I am wondering if the derivatives of the moment forms of the statistical distributions, as demonstrated below, are always zero. As far as I know, the derivative of a definite integral should be zero, but the derivative of an indefinite integral should give the integrand. In this particular case, a moment consists of a definite integral returning a constant value. Therefore, the derivative of the moment should be zero as well. However, I might still miss some fundamental knowledge here.
$\frac{d}{dx}(\int_0^\infty (x-0)^n f(x) dx) $
$\frac{d}{dx}(\int_{-\infty}^\infty (x-0)^n f(x) dx) $
I would be glad if you kindly provide some responses to the above question.
Regards,