  The directional differentiability of the solution map of obstacle type
quasi-variational inequalities (QVIs) with respect to perturbations on the
forcing term is studied. The classical result of Mignot is then extended to the
quasi-variational case under assumptions that allow multiple solutions of the
QVI. The proof involves selection procedures for the solution set and
represents the directional derivative as the limit of a monotonic sequence of
directional derivatives associated to specific variational inequalities.
Additionally, estimates on the coincidence set and several simplifications
under higher regularity are studied. The theory is illustrated by a detailed
study of an application to thermoforming comprising of modelling, analysis and
some numerical experiments.
