We investigate the Derjaguin approximation by explicitly determining the interactions between two spherical colloids using density functional theory solved in cylindrical coordinates. The colloids are composed of close-packed Lennard-Jones particles. The solvent particles are also modeled via Lennard-Jones interactions. Cross interactions are assumed to follow the commonly used Lorentz-Berthelot (LB) mixing rule. We demonstrate that this system may display a net repulsive interaction across it substantial separation range. This contradicts the Hamaker-Lifshitz theory, which predicts attractions between identical polarizable particles immersed in a polarizable medium. The source of this repulsion is traced to the LB mixing rule. Surprisingly, we also observe nonmonotonic convergences to the Derjaguin limit. This behavior is best understood by decomposing the total interaction between the colloids into separate contributions. With increasing colloid size, each of these contributions approach the Derjaguin limit in a monotonic manner, but their different rates of convergence mean that their sum may display nomnonotonic behavior.