If $x^y y^x z^z=c$, then find $\frac {\partial^2z}{\partial x \,\partial y}$ at $x=y=z$. If $x^y y^x z^z=c$, then find $\dfrac {\partial^2z}{\partial x \,\partial y}$ at $x=y=z$.
I tried taking $\log$ but that doesn't help.
Any hints will be appreciated. Thanks.
This is what I have tried:



Hint: $x^y y^x z^z=c$ gives 
$$F=y\ln x+x\ln y+z\ln z=0$$
with $\dfrac{\partial z}{\partial y}=z_y=-\dfrac{F_y}{F_z}$ and then
$$\dfrac{\partial}{\partial x}\dfrac{\partial z}{\partial y}=-\dfrac{x+y}{xy(1+\ln z)}$$
