If $F[F(x)] = f(s)$, then what is $F[\frac{d}{dx}F(x)]$?

Question

Suppose $F[F(x)] = f(s)$, then need to find $F[\frac{d}{dx}F(x)]$. Here $F$ is the Fourier transform of $f$ i.e. $$F(a) = \int_{-\infty}^{\infty}f(t)e^{-2\pi iat}dt.$$ Any idea how to proceed will be very helpful.