There are different types of the shape derivatives for the potential energies in Poisson equations with mixed boundary value problems, that is, derivatives with respect to the small change of boundaries and of the joint points of different boundary conditions. The systematic treatment in the shape derivatives is done using Generalized J-integral method proposed by author. In the shape derivative with respect to the joint points, the movements of singularities appear. This means that the numerical calculation is not easy in the shape derivatives with mixed boundary condition. It is shown that Generalized J-integral method is useful for the shape derivatives in theoretical and numerical studies of mixed boundary value problems.