So in Lee's smooth manifolds text, he says that a function $f: M \to \mathbf{R}$ on a manifold $M$ is smooth if for each $p \in M$ and a chart $(U, \phi)$ containing $p$, we have that $f \circ \phi^{-1}: \phi(U) \subset \mathbf{R}^n \to \mathbf{R}$ is smooth as a function between Euclidean spaces.
Now from the conventions in the textbook we have that the chart $(U, \phi)$ where $\phi(p)= (x^1(p), ..., x^n(p))$ puts coordinates on $U$. The inverse $\phi^{-1}$ is a local parameterization of $U$.
I'm familiar with the fact that the differential of a map between Euclidean spaces is just the Jacobian, but I don't understand why he is differentiating with respect to the coordinate charts in the formula for the differential i.e. $dF_p(\frac{\partial}{\partial x^i}\Big|_p) = \frac{\partial F^j}{\partial x^i}(p) \frac{\partial}{\partial y^j}\Big|_{F(p)}$.
I initially told myself that for Euclidean spaces the coordinate charts and local parameterizations were equal so doing this did not matter, but he uses the same reasoning for the change of coordinates for tangent vectors and I'm unsure how to justify why he is differentiating with respect to the coordinates and not local parameterizations.
Any help understanding this would be appreciated.