Find $\frac{\partial z}{\partial x}$ of equation $y z-\ln z=x+y$

Question

Find $\frac{\partial z}{\partial x}$ if the equation $y z-\ln z=x+y$ defines $z$ as a function of two independent variables $x$ and $y$ and the partial derivative exists? How to solve this problem? I do not know what to do after taking derivative with respect $z$ !

Answer 1

Consider the implicit function $$F=yz-\log(z)-x-y=0$$ Differentiate $$F'_z=y-\frac 1z$$ $$F'_y=z-1$$ Now, use the implicit function theorem $$\frac{dz}{dy}=-\frac{F'_y}{F'_z}$$