Problem: given a function $F:\mathbb{R}^2\mapsto\mathbb{R}$ of class $C^2$, with $F(0,0)=0,\nabla F=(2,3)$, shown that a surface $F(x+2y+3z-1,x^3+y^2-z^2)=0$ can be given locally at $(-2,3,-1)$ as graph of $z\stackrel{c^2}{=}z(x,y)$. then compute $\frac{\partial z}{\partial y}\bigg|_{(x,y)=(-2,3)}$, and knowing that $\frac{\partial^2F}{\partial x^2}\bigg|_{(x,y)=(0,0)}=3$,$\frac{\partial^2F}{\partial x\partial y}\bigg|_{(x,y)=(0,0)}=-1$ and $\frac{\partial^2F}{\partial y^2}\bigg|_{(x,y)=(0,0)}=5$, compute $\frac{\partial^2z}{\partial y\partial x}\bigg|_{(x,y)=(-2,3)}$
I dont known of how to solve this, i think that will need something like chain rule.