Putnam 1992 B4

Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^3 - x$. Let $$\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)}$$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.