Proving that holomorphic map $f : mathbb{C}^n to mathbb{C}^m$ maps holomorphic tangent space at point $p$ to holomorphic tangent space at $f(p)$

I’m trying to to prove proposition 2.3.1 from Jiri Lebl’s text on several complex variables here.

So far I have tried writing the jacobian in terms of $u_k$ and $v_k$ where $f_k = u_k + iv_k, k = 1,…,m$ and its partial derivatives in $x_j$ and $y_j$ where $j = 1,…,n$. However I am confused as to which basis the $T_{f(p)} mathbb{C}^m$ is using and the correspondence between the $frac{partial}{partial x}, frac{partial}{partial y}$ and $frac{partial}{partial z}, frac{partial}{partial bar{z}}$. This could be the reason as to why I’m not to sure about what the hint is saying here.

Any help is appreciated, thank you!