Gaussian process (GP) models provide non-parametric methods to fit continuous curves observed with noise. Motivated by our investigation of dark energy, we develop a GP-based inverse method that allows for the direct estimation of the derivative of a curve. In principle, a GP model may be fit to the data directly, with the derivatives obtained by means of differentiation of the correlation function. However, it is known that this approach can be inadequate due to loss of information when differentiating. We present a new method of obtaining the derivative process by viewing this procedure as an inverse problem. We use the properties of a GP to obtain a computationally efficient fit. We illustrate our method with simulated data as well as apply it to our cosmological application.