I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w) = 0$ and $H(x^{*}, y^{*},z) = 0$. Using the implicit function theorem, I can compute $\frac{\partial x^{*}(w,z)}{\partial w}$, $\frac{\partial x^{*}(w,z)}{\partial z}$, $\frac{\partial y^{*}(w,z)}{\partial w}$ and $\frac{\partial y^{*}(w,z)}{\partial z}$.For instance, for $\frac{\partial x^{*}(w,z)}{\partial w}$ I have that:
\begin{equation} \frac{\partial x^{*}(w,z)}{\partial w} = \frac{-\frac{\partial F}{\partial w}*\frac{\partial H}{\partial y}}{\frac{\partial F}{\partial x}*\frac{\partial H}{\partial y}-\frac{\partial F}{\partial y}*\frac{\partial H}{\partial x}} \end{equation}
Now I am interested in $\frac{\partial^2 x^{*}(w,z)}{\partial w \partial z}$. How can I approach this?
Thanks!!