Big degrees

Algebra Level pending

Let \(p(x)\) be a nonzero polynomial of degree less than \(2014\) that has no nonconstant factor in common with \(x^3-x\). Let \[ \frac{d^{2014}}{dx^{2014}}\left(\frac{p(x)}{x^3-x}\right)=\frac{s(x)}{k(x)}, \] where \(\frac{s(x)}{k(x)}\) is in lowest terms. Find the smallest possible degree of \(s(x)\).

(Based on 1992 Putnam Problem B4.)

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