This is motivated by a question in the Physics SE, but it's a math question not a physical one. For every function $G(x, t)$, can the function be written as a total derivative (wrt $t$) of some other function $F(x, t)$?
A simple counterexample would be fine; hopefully that would be simpler for a poor physicist to understand! A general proof/disproof would be great too.
If it would be easier to take a concrete example, if $G(x, t)$ is $x^2$ can this be written as a total derivative wrt $t$ of some function $F(x, t)$?