Derivatives of Jacobi Elliptic Functions

Here we compute derivatives of 9 of the Jacobi elliptic functions with respect to the argument \(u\). Three were derived here, and are reproduced below.

For convenience, we drop the argument and modulus.

\begin{equation}
\frac{\partial \,\mathrm{sn}}{\partial u} = \mathrm{cn}\,\mathrm{dn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cn}}{\partial u} = -\mathrm{dn}\,\mathrm{sn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{dn}}{\partial u} = -k^{2}\mathrm{cn}\,\mathrm{sn}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{ns}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sn}} = -\frac{\mathrm{cn}\,\mathrm{dn}}{\mathrm{sn}^2} = -\mathrm{cs}\,\mathrm{ds}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{nc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cn}} = \frac{\mathrm{dn}\,\mathrm{sn}}{\mathrm{cn}^2} = \mathrm{sc}\,\mathrm{dc}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{nd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{dn}} = k^{2}\frac{\mathrm{cn}\,\mathrm{sn}}{\mathrm{dn}^2} = k^{2}\mathrm{sd}\,\mathrm{cd}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cs}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sc}}
= \frac{\partial}{\partial u}\frac{\mathrm{cn}}{\mathrm{sn}} = -\frac{\mathrm{dn}}{\mathrm{sn}^2}
= -\mathrm{dn}\,\mathrm{ns}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{sd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{ds}}
= \frac{\partial}{\partial u}\frac{\mathrm{sn}}{\mathrm{dn}} = \frac{\mathrm{cn}}{\mathrm{dn}^2}
= \mathrm{cn}\,\mathrm{nd}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{ds}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{sd}}
= \frac{\partial}{\partial u}\frac{\mathrm{dn}}{\mathrm{sn}} = -\frac{\mathrm{cn}}{\mathrm{sn}^2}
= -\mathrm{cn}\,\mathrm{ns}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{sc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cs}}
= \frac{\partial}{\partial u}\frac{\mathrm{sn}}{\mathrm{cn}} = \frac{\mathrm{dn}}{\mathrm{cn}^2}
= \mathrm{nc}\,\mathrm{dc}
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{cd}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{dc}}
= \frac{\partial}{\partial u}\frac{\mathrm{cn}}{\mathrm{dn}} = -{k^{\prime}}^2\frac{\mathrm{sn}}{\mathrm{dn}^2}
= -{k^{\prime}}^2\mathrm{sn}\,\mathrm{nd}^2
\end{equation}

\begin{equation}
\frac{\partial \,\mathrm{dc}}{\partial u} = \frac{\partial}{\partial u}\frac{1}{\mathrm{cd}}
= \frac{\partial}{\partial u}\frac{\mathrm{dn}}{\mathrm{cn}} = {k^{\prime}}^2\frac{\mathrm{sn}}{\mathrm{cn}^2}
= {k^{\prime}}^2\mathrm{sn}\,\mathrm{nc}^2
\end{equation}

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