$A(t)= \pmatrix{\alpha(t)^2&2\alpha(t)\beta(t)&...&...\\...&\alpha(t)\bar{\alpha}(t)-\beta(t)\bar{\beta}(t)&...&...\\...&...&2\alpha\bar{\alpha}\beta-\beta^2\bar{\beta}&...}$
I am meant to find A'(t) and thus A'(0). The $...$ are just other terms that I did not write out as its not necessary for my question below (some are quite long!).
So obviously for A'(t), terms like $\alpha(t)^2$ become 2\alpha'(t), and I used the product rule to differentiate $2\alpha(t)\beta(t)$ and it becomes 2(\alpha'(t)\beta(t)+\alpha(t)\beta'(t)) etc..
The problem is when I come across $2\alpha\bar{\alpha}\beta-\beta^2\bar{\beta}$, how would I find A'(t)? Since there are 3 terms $\alpha$, $\bar{\alpha}$ and $\beta$.