Let $p(x)$ be a polynomial of degree strictly less than 100 and such that it does not have $x^{3} - x$ as a factor. If
$$\frac{d^{100}}{dx^{100}} \left( \frac{p(x)}{x^{3} - x} \right) = \frac{f(x)}{g(x)}$$
for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$. Here $\frac{d^{100}}{dx^{100}}$ means taking the 100th derivative.