Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. Apparently the converse is also true (at least for finite extensions). It is set right at the end of this answer as an "easy exercise", but I'm really struggling to show it. I suppose one way of possibly doing it is assuming the extension is inseparable and exhibiting a nontrivial $K$-derivation $L\to L$ to obtain
$$ \dim_L \Omega_{L/K}=\dim_L \Omega_{L/K}^\vee = \dim_L\text{Der}_K (L,L)>0,$$ but I can't come up with one. Can anyone help?