Pushforward of vector field multiplied by real valued function

Question

Let M and N be smooth manifolds: $$\mathbb{R}\xleftarrow{g} M \xrightarrow{h}N\xrightarrow{f}\mathbb{R}$$ and of course: $$TM\xrightarrow{h_*}TN$$ I need to show that: $$h_{*}(gX)f=(g\circ h^{-1})h_{*}Xf$$ where $X\in TM$. Tried by substituting $f\rightarrow g\circ h^{-1}$ but it doesn't make sens. Any suggestions?

Answer 1

Substitute $Y = gX$. The pushforward then becomes $(h_*Y)_q(f) = Y_{h^{-1}(q)}(f\circ h)$. Then you can reenter your substitution to obtain: $g(h^{-1}(q))X_{h^{-1}(q)}(f\circ h)$ or equivalently $(g\circ h^{-1})h_*X(f)$.