19.i.1 Generalized derivatives

condition: Show that the following mapping is differentiable and find its strong derivative: \[ f: \mathbb{R}_{2}^{3} \rightarrow \mathbb{R}_{2}^{3}, \quad\left\{\begin{array}{c} y_{1}=2 x_{1}^{2}+x_{1} x_{2}^{2}-x_{1} x_{3} \\ y_{2}=x_{1} x_{2}-3 x_{2}^{2}+x_{1} x_{3}^{2} \\ y_{3}=2 x_{1}^{2} x_{3}^{2}-4 x_{2}^{2} x_{3} \end{array}\right. \] at the point \( (1,1,1) \).