This question is an exercise from Jeff Lee's Manifolds and Differential Geometry
Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). Furthermore, let $f$ be a linear functional on $L(V,V)$ and define the linear functional $f_{,X}$ by $f_{,X}(A)=f(A\circ X)$.
We are required to show $\tilde{X}f=f_{,X}$. He then says the hint/solution is
$(\tilde{X}f)(A)=df|_A(\tilde{X}_A)=f(A\circ X)$.
I think I am okay with the first equals sign in the above line but I have no idea how he shows the middle equals the right hand side of the above equations.
Any help would be much appreciated