Some details on the tangent space of a manifold

Question

Let $x$ be a point of some smooth manifold $(M,\mathcal U)$ of dimension $n$. Let $I_x$ be the set of all $U\in \mathcal U$ containing $x$. Define the relation "$\sim_x$" on $I_x\times \mathbb R^n$ by $$ (U,u)\sim_x (V,v) \Longleftrightarrow u= (d\phi_{UV})_{\phi_V(x)}(v)$$ where $\phi_{UV}$ is the coordinate change function. I need some explanations to understand this relation -in particular, what is $(d\phi_{UV})_{\phi_V(x)}(v)$- and why it is an equivalence relation?