Suppose that $f(x,y,\ldots)$ is a multivariable non-constant rational function. When $f/\frac{\partial f}{\partial x}$ does not depend upon $x$?
My attempt: The question equivalent to when $$ \frac{\partial}{\partial x}\left(f/\frac{\partial f}{\partial x}\right)\equiv0 $$ that is $$ \left(\frac{\partial f}{\partial x}\right)^2\equiv f\cdot\frac{\partial^2 f}{\partial x^2} $$ Is this equation solvable? I know that if $f$ is a single variable function, then the solution of $(f')^2=ff''$ is $f=Ae^{Bx}$, so $f$ is not rational.