I have been trying to work out the following problem:
Problem:
Let $M$ be a smooth manifold and $x\in M$ and $X,Y$ be smooth vector fields with local one parameter groups $T^X_t$ and $T^Y_t$ for small enough $t$.Prove that if $f$ is a smooth function on $M$,then $f(T^X_t(x))=f(x)+t(Xf)(x)+\frac{t^2}{2!}(X^2f)(x)+\dots+\frac{t^n}{n!}(X^nf)(x)+o(t^n)$ for $n\geq 1$ and $x\in M$ using this further show that $[X,Y](f)(x)=\lim\limits_{t\to 0}\frac{f(\Theta_t(x))-f(x)}{t^2}$ where $\Theta_t=T^Y_{-t}\circ T^X_{-t}\circ T^Y_t\circ T^X_t$.
I am not able to prove the second part as I do not exactly get the geometric meaning hidden underneath.We have not been taught Lie Derivatives $\mathcal L_XY(f)=[X,Y](f)$ and hence I am not comfortable with what is going on actually.Everything seems to be wrapped around by a veil of mystery here.I am looking for an explanation of the geometric principle lying beneath as well as a formal approach towards proving this result.