Let $K$ be a field of characteristic $p>0$, and let $f(x)$ be a polynomial in $K[x] - K$. We know $f' = 0$ implies that $f$ is not separable. A proof that works is that if it were separable then we could write it in distinct linear factors over an algebraic closure, and substituting a root of $f$ in $f'$ gives a nonzero answer, hence $f'$ could not have been zero.
But I don't intuitively see why $f' = 0$ implies inseparability.
An argument which makes it bit more intuitive for me is that if $f' = 0$ then $f = g(x^p)$ for some polynomial $g(x)$. So when we look in an algebraic closure, we find that if $g(x)$ has $(x-a)$ as a linear factor, then $f$ has $(x - a^{1/p})^p$ in its factorization, hence not separable.