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