Partial derivative of an integral

Question

I have the following function (it is the incomplete elliptic integral of first kind) $$ F(b,g) = \int_{0}^{b} \frac{dx}{\sqrt{(1-x^2)(1-gx^2)}} $$ I would like to compute $$\frac{\partial F}{\partial g} \ ,\ \frac{\partial F}{\partial b} \ ,\ \frac{\partial^2 F}{\partial g^2} \ ,\ \frac{\partial^2 F}{\partial b^2} \ ,\ \frac{\partial^2 F}{\partial b\partial g}$$ so I defined

F[b_,g_]:= Integrate[1/Sqrt[(1 - x^2)*(1 - g*x^2)], {x, 0, b}]

and tried the command

D[F[b,g],g]

but Mathematica cannot compute it. Am I doing something wrong or is there a way to do it?

Answer 1

I don't know if you want to be this restrictive, but you can avoid singularities that Mathematica evidently has trouble with by making assumptions.

F[b_,g_]:= Integrate[1/Sqrt[(1 - x^2)*(1 - g*x^2)], {x, 0, b}]

$Assumptions = 0 < b < 1 && 0 < g < 1

D[F[b, g], g]

$\frac{\sin \left(2 \sin ^{-1}(b)\right)}{4 (g-1) \sqrt{1-b^2 g}}-\frac{F\left(\left.\sin ^{-1}(b)\right|g\right)}{2 g}-\frac{E\left(\left.\sin ^{-1}(b)\right|g\right)}{2 (g-1) g}$

The other derivatives work also.