I found this interesting result in the work of D. B. Fuchs and T. Evans(2002), and also I have proved this result using Bell polynomials. However, I suppose that there should be more easier solution. Any ideas?
Let $D=f\frac{d}{dx}$ be an operator and $f$ be a function of $x$, so
$D^{p-2}(f)+\frac{d^{p-2}}{dx^{p-2}}\left(f^{p-1}(x)\right)\equiv 0\;(mod\; p)$