We have a function $F(x, y, z) = 0$ and interested in $\frac{ \partial^2 x }{ \partial y \partial z} $. Would simply using partial differentiation after using implicit function theorem work? More precisely;
$\frac {\partial (-F_y / F_x)} { \partial z} = \frac{\partial (\partial x / \partial y)} {\partial z}=\frac {\partial^2 x} {\partial y \partial z} $
Is this correct?
Edit: Let me clarify with a quick example.
Suppose we have $ x^2 + y^2z + z^3 = 0 $. Is the following correct?
$ F_x = 2x $
$ F_y = 2yz $
$ \frac{\partial x}{\partial y} = \frac{-F_y}{F_x}= \frac{-yz}{x} $
$ \frac {\partial^2 x} {\partial y \partial z} = \frac{-y}{x} $