In this paper, some evaluations of the directional Rockafellar derivative of a function defined by $\varphi_F(x)=\mathrm{inf}\{f(x,y)\,|\, y\in F(x)\}$ via the directional Clarke and Rockafellar derivative of $f$ are shown. These evaluations are applied to find necessary optimality conditions for generalized optimal control problems governed by hemivariational inequalities.