Let $\phi:M\rightarrow N$ be a smooth map between smooth manifolds, $v \in \operatorname{Vect}(M)$. Let $\{x^\mu\}$ and $\{y^i\}$ be local coordinates on $M$ and $N$ respectively.
How can I show that the components of the pushforward vector $\phi_*v$ in local coordinates is given by the following: $$(\phi_*v)^i(y) = \frac{\partial y^i}{\partial x^\mu}v^\mu(x)$$
Differential geometry in general is rather new to me so any help would be appreciated.