show that $f'(x)=0$ if and only if there is $g(x) \in F[x]$ such that $f(x)=g(x)^p$, for characteristic $p$

Question

Let $F$ be a finite field of characteristic $p$.

What i want to prove is following statement.

Let $f(x) = F[x]$, show that $f'(x)=0$ if and only if there is $g(x) \in F[x]$ such that $f(x)=g(x)^p$.

One side is clear. $f(x)=g(x)^p$, then $f'(x) = p g(x)^{p-1} g'(x)$ and since $p$ is char, it clearly zero. But how about the other side?