An extension $E/F$ is separable iff every element $\alpha \in E$ is separable, meaning that its irreducible polynomial $f \in F[x]$ (a monic generator of the kernel of $F[x] \to E: x \mapsto \alpha$) has no multiple roots. Of course $f$ has a multiple root only if its derivative satisfies $f'(\alpha) = 0$, which means $f' \in (f)$: by degree considerations this can happen only if $f'$ is the zero polynomial. Notice this cannot happen in characteristic zero.