The Lie derivative is defined as: $$L_v(fX)=[V,fX]=(Vf)(X)+f[V,X]$$
Show that $L_v(df)=d(Vf)$
Applying the formula straightforwardly:
$$L_v(df)(X)=V df(X)+df[V,X]=(Vdf)(X)+df(V(X)-X(V))=VdfX+df(V(X))-df(X(V))$$
I suspect that the first and last term would cut, but how do I justify that?
How do I handle $VdfX$?
Question:
Can someone help me solve and understand this?