Suppose one is given
$f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$
$f(z_0, z_1) = (z_0z_1, \ 2z_0z_1)$
How would one find a total differential of such function? My best guess is
$ df = \ (\frac{\partial(z_0z_1)}{\partial{z_0}}, \ \frac{\partial(2z_0z_1)}{\partial{z_0}})dz_0 + (\frac{\partial(2z_0z_1)}{\partial{z_1}}, \ \frac{\partial(2z_0z_1)}{\partial{z_1}})dz_1$
However, my understanding is limited and I'm not sure if it is correct. If somebody could confirm/deny that this is correct or provide more information, it would be great!