In DoCarmo’s Riemannian Geometry, a vector field $X$ on a manifold $M$ is interpreted both as a map from $M$ to its tangent bundle $TM$, and when smooth as an operator on the set of smooth real-valued maps $\mathscr D$ of $M$.
After defining the bracket of two smooth vector fields $[X,Y]$, there comes the following theorem: If $X,Y$ are smooth vector fields on $M$ and $f,g$ are smooth real-valued maps on $M$, then $[fX,gY]=fg[X,Y]+fX(g)Y-gY(f)X$.
My problem is, what does $fX$ mean?