Many applications require the minimization of a smooth function f: Rn → R whose evaluation requires the solution of a system of nonlinear equations. This system represents a numerical simulation that must be run to evaluate f. We refer to this system of nonlinear equations as an implicit constraint. In theory f can be minimized using the steepest descent method or Newton-type methods for unconstrained minimization. However, for the practical application of derivative based methods for the minimization of f one has to deal with many interesting issues that arise out of the presence of the system of nonlinear equations that must be solved to evaluate f. This article studies some of these issues, ranging from sensitivity and adjoint techniques for derivative computation to implementation issues in Newton-type methods. A discretized optimal control problem governed by the unsteady Burgers equation is used to illustrate the ideas.