On p.30 of Silverman's The Arithmetic of Elliptic curves, the notion of a meromorphic differential form on a curve $C$ is defined as follows:
Definition. Let $C$ be a curve. The space of (meromorphic) differential forms on $C$, denoted by $\Omega_C$ , is the $\overline{K}(C)$-vector space generated by symbols of the form $dx$ for $x \in \overline{K}(C)$, subject to the usual relations:
(i) $d(x + y) = dx + dy$ for all $x, y \in \overline{K}(C)$.
(ii) $d(xy) = x dy + y dx$ for all $x, y ∈ \overline{K}(C)$.
(iii) $da = 0$ for all $a \in \overline{K}$
However as far as I can tell, meromorphic functions in their full generality don't really appear as coefficients here. For example, we're probably not going to have $e^x dx$ in there. If so, then why is this terminology used? Shouldn't they be called rational differential forms?