This is question 7 of Do carmo's differenital geometry of curves of surfaces 3.4.
Define the derivative $w(f)$ of a differentiable function $f:U \subset S\rightarrow R$ relative to a vector field $w$ in $U$ by \begin{equation}w(f)(q)=\frac{d}{dt}(f\circ\alpha)|_{t=0}, q \in U \end{equation} where $\alpha:I \rightarrow S$ is a curve such that $\alpha(0)=q,\alpha'(0)= w(q).$ Prove that
$w$ is differentiable in $U$ if and only if $w(f)$ is differentiable for all differentiable $f$ in $U$.
By definition , $w$ is differentiable at $q \in U$ if for some parametrization $x(u,v)$ at q, the functions $a(u,v)$ and $b(u,v)$ given by $w(q)=a(u,v)x_u+b(u,v)x_v$ are differentiable fucntions at q. Since the composition of differentiable functions is diferentiable function, $w$ is differentiable should give that $\frac{d}{dt}(f\circ\alpha)|_{t=0}$ is differentiable. however, for the reverse direction , we need $\frac{d}{dt}(f\circ\alpha)|_{t=0}$ is differentiable implies $\alpha'(0)$ is differentiable and the 1-1 correspondence of $\alpha(0)$ and $w(q)$. I don't know whether the argument should proceed like this and stuck at the reverse direction. Thanks in advance for any help!