Let $X$ be a manifold and $X_1$ and $X_2$ two vector fields over $X$
Prove that: $[f X_1 , X_2 ] = f · [X_1 , X_2 ] − X_2 (f ) · X_1$
where [] is the Lie brakets and $D_{X_2}$ the derivative operator acting on $ C^∞ (X, \mathbb{R})$.
I don't really find any clear definition of the derivation operator, morover, [] is bilinear so $[f X_1 , X_2 ] = f · [X_1 , X_2 ]$.
Thank you for your help.