Uniformizer $x$ on a curve then $x - x(p)$ uniformizer at all but fin. many points $p$?

Question

Is it true that on a projective curve $C$, we have that if $x$ is a uniformizer at some point, then $x - x(p)$ is a uniformizer at $p$ for all but finitely many $p \in C$? If so can u give a proof? Thanks