I am trying to prove $L_X(\omega(Y))=(L_X\omega)(Y)+\omega(L_XY)$, where $X,Y$ are vector fields are vector field, and $\omega$ is a covector field. $\omega(Y)$ is a function, and for functions,
$$ L_Xf(p)=\lim_{h\rightarrow 0} \frac{f(\phi_h(p)) - f(p)}{h}, $$
where $\phi_t$ is the flow generated by $X$. For covector fields, we have
$$ (L_X\omega)(p)=\lim_{h\rightarrow 0}\frac{({\phi_h}^*\omega)(p)-\omega(p)}{h} $$
and for vector fields,
$$ (L_XY)(p)=\lim_{h\rightarrow 0} \frac{Y(p)-({\phi_h}_*Y)(p)}{h}=\lim_{h\rightarrow 0}\frac{({\phi_h}^*Y)(p)-Y(p)}{h}. $$
I can start with the definition, to find
$$ L_X(\omega(Y))(p)=\lim_{h\rightarrow 0}\frac{(\omega(Y))(\phi_h(p))-(\omega(Y))(p)}{h}\\ =\lim_{h\rightarrow 0}\frac{\omega(\phi_h(p))(Y(\phi_h(p)))-\omega(p)(Y(p))}{h} $$
but I'm not sure where to go next. I think the idea is to add 0 in the right way, but I'm not sure which terms to add. If I start from the result,
$$ (L_X\omega)(Y)(p)+\omega(L_XY)(p) \\=\lim_{h\rightarrow 0}\frac{({\phi_h}^*\omega)(p)(Y(p))-\omega(p)(Y(p))}{h} + \lim_{h\rightarrow 0} \frac{\omega(p)(Y(p))-\omega(p)(({\phi_h}_*Y)(p))}{h}\\ =\lim_{h\rightarrow 0}\frac{({\phi_h}^*\omega)(p)(Y(p))-\omega(p)(({\phi_h}_*Y)(p))}{h}\\ =\lim_{h\rightarrow 0}\frac{\omega(\phi_h(p))(({\phi_h}_*Y)(\phi_h(p)))-\omega(p)(({\phi_h}_*Y)(p))}{h}, $$ it looks like I should add some pushforward terms, but I still can't see how to do it.