An approach to determining the derivative of a signal which is an exponentially-decaying function of time is presented. The case of a waveform sampled in time is treated. The method is applicable to functions which are complicated combinations of decaying exponentials. Robust behavior in the presence of additive noise is also obtained. It is shown that computing the derivative of the waveform by Fourier transforming it, using the DFT derivative theorem, and taking the inverse Fourier transform yields a much more accurate approximation to the derivative in the presence of noise than does computing the forward difference. A recursive discrete Fourier transform is then used, to permit computing the derivative of a sampled waveform as the samples are taken. A recursive DFT derivative theorem is derived. This is then applied to numerical simulations using a function which is the sum of three unevenly-weighted exponentials with different decay constants, and it is shown that useful results for the derivative are obtained even when the randomly-distributed additive noise has a peak value as high as 0.25 times the peak value of the original waveform. Extensions of the work to obtain a recursive chirp-Z transform and derivative theorems, and applications to resonance extraction, are also presented.