Let $\phi:M\to N $ be a smooth map and let $p\in M$ be a point at which $d\phi_p$ is a linear isomorphism. Given a coordinate system $\zeta$ at $p$ or $\eta$ at $\phi(p)$ the other can be selected so that $\phi$ preserves coordinates $y^i(\phi_q)=x^i(q)$ for $q$ near $p$. Then $d\phi\left(\frac{\partial}{\partial x^i}\right)=\frac{\partial}{\partial y^i}$ and if $f\in\mathscr{T}(N)$
$$\frac{\partial f\circ \phi}{\partial x^i}=\frac{\partial f}{\partial y^i}\circ \phi$$ near $p$
In the first part, I used the inverse function theorem. Since we have a linear isomorphism, there is a neighbourhood around p and one around $\phi(p)$, where phi is a diffeomorphism.
Then if I pick $\eta=\phi\circ\zeta^{-1}$ is a coordinate system in the neighbourhood of $\phi(p)$, the same applies if I choose $\zeta=\phi^{-1}\circ\eta^{-1}$. I think this proves the assertion.
Regarding the second part, I am confused once I am not able to relate to the specific coordinates $x^i$ and $y^i$
Question:
Can someone help me solve this problem?
Thanks in advance.