A derivative formula for the free energy function

We consider bond percolation on the ${\mathbf{Z}}^d$ lattice. Let $M_n$ be the number of open clusters in $B(n)=[-n,n{]}^d$. It is well known that $E_pM_n / (2n+1)^d$convergestothefreeenergyfunction$\kappa(p)$ at the zero field. In this paper, we show that ${\sigma }_p^2(M_n)\mathrm{/}(2n+1{)}^d$ converges to $-(p^2(1-p)+p(1-p{)}^2){\kappa }^{\mathrm{\prime }}(p)$.