For an implicit multifunction Phi (p) defined by the generally nonsmooth eq
uation F(x, p) = 0, contingent derivative formulas are derived, being simil
ar to the formula Phi' = -F-x(-1) F-p in the standard implicit function the
orem for smooth F and Phi. This will be applied to the projection X (p) = {
x \ There Existsy: (x, y) is an element of Phi (p)} of the solution set Phi
(p) of the system F(x, y, p) = 0 onto the x-space. In particular settings,
X (p) may be interpreted as stationary solution sets. We discuss in detail
the situation in which X(p) arises from the Karush-Kuhn-Tucker system of a
nonlinear program.