Show that $x\frac{\partial g}{\partial x}+y\frac{\partial g}{\partial y}=0$ when g$\begin{pmatrix}x \\ y\end{pmatrix}$ = $f({\frac{x}{y}})$

Question

Can I compute $\frac{\partial g}{\partial x}$ to be $\frac{df}{dx}$? The reason is I think $\mathit{f}$ is a one variable function. So $$\frac{\partial g}{\partial x}=\frac{df}{dx}=\frac{1}{y}f'$$ and $$\frac{\partial g}{\partial x}=\frac{df}{dy} = \frac{-x}{y^2}f'$$