Reading the following paragraphs from Peter Peterson's Riemannian geometry
Let $c : I \to M$ be a curve, where $M$ is a manifold. We define $\dot c(t) = \frac{dx^{i}}{dt} \partial_{x_{i}}$. He justifies that this is independent of the coordinate by saying that this is simply because the chain rule tells us that $\dot x(t) = dx^{i}(\dot c(t))$. I am not entirely sure what he means here because in order to show coordinate independence shouldn't you just take another coordinate and prove that the definition yields the same result? Doing this is not hard in this case but I want to understand the book's reasoning.