The $100$th derivative of $(x^2 + 1)/(x^3 - x)$
Solution 1:
We have a partial fraction decomposition $$ \frac{x^2+1}{x^3-x}=\frac{-1}{x}+\frac{1}{x+1}+\frac{1}{x-1} $$
It follows that $$ \left(\frac{d}{dx}\right)^{100}\frac{x^2+1}{x^3-x}=\frac{-100!}{x^{101}}+\frac{100!}{(x+1)^{101}}+\frac{100!}{(x-1)^{101}} $$