27;s clear. So when you read here, so we have f of X is equal to X over X square minus one. We&#x27;re gonna use the quotient role. We&#x27;ll get X square minus one D acts d over DX minus x times D over D X X square minus one well over X square minus one square just becomes equal to X square minus one minus two X square over X square minus one square, which is equal to negative X square minus one over X square minus one square. We&#x27;re gonna find the second derivative by using the Kocian rule, which is X square minus one square D over DX your negative X square minus one minus negative X square minus one de over de x X square minus one square All over X square minus one square. You square that again, this becomes equal to that&#x27;s square minus one times negative. Two acts less negative. Two X square minus two turns negative. Two acts well over X square minus one. Cute. You could simplify this to two x times X square plus three all over X square minus one. Cute

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

Ankit Gupta · Answer

in this video, we&#x27;re going to find a valuable for minus one, for which we have our f X equals two weeks squared, minus X. So basically, we&#x27;re going to do me a different value of minus one straight off X. So it&#x27;s like four minus. Will that be able to pull games minus one? Holds where? Minus off. Minus one. Right? So it&#x27;s like two days minus one word that people one so two multiplied by one minus of minus one so that they become plus off. So it would like too much pride by one it will be too, plus one, because we&#x27;re pretty close to three.

Carson Merrill · Answer

this problem. Number thirty four of the Stuart documents a tradition section two point seven. Find a friend and proud of a were any this definition number one here to the right shop Green for the function after cools exterminated too the same one over x squared. So Oh, and the definition two no perches, eh? I was a function after the X, which is one over X squared finest function that vaccine Hollywood in a so won over a squared. Although by that point, in order to simplify this fraction according to multiply that bright at the mermaid er Pipi, he&#x27;s come out maybe two for the squared. No, pray during that. Him Eliminate the fractions. This first faction becomes X squared a score ten one or exploring which is just a square and then explain a score. Times wanna raise square, but it&#x27;s just export. It&#x27;s all divided by the quantity X minus a I&#x27;m X squared eight square. Hey, what we notice is that the numerator a squared minus X word is indifference of squares. It could be fact throughout eight minus b plus denominators. Still saying where&#x27;d a square here We notice that a minor sex is Just sing it one times X men say so you can write it like that. Then here we see that we can eliminate the next will determine the rate of a little later leaving us with Well, a minute&#x27;s expert, eh? Negative B plus blinded by a squared X squared. And if x approaches aim the numerator becomes thinking to eight plus a our native to aim And then the nominators a sward a squared or two before and the simplifying Give this our final look equation Negative too over. Thank you. So this is the result for the derivative of the function Have evaluated I&#x27;m a baby.

Carson Merrill · Answer

So in this problem we are asked to find f prime at a when half of x is X to the -2. Well, by definition, if private A Is the limit as H approaches zero of F. Of a plus H minus F of a over H. All right. So that means that this is now the limit as h approaches zero F of a plus age and in this function F of X. So that is one over a plus H squared minus one over a squared all that is over eight. All right. So that means I have a limit as H goes to zero then of what? When I get a common denominator in the top. All right. And so I&#x27;ll have a squared coming on top and then combine the fractions. That means I&#x27;ll have a squared minus a squared plus two. A H plus H squared all over a squared times A plus H squared times H from the from the denominator previously. All right. So like this a squared minus at a square is but those two are gonna be gone, aren&#x27;t they? So that means I have the limit as h goes to zero of minus to a H plus H squared over a squared times A plus h squared H. But I have an H in each term in the numerator so I can cancel one of those out. Can&#x27;t I like that? So, that means I now have the limit As a joke goes to zero of minus to a plus H over a squared times A plus H squared. Okay, so taking now the limit as H goes to zero in the numerator, I&#x27;m just left with minus two way, aren&#x27;t I? And the denominator. I have a squared times A squared, right? Because H goes to zero. So A plus H just goes to a square that let&#x27;s just a square. So that means I&#x27;m left with -2 A over eight of the 4th. Well I can cancel 1, 1 of those AIDS out. and so that means I&#x27;m left with -2 over a cute is our derivative If prime it ain&#x27;t.

Problem

Find $ f'(a) $. $ f(x) = \sqrt{1 - 2x} $

Problem 34 Medium Difficulty

Find $ f'(a) $.

$ f(x) = x^{-2} $

Answer

$f^{\prime}(a)=-2 a^{-3}$

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$f^{\prime}(a)=-2 a^{-3}$

Calculus

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Kristen K.

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Precalculus Review - Intro

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James StewartCalculus

Catherine R.

Kristen K.

Samuel H.

Michael J.

James Stewart

Calculus