\[\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{dc}\left(z,k\right)\right)^{%
2}=\left({\operatorname{dc}}^{2}\left(z,k\right)-1\right)\left({\operatorname{%
dc}}^{2}\left(z,k\right)-k^{2}\right),\]