Let $f\left(x,y\right)$ be an elementary function of $x$ and $y$.
Examples:
$$4x^{3}-ax-b-y^{2}$$ $$a^{x}-y^{2}+x^{3}y-1$$ $$x^{3}-3xy+y^{3}$$ $$\frac{\sin^{2}x}{x^{2}+\sin^{2}y}-\frac{1}{3}$$
Let $C$ denote the zero-locus of $f\left(x,y\right)$ over $\mathbb{C}$ (affine and/or projective—whatever makes sense for the given $f\left(x,y\right)$).
I know that any holomorphic or meromorphic differential 1-form defined on $C$ can be written as $g\left(x,y\right)dx$ for some expression $g\left(x,y\right)$.
My question is, given $f\left(x,y\right)$, $C$, and $g\left(x,y\right)$ as described above, what computation do I perform to check if $g\left(x,y\right)dx$ defines a holomorphic or meromorphic differential 1-form on $C$?
Heads up: the terms "chart" and "transition map" give me a headache (they're too abstract), so just speak of "formulas"/"expressions" and what I need to do with them.
Thanks!