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Find the derivative of each function by using the quotient rule.$$y=\frac{2 x^{2}-x-1}{x^{2}(x+2)}$$

$\frac{d y}{d x}=\frac{-2 x^{4}+2 x^{3}+5 x^{2}-4 x}{\left(x^{3}+2 x^{2}\right)^{2}}$

Calculus 1 / AB

Chapter 23

The Derivative

Section 6

Derivatives of Products and Quotients of Functions

Derivatives

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alright in this problem, I think they're trying to get you to do the product rule and the quotient rule at the same time, uh which I'll go ahead and do it that way. I probably wouldn't if I were teaching this. So because if you notice in the denominator of this quotient, so there's your quotient, but down here you have um you have a product. So the way I would probably do is think about in the denominator, the denominator would be equal to X to the third power plus two X squared when you distribute. Um So I guess I could do both. Thinking about it where I show you dy dx where the drift of of the top is For X -1, then you leave the bottom alone, so I'm going to leave it as an X squared X plus two minus now. You leave the top alone and the drift of the bottom, I mean they use this piece. So I don't have to do the product rule is three X squared plus four X. All over the denominator squared. Well if you're going to square each, you know, square this function, you can also just write it as X squared squared. So X to the fourth and then X plus two that quantity squared. Um So as you're trying to simplify this, which I don't know if that's really advantageous to do that, but what you could actually do is factor male, do this in red. Um That you could factor out one of these exes. Uh So the reason why that would be beneficial is then I could cancel out uh an ex from all of those pieces. So now in the denominator them down to X cubed. Just think I divide everything by X. Um And then the other thing is you can distribute for X to each term negative one to each term, distribute X to each term, same thing over here as a distribute, distribute, distribute and also distribute them negative to everything. And at the end, when you combine like terms, uh you'll end up with Negative -2 x cubed Plus two x squared plus five X plus four, all over X to the 3rd power, X plus two to the second power, and nothing else can be done. So this is a good answer. Will circle it and move on.

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