Computing dq/dk by using q = exp( -Pi K'/K ) and \diff{K}{k} = \frac{E - k'^2 K}{k k'^2}, \diff{K'}{k} = \frac{k^2 K'-E'}{k k'^2} (LaTeX-note: read \diff{y}{x} as \frac{dy}{dx}) one obtains a somewhat lengthy expression. Simplifying using k^2+k'^2 = 1 and then (surprise!) using E K' + E' K + K K' = Pi/2 (Legendre's relation) one can obtain (setting T := (2 K^2)/(\pi^2 q) = \Theta_3(q)^4 / (2 q) ) \diff{k}{q} = + T k k'^2 \diff{k'}{q} = - T k^2 k' \diff{K}{q} = + T (E - k'^2 K) \diff{K'}{q} = - T (E' - k^2 K') \diff{E}{q} = + T k'^2 (E - K) \diff{E'}{q} = - T k^2 (E' - K') I have not seen any of these before (pointers, anyone?).
From these follow relations involving theta derivatives such as \frac{2 E}{\pi} = \frac{1}{\Theta_3^3(q)} ( \Theta_4^4(q) \Theta_3(q) + 4 q \diff{\Theta_3(q)}{q} ) (Cf. http://oeis.org/A194094 )
(End of bore). Best regards, jj