We analyze the nonlinear voltage dependence of electrochemical capacitance for nanoscale conductors. This voltage dependence is due to the finite density of states of the conductors. Within Hartree theory we derive an exact expression for the electrochemical capacitance-voltage curve for a parallel plate system. The result suggests a quantum scanning capacitance microscopy at the nanoscale: by inverting the capacitance-voltage expression one is able to deduce the local spectral function of the nanoscale conductor.