How to calculate the derivative at a point of a non-integrable multivariable function?

Question

I want to integrate this function with respect to θ1 and then take the derivative of that function with respect to n. Since the integral is not a known function I am having trouble with this operation. Does anyone have any ideas? I know that I can perform numerical integration on the function. The limits of integration that I'm interested in are 0 and ArcSin[0.2/n].

1/2 ((n Cos[θ1] - 
 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(n Cos[θ1] + 
 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2 + (-0.2 Cos[θ1] + 
 n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(0.2 Cos[θ1] + 
 n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2)

Answer 1

If numerical results are acceptable;

Needs["NumericalCalculus`"]

func[θ1_,n_]:= 1/2 ((n Cos[θ1] - 
 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(n Cos[θ1] + 
 0.2 Sqrt[1 - 25. n^2 Sin[θ1]^2])^2 + (-0.2 Cos[θ1] + 
 n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2/(0.2 Cos[θ1] + 
 n Sqrt[1 - 25. n^2 Sin[θ1]^2])^2)

int[n_?NumberQ] := 
 int[n] = NIntegrate[func[\[Theta]1, n], {\[Theta]1, 0, ArcSin[.2/n]},
    AccuracyGoal -> 10]
der[n0_?NumberQ] := der[n0] = ND[int[n], n, n0]

der[4]