Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?

Can anyone find an approximate closed form for $$ \frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}\frac{1}{2-z^t} $$ by approximating the product (which we then derive $k$ times) for large $N$? The answer is $$\frac{\mathrm{d}^k}{\mathrm{d}z^k}\left(\frac{1}{2^{1+N}(\frac{1}{2} ; z)_{1+N}}\right)$$ which involves the q-Pochhammer symbol, but the closed form for the $k^{\text{th}}$ derivative w.r.t. $z$ is impossible to find due to difficulty deriving the q-Pochhammer symbol.

Is the previous question mentioned at the beginning a good way to simplify this?