It is known that when we look for sufficient conditions of local extremum for G\^ateaux functionals (G.f) associated to Dirichlet problem of second order in $\R^2$, the (G.f) is not necesseraly Frechet differentiable. In this note, using a recent extension of Frechet Differentiability, (see \cite{s}), we obtain that the (G.f) is differentiable with respect to the new notion. Thus we can give sufficient conditions for obtaining local minimum.