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 $f(x)$.

Let's think step by step and output the final answer within \boxed{}.
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