The main emphasis in statistical work on the design of experiments has been on the comparison of treatments and, especially, on the estimation of treatment contrasts. An exception is in the study of response surface designs, that is in the design of experiments in which the treatments are identified by the values of quantitative variables and in which the expected response is a smooth function of these variables. The emphasis in developing these designs has been on the estimation of the height of the response surface at points in the factor space, i.e. on the estimation of absolute response rather than of differences in response. In particular, in the work of Box & Draper (1959) on the bias introduced by fitting a polynomial of too low a degree, the criterion used for selecting a design is the mean square error in estimating the height of the surface averaged over some region in the factor space. But even in such experiments, differences in response will often be of more importance than the absolute response. If differences at points close together in the factor space are involved, this implies that estimation of the local slope of the response surface is of interest. In what follows the choice of designs to estimate the slope of response surfaces is therefore considered. In designing these experiments, allowance is made for bias due to an inadequate model. Theexperimental errors are assumed to be independently and identically distributed. A first order polynomial is fitted to the results by standard least squares methods and the fitted coefficients used to estimate the slope of the response surface, either at a specified point or somewhere within a given region of interest. This region is not necessarily the same as the experimental region, that is the region in factor space in which it is feasible to perform trials. If the true response surface is quadratic, how should experiments be designed so that the estimate of the slope is as precise as possible? Designs for one factor are the subject of the following section. In ? 3 the theory of designs for any number of factors is developed and, in ? 4, applied to a two-dimensional example.