I don't see much of a "problem statement". If I were asked to find the derivative of [itex]\frac{1}{x^2}- x[/itex], I would write it as [itex]x^{-2}- x[/itex], and then use the power rule: the derivative is [itex]-2x^{-3}-1= \frac{-2}{x^3}- 1[/itex].
If the problem itself requires that you use the definition of the derivative, say so!
What is [tex]\frac{1}{(x+h)^2}- \frac{1}{x^2}[/tex]?