Gauvin and Tolle have obtained bounds on the directional derivative limit quotient of the optimal value function for mathematical programs containing a right-hand side perturbation. In this paper, we extend the results of Gauvin and Tolle to the general mathematical program in which a parameter appears arbitrarily in the constraints and in the objective function. An implicit function theorem is applied to transform the general mathematical program to a locally equivalent inequality constrained program, and, under conditions used by Gauvin and Tolle, their upper and lower bounds on the optimal value function directional derivative limit quotient are shown to pertain to this reduced program. These bounds are then shown to apply in programs having both inequality and equality constraints where a parameter may appear anywhere in the program. (Author)