I am trying to understand John Lee's derivation of equation 3.11, which shows how tangent basis vectors on a smooth manifold are transformed under a change of coordinates.
He starts with the following setup: let $M^n$ be a smooth manifold and $(U,\varphi)$ and $(V,\psi)$ be charts with nonempty intersection and $p\in U\cap V$. Then let $x^i$ and $\tilde{x}^i$ be the components of the functions $\varphi:U\to \mathbb{R}^n$ and $\psi:V\to \mathbb{R}^n$ so that $\forall q\in U\cap V,\: \varphi(q) = (x^1(q),\ldots,x^n(q))$ and $\psi(q) = (\tilde{x}^1(q),\ldots,\tilde{x}^n(q))$.
The next step is what confuses me. John Lee says with "abuse of notation" that $\psi\circ \varphi^{-1}(x)$ can be written as $(\tilde{x}^1(x),\ldots,\tilde{x}^n(x))$, where the $\tilde{x}^i$ are coordinate functions defined on an open subset of $M$ identified with an open subset of $\mathbb{R}^n$ and $x$ is a point in $\varphi(U\cap V)$.
Is the following interpretation of the above statement correct? Let $x\in \varphi(U\cap V)$. Then $\psi\circ \varphi^{-1}(x) = (\tilde{x}^1(\varphi^{-1}(x)),\ldots,\tilde{x}^n(\varphi^{-1}(x))) = ((\tilde{x}^1\circ \varphi^{-1})(x),\ldots,(\tilde{x}^n\circ \varphi^{-1})(x))$. Then with abused notation, $\tilde{x}^i$ (whose domain is "identified with" a subset of $\mathbb{R}^n$) really refers to the function $\tilde{x}^i\circ \varphi^{-1}: \varphi(U\cap V)\to \mathbb{R}$.
After this, he writes the following equation: $$d(\psi\circ\varphi^{-1})_{\varphi(p)}\left(\frac{\partial}{\partial x^i}\Bigr|_{\varphi(p)}\right) = \frac{\partial \tilde{x}^j}{\partial x^i}(\varphi(p)) \frac{\partial}{\partial \tilde{x}^j}\Bigr|_{\psi(p)}$$
I think that the above follows from the formula for the differential defined in Euclidean space (3.9) plus the interpretation of $\tilde{x}^i$ as a map between Euclidean spaces. Is this correct?
Thanks in advance