In order to obtain some theorems of local existence and uniqueness for a non-linear differential problem and investigate about the significance of its linearization, we study the differentiability of a non-linear operator acting among Banach spaces for which the (formally) linearized operator is an isomorphism. We give a definition of “admissible linearization” with respect to a pair of Banach spaces. We consider a differential operator $T$ of the form $Tu = D^{*} a(x,Du)$, [where $Du$ is the gradient of $u$ and $D^{*}$ is the formal transpose of $D$] and by using the results of [7] we give examples of admissible and non-admissible linearizations for the Dirichlet problem.