The author introduces a new set-valued derivative which is called the TP- derivative; it is weaker than the known contingent derivative. The author establishes results on the relationship between the contingent derivatives of \(F+K\) and \(F\) by using the TP-derivative, where \(F\) is a set-valued map and \(K\) is the positive cone of the considered space. The author considers also the relationship between the contingent derivative of the objective space image of the feasible region and the efficient set in the objective space for a parametrized multiobjective optimization problem.