Partial derivates of smooth functions defined on a manifold

Question

Let $M$ be a manifold and $f:M\longrightarrow \mathbb{R}$ a smooth map. Given a chart $(\mathbb{U},\phi=(x^1,x^2,\ldots,x^n))$ on $p\in M$, I define $\dfrac{\partial f}{\partial x^k}|_p:=\dfrac{\partial (f\circ \phi^{-1} )}{\partial r^k}\vert_{\phi(p)}$, where $(r^1,r^2,\ldots,r^n)$ are the standard coordinates of $\mathbb{R}^n$. I think that the value of the derivate at $p$ is indipendent from the choice of the chart $(\mathbb{U},\phi)$. How I can prove this assertion, if it is true?