$z(x,y)$ is a function defined implicitly by the equation $$F(x,y,z)=5x+2y+5z+5\cos(5z)+2=0$$
I'm trying to find $\frac{\partial^2z}{\partial x \partial y}$ at the point $(\frac{\pi}{5},\frac{3}{2},\frac{\pi}{5})$.
As far as I can tell I need to use the implicit function theorem, and I'm able to find both $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$, but the second derivative becomes 0, which is incorrect.
How can I find $\frac{\partial^2z}{\partial x \partial y}$ ?
Note that $$\frac{\partial^2F}{\partial x \partial y} =5\frac{\partial^2z}{\partial x \partial y}+5\frac{\partial}{\partial x }\left(-5\sin(5z)\frac{\partial z}{\partial y}\right) =5\frac{\partial^2z}{\partial x \partial y}-125\cos(5z)\frac{\partial z}{\partial x}\cdot\frac{\partial z}{\partial y}-25\sin(5z)\frac{\partial^2z}{\partial x \partial y}=0.$$ Letting $(x,y,z)=(\frac{\pi}{5},\frac{3}{2},\frac{\pi}{5})$, we find $$5\frac{\partial^2z}{\partial x \partial y}+125\frac{\partial z}{\partial x}\cdot\frac{\partial z}{\partial y}=0.$$ Hence knowing $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ at $(\frac{\pi}{5},\frac{3}{2})$, you should be able to find $\frac{\partial^2z}{\partial x \partial y}$.