The discrete-time version of Levant's arbitrary order robust exact differentiator, which is a forward Euler discretized version of the continuous-time algorithm enhanced by linear higher order terms, is extended by taking into account also nonlinear higher order terms. The resulting differentiator preserves the asymptotic accuracies with respect to sampling and noise known from the continuous-time algorithm. It is demonstrated in a simulation example and by differentiating a measured signal that the nonlinear higher order terms allow reducing the high-frequency switching amplitude whenever the (n+ 1)th derivative of the signal to be differentiated vanishes, leading to an improvement in the precision.