Assuming[{a > 0, b > 0, n > 0, sse > 0, b > a},
 Integrate[(1/(Sqrt[2 Pi sigma^2]))^n
 Exp[-(x - mu)^2/(2 sigma^2)] PDF[UniformDistribution[{a, b}], sigma], {sigma, a, b}]]

([Pi]^(-n/2) (1/(mu - x)^2)^(-(1/2) + n/2) (Gamma[1/2 (-1 + n), (mu - x)^2/(2 a^2)] -
Gamma[1/2 (-1 + n), (mu - x)^2/(2 b^2)]))/(2 Sqrt[2] (a - b))

FullSimplify@D[g, mu]

fDeriv2[x_, mu_, n_, a_, b_] := (-2^(3/2 - n/2) E^(-((mu - x)^2/(2 a^2))) ((mu - x)^2/a^2)^(1/2 (-1 + n)) +
   2^(3/2 - n/2) E^(-((mu - x)^2/(2 b^2))) ((mu - x)^2/b^2)^(
   1/2 (-1 + n)) - (-1 + n) (Gamma[1/2 (-1 + n), (mu - x)^2/(2 a^2)] -
   Gamma[1/2 (-1 + n), (mu - x)^2/(2 b^2)]))/((mu - x) (Gamma[1/2 (-1 + n), (mu - x)^2/(2 a^2)] -
   Gamma[1/2 (-1 + n), (mu - x)^2/(2 b^2)]))

fDeriv2[0.1, 0.2, 100, 2, 4]

ComplexInfinity