This paper presents a superconvergence analysis for the Shortley-Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h(1.5))-order superconvergence of the solution derivatives in a discrete H-1 norm is obtained. Finally, numerical experiments art provided to support the theoretical convergence rate obtained.