I have the following definition of the Lie derivative $L_X$, for a vector field $X$ on a manifold $M$: \begin{align} L_X:\mathcal{C}^\infty (M) \rightarrow \mathcal{C}^\infty (M) \end{align} is a map defined by \begin{align} (L_Xf)(p)=(D_pf)(X(p)). \end{align}
I then have a definition of the commutator of vector fields $[X,Y]$ defined as the vector field satisfying ($f:M \rightarrow \mathbb{R}$ is a function) \begin{align} L_{[X,Y]}=L_X\circ L_Y - L_Y \circ L_X. \end{align}
What I want to be able to do is find \begin{align} [X,Y](f)=X(Y(f))-Y(X(f)) \end{align} from this, but I'm not quite sure how (if it's possible). My attempt starts by looking at: \begin{align} (L_X (L_Yf))(p)&=(L_X((D_pf)(Y)))(p)\\ &=(D_p((D_pf)(Y)))(X)(p), \end{align} but I get stuck here. If $\gamma_X$ is an integral curve of $X$ at the point $p\in M$, then I am using the definition \begin{align} (D_pf)(X(p))=\left.\frac{d}{d t}\right|_{t=0}(f\circ \gamma_X)(t), \end{align} but if this is the correct way to proceed, then I'm struggling to see how to use this a second time.
If there is an answer, I'd appreciate as much detail as possible about all the definitions and steps used.
To answer your comment: in general $f$ is not a diffeomorphism, as $M$ may have dimension higher than one.
To answer your question: at a point $p \in M$, $D_p(f)$ is equivalent to the gradient of $f$, so $D_p(f)(X_p)$ is the directional derivative $X_pf$ of $f$. Hence $$(L_{[X,Y]}f)_p = [X,Y]_pf,$$ and \begin{align*} ((L_X\circ L_Y)f)_p-((L_Y\circ L_X)f)_p & = (L_XY(f))_p-(L_YX(f))_p \\ & = X_p(Y(f))-Y_p(X(f)). \end{align*}