The setup is as follows: $u=f(x-ct)$ where $f$ is some arbitrary function and $x-ct$ is its argument.
Differentiation of $u$ yields: $$du = f'dx-cf'dt$$
It's completely unclear to me with respect to what variable the expression above is being differentiated.
If $x$ and $t$ in turn were functions of the same variable then this would make more sense to me, but they are not ($x$ is space and $t$ is time).
I can provide more context if needed - it's from an acoustics textbook.