1. Hard derivative

Differentiate the fnction
$\displaystyle f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^{3} - 2x^{-2})^{-2}}$

Differentiate the fnction
$\displaystyle f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^{3} - 2x^{-2})^{-2}}$
Someone wants to give you an ulcer, don't they?

Simplify first:

$\displaystyle f(x) = \frac{(x^{-1} + x^{2})^{-1}}{(x^3 - 2x^{-2})^{-2}}$

$\displaystyle f(x) = \frac{(x^3 - 2x^{-2})^2}{x^{-1} + x^2}$

$\displaystyle f(x) = \frac{\left ( x^3 - \frac{2}{x^2} \right )^2}{\frac{1}{x} + x^2}$

I'm going to expand the numerator so my next step will be a bit clearer:
$\displaystyle f(x) = \frac{x^6 - \frac{4x^3}{x^2} + \frac{4}{x^4}}{\frac{1}{x} + x^2}$

$\displaystyle f(x) = \frac{x^6 - 4x + \frac{4}{x^4}}{\frac{1}{x} + x^2}$

Now mulitply the numerator and denominator by x^4. We would normally need to be wary of doing this, but the domain of the original function excludes x = 0 anyway in this case.
$\displaystyle f(x) = \frac{x^{10} - 4x^5 + 4}{x^3 + x^6}$

You should be able to take the derivative now.

-Dan