Pull Back (change of variables)

Question

Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. Define $(Xf):\mathbb{R^2}\rightarrow\mathbb{R}$ by $(Xf)=<X,\nabla f>$. Being $h$ a change of variables, $h_{*}X=\widetilde{X}$ where $$ (h_{*}X)(p)=Dh(h^{-1}(p))X(h^{-1}(p)) $$ and $Dh$ denotes the differential of $h$. Given a function $f:\mathbb{R^2}\rightarrow\mathbb{R}$, $h_{*}f=f\circ h^{-1}$. Is that true $$ h_{*}(Xf)(\widetilde{p})=h_{*}Xh_{*}f(\widetilde{p}) $$ where $\widetilde{p}=h(p)$?