Let $\Sigma^m \subset (M^{m+1},g)$ be a two-sided minimal hypersurface with unit normal $\nu$, second fundamental form $A$, and induced measure $d\mathcal H^m$. Let $F:(-\varepsilon,\varepsilon)\times\Sigma\to M$ be a compactly supported normal variation, meaning that there is a compact set $K\subset\Sigma$ such that $F_t(p)=p$ for every $p\in\Sigma\setminus K$ and every sufficiently small $|t|$, and such that its initial variational vector field is $\left.\partial_tF_t\right|_{t=0}=\phi\nu$ for some $\phi \in C_c^\infty(\Sigma)$. Then, for every relatively compact smooth domain $\Omega\subset\Sigma$ containing $K$ in its interior,

\begin{align*} \frac{d^2}{dt^2}\bigg|_{t=0}\mathcal H^m(F_t(\Omega)) =\int_\Sigma \left(|\nabla^\Sigma \phi|^2-\left(|A|^2+\operatorname{Ric}_M(\nu,\nu)\right)\phi^2\right)\,d\mathcal H^m. \end{align*}

Equivalently, the same formula gives the second derivative of the area change of the compactly supported variation, since $\mathcal H^m(F_t(\Omega))-\mathcal H^m(\Omega)$ is independent of the chosen such $\Omega$.