Let $M, N$ be smooth manifolds, and let $f: M \to N$ be a smooth mapping. I now want to prove:
If $(U, \phi = (x_1, ..., x_m))$ and $(V, \psi = (y_1, ..., y_n))$ are charts for $M$ and $N$ respectively, with $p \in U \subseteq M$ and $f(p) \in V \subseteq N$, then for the function $F$ (that we call the coordinate representation of $f$), given by $F = \psi \circ f \circ \phi^{-1}$, the following formula holds:
$$d f_p \left(\frac{\partial}{\partial x_i} \mid_p\right) = \sum_{j=1}^n \frac{\partial F_j}{\partial x_i} (\phi(p)) \cdot \frac{\partial}{\partial y_j} \mid_{f(p)}$$
I must admit that I don't have much of an idea know how to approach this. I probably have to work with the fact that I have the derivative of $f$ on the left: per definition, $f$ is differentiable in $p$ if for every chart $\xi$ of a neighborhood of $p$, we have that $f \circ \xi$ is a differentiable function in $\mathbb{R}$. Now in my case, I have to charts, one for a neighborhood of $p$, another one for a neighborhood of $f(p)$, and a function $F$ that represents $f$ but goes from $\mathbb{R}^m$ to $\mathbb{R}^n$. I don't really know where to go from there.