Differential of Verschiebung morphism

Let $G=\operatorname{Spec}(R)$ be a finite flat commutative group scheme over $S=\operatorname{Spec}(A)$ of characteristic $p>0$. Suppose the $p$-Lie algebra $\operatorname{Lie}(G/S)$ is locally free. I would like to know the $p$-mapping on $\operatorname{Lie}(G/S)$ coincides with differential of the Verschiebung morphism $V_G: G^{(p/S)}\rightarrow G$.

That is, if $D$ is an invariant derivation of $\operatorname{Der}_S(A^{(p/S)}, S)$, then $dV_G(D)=D\circ V$ should be $D^p$.

It is embarrassing that I do not know how to show this by hand. Any help will be appreciated.