Identity on vector fields.

Question

Given $X,Y$ smooth vector fields on a manifold $M$ and $f\in C^{\infty}(M)$ I want to prove that $X(f)Y=Y(f)X$, because if that's true it will make some calculations a lot easier in another problem I'm solving, but i don't even know if it's true, by now what I've been using that $Y=\sum_{i=1}^n Y^{i}\frac{\partial}{\partial x^{i}}$ and similar for $X$, but that doesn't get me to anything. If it's true i would appreciate a hint to get the result.

Answer 1

Counter example: $M = \Bbb R^2$, $X = \partial_x$, $Y = \partial_y$ and $f(x,y) = x$. One has $X(f)Y = \partial_y$ and $Y(f)X = 0$.