Why only the numerator is derived? Why the derivative of $y = \frac{x^5}{a+b}-\frac{x^2}{a-b}-x$ is solved by deriving just the numerators?
The solution is $\frac{dy}{dx}=\frac{5x^4}{a-b}-\frac{2x}{a-b}-1$.

If the denominator is constant, it's really easiest to apply the constant multiple rule.  That is, $\frac{d}{dx}\frac{f(x)}{c}=\frac{d}{dx}\frac{1}{c}f(x)=\frac1c f'(x)=\frac{f'(x)}{c}$
If you insist on using the quotient rule, you get the same answer, as shown in marwalix's post.
