If $\psi:M\to N$ is a diffeomorphism between smooth manifolds $M,N$ and $f:N\to\mathbb{R}$ is a smooth function , then prove that for a vector field $X$ in $M$ $$(\psi_*X)(f)(p)=X(f\circ\psi)(\psi^{-1}(p))$$ for $p\in N$ , where $\psi_*$ is the pushforward .
I understand that the pushforward of $X$ as a vector field $Y$ in $N$ can be defined as $$Y(p):=(\psi_*X)(p)=d\psi(X(\psi^{-1}(p)))$$ Hence $$(\psi_*X)(f)(p)=(Yf)(p)=d\psi(Yf(\psi^{-1}(p)))$$ Rest I am confused how to equate it to the required expression on RHS . Any help is appreciated .