Approximate analytical expressions for the values of the derivatives of the radial distribution function at contact in a hard-sphere fluid are derived. These expressions are obtained within a rational function approximation (RFA) method recently developed. The results are compared with those based on the Percus-Yevick approximation, as well as with those obtained from expressions for the cavity function inside the core or through computer simulations. While the qualitative dependence on the packing fraction is similar for the values of the first two derivatives at contact in all cases, tile one corresponding to the third derivative is definitely different in our approach from that previously reported. The integral moments of the total correlation function as given by the RFA method are also computed. (C) 1997 American Institute of Physics.