I have a quite basic question on the differential of a function and the Lie bracket. Given a smooth map \begin{equation} f:M\rightarrow{N} \end{equation} between to manifolds. The vector fields on M are closed under the Lie bracket so they form a Lie Algebra. In books I've often seen that the differential $df$ is a Lie algebra homomorphism, i.e. \begin{equation} df([X,Y])=[df(X),df(Y)] \end{equation} for $X,Y$ vector fields in a region of M. Could someone explain if and why this equation always holds? I've tried to use the definition of the differential with curves but unfortunately I got stuck.
Thanks a lot.