This paper presents D-optimal experimental designs for a variety of non-linear models which depend on an arbitrary number of covariates but assume a positive prior mean and a Fisher information matrix satisfying particular properties. It is argued that these optimal designs can be regarded as a first-order approximation of the asymptotic increase of Shannon information. The efficiency of this approximation is compared in some examples, which show how the results can be further used to compute the Bayesian optimal design, when the approximate solution is not accurate enough.