I recently stumbled upon the Gaussian $q$-binomail coefficients. The generalization is obtained using the following formula:
$$\binom{n}{k}_q = \frac{(1-q^n)(1-q^{n-1})\cdots(1-q^{n-k+1})}{(1-q)(1-q^2)\cdots(1-q^k)}$$
and these $q$-binomails obey many of the nice properties that we expect from anything which we might call a "binomial". For example, the following recursive relationship holds:
$$\binom{n}{k}_q = q^k\binom{n-1}{k}_q + \binom{n-1}{k-1}_q$$
I am interested in taking the derivative of the $q$-binomial with respect to $q$. Specifically, I would like to know if there is a nice recursive relationship, or, better yet, a closed formula for
$$\frac{d}{dq}\left[\binom{n}{k}_q\right] = \,???$$
I have done some digging around, and there seem to be some nice things that pop up when you consider Jackson derivatives of such quantities, but I was unable to find anything nice for the regular derivative given above, and my own investigations have not borne any convenient truths either, though this is not my area of expertise, so I could be missing something rather simple. Any help is appreciated; thank you!