TITLE: Show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.
QUESTION [5 upvotes]: If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that
$$[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X.$$
This is a proposition in a book. But I cannot prove this.

REPLY [7 votes]: By definition, for any two vector fields $X,Y$, we have 
$$[X,Y]=XY-YX.$$
Note also that for any two vector fields $X,Y$ and a smooth function $g$, we have
$$X(gY)=(Xg)Y+gXY.$$
Using these, we have 
$$[fX,gY]=fX(gY)-gY(fX)=
f(Xg)Y+fgXY-g(Yf)X-gfYX$$
$$
=f(Xg)Y-g(Yf)X+fg(XY-YX)=f(Xg)Y-g(Yf)X+fg[X,Y],$$
as required.