High-gain observers have been used in non-linear control to estimate derivatives of the output. In this paper, we study discrete-time implementation of high-gain observers and their use as numerical differentiators, in noise-free as well as noisy measurements. We show that discretization using the bilinear transformation method gives better results than other discretization methods. We also show that many of the available numerical differentiators are special cases of the bilinear discrete-time equivalents of full-order or reduced-order high-gain observers.