My knowledge of Differential Geometry is quite limited, one defines a Vectorfield $X$ as the ordering: $ p \mapsto X_p$ as a mapping between $ M \rightarrow TM$ with additional smoothness condition of: For all locally smooth functions $f$ the function : $$X(f): M \rightarrow \mathbb{R} \; p \mapsto X_p (f) $$ is again smooth.
In this light, one presents an example for a chart $(U,\phi), \; \phi= (x_1,\cdots x_n)$ one defines vectorfields on $U$ by: $$\frac{\partial}{\partial x_i}_p$$ taking a $p$ from $U$ and sending it to the tangent vector. One additionally defines: $$ \frac{\partial}{\partial x_i}_p [f] := \frac{\partial (f \circ \phi^-1)}{\partial y_i} (\phi(p))$$ As a derivation on the germs of functions around $p$. The $y_i$ are standard coordinations on the reals.
Given these terms, i would be intereseted to understand. why is the defined field smooth?
I would then have $$ \frac{\partial}{\partial x_i} [f] := \frac{\partial (f \circ \phi^-1)}{\partial y_i} (\phi(-))$$ As a function from $U$ to the reals. How can we see that this is a smooth function? i feel that the answer might be trivial, nevertheless missing it for my infintial knowledge in this subject.