We prove $q$-analogues of identities that are equivalent to the functional equation of the arithmetic-geometric mean. We also present $q$-analogues of $F(\sqrt{k},\fracπ{2})$, the complete elliptical integral of the first kind, and its derivatives evaluated at $k=\frac{1}{2}$. These $q$-analogues interpolate those $n$th derivative evaluations by extending $n$ to a complex variable $s$, and we prove that they can be expressed as an infinite product.