Modeling chemical reactions and complicated molecular systems has been proposed as the“killer application”of a future quantumcomputer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometryoptimization, transition state searches, predictions of the response to an applied electric or magneticfield, and molecular dynamicssimulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based onquantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds andapproximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimizationon an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0:014 Å ofthe full configuration interaction value. Within the same experiment, we estimate the polarizability of the H2molecule,findingagreement at the equilibrium bond length to within 0:06 a.u. (2%relative error)