On Page 171 of Rick Miranda's Algebraic Curves and Riemann Surfaces, he gave Theorem 1.9 without a proof.
Definition 1.1. Let $S$ be a set of meromorphic functions on a compact Riemann surface $X$. We say that $S$ separates points of $X$ if for every pair of distinct points $p$ and $q$ in $X$ there is a meromorphic function $f \in S$ such that $f(p) \ne f(q)$. We say that $S$ separates tangents of $X$ if for every point $p \in X$ there is a meromorphic function $f \in S$ which has multiplicity one at $p$. A compact Riemann surface $X$ is an algebraic curve if the field $\mathcal{M}(X)$ of global meromorphic functions separates the points and tangents of $X$.
The basic analytic result from which we will proceed is the following.
Theorem 1.9. Every compact Riemann surface is an algebraic curve.
By reading Otto Forster's text I found the proof of separating points part (Corollary 14.13, Page 116):
14.13. Corollary. Suppose $X$ is a compact Riemann surface and $a_1, \dots, a_n$ are distinct points on $X$. Then for any given complex numbers $c_1, \dots, c_n \in \mathbb{C}$, there exists a meromorphic function $f \in \mathscr{M}(X)$ such that $f(a_i) = c_i$ for $i = 1, \dots, n$.
but can anyone tell me how to prove the separating tangents part?