How to Correctly Take the Derivative of a Fraction with a Quadratic Denominator

[PhizKid]

Homework Statement

Derivative of $$\frac{y - 1}{y^2 - y + 1}$$

Homework Equations

The Attempt at a Solution

The solution is $$\frac{y^2 - 2y}{(y^2 - y + 1)^2}$$ but in my work, the answer will have something to the 4th power on the top which will be impossible to cancel out. What have I done wrong?

Edit: Never mind, I see my mistake

 

[Scootertaj]

Remember: $$\frac{d}{dx}\frac{f(x)}{g(x)} = \frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}$$

 

[PhizKid]

Yes but I don't like to work with the quotient rule. I should be getting the same answer using the product rule anyway, right?

Edit: Never mind, I see my mistake

 

[Scootertaj]

That's fine.

Then, remember: $$-(y^2-y+1)^{-2} = -\frac{1}{(y^2-y+1)^2}$$
(An expression to the -2 power doesn't equal 1/sqrt(expression))