In John M. Lee's Introduction to Smooth Manifolds book, I have a small problem following through his proof of smoothness of vector fields.
In the last paragraph, he claims that $Yf$ is smooth since the component functions are smooth. However, how does he know that the function $x \mapsto \frac{\partial f }{\partial x^i}|_x $ is also smooth? It is clear how that would hold in the context of analysis on $\mathbb{R}^n$ but I don't see any reason why $x \mapsto \frac{\partial f }{\partial x^i}|_x $ is also smooth in the context where we define a derivation at a point as a map $f \mapsto \frac{\partial f }{\partial x^i}|_x$ instead of $x \mapsto \frac{\partial f }{\partial x^i}|_x$.
Here, $$\frac{\partial f}{\partial x^i} $$ means the partial derivative in the $i$th coordinate of the function $f$, in the coordinates $(x^1, \cdots, x^n)$. If you were to write this in the "standard" way, using a chart $(U, \varphi)$, then $$ \frac{\partial f}{\partial x^i}(p) = \frac{\partial (f \circ \varphi^{-1})}{\partial x_i}(\varphi(p)), $$ where the right hand side is taken in $\mathbb{R}^n$, with $(x_1, \cdots, x_n)$ being the standard coordinates of $\mathbb{R}^n$.
Moreover, $f \circ \varphi^{-1}$ is smooth by definition.