find-the-derivative-of-each-function-by-using-the-quotient-rule-fxfrac3-x8x24-x2-2

Question

Find the derivative of each function by using the quotient rule.
$$f(x)=\frac{3 x+8}{x^{2}+4 x+2}$$

Gregory Higby · Accepted Answer

Alright, We&#x27;re given this function defined as a quotient. So look in the numerator, you have three explicit all over a giant quantity X squared plus four X plus two. So as I mentioned, the quotient, that&#x27;s just the piece where we are dividing. So we&#x27;re going to do the quotient rule while the quotient rule tells you to take the derivative of the top, which is just three. The linear term. You leave the bottom alone and then minus. Now you take the derivative of the bottom, which is the quantity of two X plus four. While you leave the top alone all over the denominator squared. Now, some teachers will be generous and let you leave your answer like that. But I know a lot of other teachers would not. They would expect you to simplify by distributing and foiling in the denominator or start boiling over here. It&#x27;s all distributing. So in blue, I&#x27;m just showing you my work for distributing. That&#x27;s a plus right there. And then if you were to uh I&#x27;ll do this piece by piece. But you can see you have six X squared. The outside would be 16 X. The inside would be 12 X. That&#x27;s a total of 28 X. And then four times 8 is 32. But at the same time, you need to distribute that minus this minus. Right here. If you&#x27;re really trying to simplify, I wish I&#x27;ll go ahead and give this simplified answer. That three X squared minus six X squared is negative three X squared 12 X minus 28 X would be minus uh 16 X And 6 -32 would be -26. And the denominator. I can&#x27;t simplify that at all. So I&#x27;m just going to rewrite it. I&#x27;m sure it could if I really wanted to, but I don&#x27;t want to. So that&#x27;s a good answer. And it&#x27;s correct.

Yingtai Xiao · Answer

out. Let&#x27;s look at another function. Is this eight X square plus five over X square plus three. Here you X, you see goes to eight x square plus five the excess because two X square plus three Did your Ativan, you ISS 16 X curative evidence X two X Grissom Why I right on you accent P X is that we want to use the caution through clearly so we don&#x27;t mix. There&#x27;s no patience. Curative of affects is he goes to X square plus three square in the numerator and in the numerator uh, saying is that the directive of you just 16 annex times need minus dura Tear off be just two x times you and simplify the expression get 48 x minus 16 cubic minus 10 x His two terms Kansas out and in the denominator, Mr have this term, we get the results. This follows

Ernest Castorena · Answer

be function so we could more easily apply power. And now that&#x27;s the plan. Well, in the boy&#x27;s negative three x to the negative two minus eight and the constant goes away.

Ernest Castorena · Answer

thank you. This problem. They give us the function after the X is equal to 1/8 x cubed plus for it and all we can do. General. Let&#x27;s see, we&#x27;ll get a Bible brain because you have invented the with. So take the input to the fourth power and out and then take the word of the input, which is the expert Nagel.

Todd Vawdrey · Answer

for this problem. We&#x27;re gonna go ahead and calculate a derivative right here. So we&#x27;re looking for D Y D X. Now, In order to do that, we can see that the original thing is a composition. So we have this piece right here, sitting inside oven X to the fourth function. So the chain rule says we&#x27;re gonna treat it as if that red thing were just its own little piece right here. So that gives us, um, it&#x27;s using the power rule four times. This green stuff cube was just the power or right here, times the derivative of the green stuff and the derivative of that green stuff again I can use. The power rule is just 12 x to the third, minus four X and then there isn&#x27;t any. It&#x27;s plus zero technically right here. Um, so I can write this all together as four times the quantity. Three extra, The fourth minus two x squared plus eight all being cubed times 12 X cube minus four x. And that would be the derivative of the whole thing. Using the chain rule

Ernest Castorena · Answer

they give us this function and we should write it into a form we can more easily handle. And now we can take the derivative. Well, let&#x27;s see. Pyro list Spring the power down. Well, Underwood negative. Eight of the negative fifth, and for the second time we&#x27;ll have negative X squared.

Problem

Find the derivative of each function by using the…

Problem 21 Easy Difficulty

Find the derivative of each function by using the quotient rule.
$$f(x)=\frac{3 x+8}{x^{2}+4 x+2}$$

Answer

$f^{\prime}(x)=\frac{-3 x^{2}-16 x-26}{\left(x^{2}+4 x+2\right)^{2}}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$f^{\prime}(x)=\frac{-3 x^{2}-16 x-26}{\left(x^{2}+4 x+2\right)^{2}}$

Basic Technical Mathematics with Calculus

Grace H.

Heather Z.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Heather Z.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus