2-form on a smooth manifold

Question

Let $M$ be a smooth manifold, $f:M$ $\rightarrow \mathbb{R}$ differentiable and $p\in M$ with $df(p)=0$. I am trying to show that the application, $$\begin{matrix}\mathfrak{X}(M)\times \mathfrak{X}(M):&\rightarrow&\mathbb{R}\\(V,W)&\rightarrow&V(W(f))(p)\end{matrix}$$ is symmetric. I manage to do some manipulations but I always end up coming back to the first expression.

Answer 1

We have that, $$V(W(f))(p)-W(V(f))(p)=[V,W](p)(f)=df(p)[V,W]=0$$ so, $$V(W(f))(p)=W(V(f))(p)$$

Answer 2

By definition of differential, for any vector field $W$, we have: $$W_p f=df_p W_p=0$$so your form is not only symmetric, it is identically null.

My intuition is that a vector is a sort of partial derivative. But the total differential vanishes, so any partial derivative must also be zero.