The effective Hamiltonian of our field theoretical model yields the linearized Debye-H\"uckel theory in the bulk. Here we consider it for a confined geometry of a slab. The differential capacitance is computed using two ways. The first relates it to the charge-charge correlation function by calculating the response to charging of the slab walls. The other relates it to the fluctuation of charge in one half of the slab. Since the model is exactly soluble in terms of the diagonal representation we can verify the fact that the two ways are consistent with each other and with the Gouy-Chapman differential capacitance in the limit of a thick slab. The interest of this derivation is to relate a measurable quantity to the structural information contained in the charge-charge correlation function.