Problem in proving the property of Lie bracket of vector fields

Question

Let $M$ be a Riemannian manifold, $f \in C^{\infty}(M)$, $X,Y$ vector fields on $M$. Then i have to prove $[X,f\cdot Y]=f\cdot [X,Y]+X(f)\cdot Y$. First i use the definition of Lie bracket: $[X,f\cdot Y]=X\circ (f\cdot Y)-(f\cdot Y)\circ X$. After that i am stuck, because i must somehow pull $f$ from the composition and i do not know how. Please help.

Answer 1

Since the vector field $X$ acts as a derivation, you can use the product rule for the first composition:

$$\left(X\circ (fY)\right)(h) = X\left(fY(h)\right) = X(f)Y(h) + f\left(XY(h)\right)$$

Your result should follow quickly from this.