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

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

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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Alright, We'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's just the piece where we are dividing. So we'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's all distributing. So in blue, I'm just showing you my work for distributing. That's a plus right there. And then if you were to uh I'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'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're really trying to simplify, I wish I'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't simplify that at all. So I'm just going to rewrite it. I'm sure it could if I really wanted to, but I don't want to. So that's a good answer. And it's correct.

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