$\newcommand{\dd}{\partial}$Let $f(u,v)$ be a smooth bivariate polynomial over $\mathbb{C}$. Let $X=\{(u,v)\in\mathbb{C}^2\vert f(u,v)=0\}$ be the corresponding affine smooth plane curve. In this answer by Andrew D. Hwang it is claimed that on $X$ we have
$$ \frac{\dd f}{\dd u}\, du + \frac{\dd f}{\dd v}\, dv = 0. $$
Why is that true? It seems to somehow come from "differentiating" the defining equation of $X$. I am struggling to derive the details. My definition of differentials is taken from Forster's "Lectures on Riemann Surfaces".
EDIT: In Forster's book the differential is defined as follows. Let $f$ be a smooth function on an open subset $U$ of a Riemann surface. Let $a\in U$. Then the differential of $f$ at $a$ is defined as $d_af:=(f-f(a))\operatorname{mod} \mathsf{m_a}^2 \in T^{(1)}_a$. Here $T^{(1)}_a:=\frac{\mathsf{m_a}}{\mathsf{m_a}^2}$ denotes the cotangent space at $a$, where $\frac{\mathsf{m_a}}{\mathsf{m_a}^2}$ is the quotient vector space of the vector space $\mathsf{m_a}$ of function germs at $a$ that vanish at $a$ and the vector subspace $\mathsf{m_a}^2\subset \mathsf{m_a}$ of function germs at $a$ that vanish at $a$ to the second order.