Relation between $\frac{\partial}{\partial x_i}$ and $\frac{\partial}{\partial y_i}$

Question

Let $M$ be a real manifold of dimension n. Let $U, V$ be two chart domains on M such that $U \cap V$ is nonempty. Let $\varphi:U \rightarrow \mathbb{R}^n$, $\varphi = (x_1, \ldots, x_n)$ and $\psi:V \rightarrow \mathbb{R}^n$, $\psi = (y_1, \ldots, y_n)$ be the corresponding charts. I know that the vector fields $\frac{\partial}{\partial x_i}$ form a basis of the $C^{\infty}(U)$-module $\Gamma(TU)$, so on $U \cap V$ there exists some functions $f_{ij}$ such that $$\frac{\partial}{\partial y_i} = \sum_{j=1}^n f_{ij} \frac{\partial}{\partial x_j}$$ My question is: How can I find these functions $f_{ij}$ ? I'm pretty new in this stuff. Thanks a lot!