THIS QUESTION WAS MIGRATED PROM PHYSICS SE, SO THE ANSWERS MAY NOT BE APPLICABLE TO MATHEMATICS
$\left(\frac{\partial f}{\partial x}\right)_y$ means to differentiate $f(x,y)$ with respect to $x$, while holding $y$ constant. Would it be possible to write this simply as $\frac{\partial f(x)}{\partial x}$ instead, or are there any other alternatives in use?
Writing $\frac{\partial f(x)}{\partial x}$ is misleading because it suggests that $f$ does not depend on $y$. This can be harmful because in the reader's mind, this means $\frac{\partial f}{\partial y}=0$. This is why @Farcher's suggestion to write $\frac{\partial f(x,y)}{\partial x}$ is a better option. but once again, this is likely to imply that $f$ does not depend on anything else ($z$...). So unless you are sure to list all possible variables, I would recommend to stick to the standard $\left(\frac{\partial f}{\partial x}\right)_y$.