Modulus derivative of the incomplete elliptic integral of the second kind (${\partial }_{k}E(\varphi ,k)=[E(\varphi ,k)-F(\varphi ,k)]/k$)

For real $|k|<1$, define $E(\varphi ,k)={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^{2}x}dx$ and $F(\varphi ,k)={\int }_0^{\varphi }(1-k^2{\sin }^{2}x{)}^{-1/2}dx$. Smoothness of the integrands permits differentiation under the integral sign. Subtracting the two integrands gives $E-F=-k^2{\int }_0^{\varphi }{\sin }^{2}x/\sqrt{1-k^2{\sin }^{2}x}dx$, which is $k{\partial }_{k}E$. This proves the displayed identity for $k\ne 0$. At $k=0$, both sides extend continuously with value zero. The variable $k$ is the modulus; derivatives with respect to the parameter $m=k^2$ instead obey ${\partial }_{m}E=(E-F)/(2m)$, with removable value $-\frac{1}{2}{\int }_0^{\varphi }{\sin }^{2}xdx$ at zero.