Silicon drift detectors have been successfully employed in both soft and hard X-ray spectroscopy. The response function to incident radiation at soft X-ray levels has been well studied and modeled, but less research has been published on response functions for these detectors to hard X-ray input spectra above 20 keV. When used with hard X-ray sources a significant low energy, non-peak response exists which can adversely affect detection limits for lighter elements in, for example, X-ray fluorescence spectroscopy. We present a numerical model that explains the non-peak response function of silicon drift detectors to hard X-rays based on incoherent Compton scattering within the detector volume. Experimental results are presented and numerically compared to model results.