Let $$g = \frac{27J}{1 - J},$$ where $J$ is the absolute invariant, and define $$\Omega = \int_{\gamma(J)} \frac{dz}{\sqrt{4z^3 + g(z + 1)}}.$$ Here, $\gamma(J)$ is a contour in the complex plane that encloses any two of the three roots of $4z^3 + g(z + 1)$ and does not go through the third. Since the roots are given in terms of $J$, the path $\gamma(J)$ is a function of $J$ (or $g$).
I would like to differentiate $\Omega$ under the integral sign with respect to $g$. Is it permissible? Could someone point me to a theorem I could use or help me prove it directly?