The math below was written in a thread I needed help in.
Define $$F(x,y,z)=x^2y+xz^2-5.$$
Hence $$\frac{\partial x}{\partial y}=-\frac{F’_y}{F’_x}=-\frac{x^2}{2xy+z^2}\,,$$$$\frac{\partial x}{\partial z}=-\frac{F’_z}{F’_x}=-\frac{2xz}{2xy+z^2}\,.$$
It seems more intuitive to me that $\frac{\partial x}{\partial y} = \frac{\frac{\partial F}{\partial y}}{\frac{\partial F}{\partial x}} = \frac{F'_y}{F'_x}$
I don't see where the negative sign comes from!