Differentiating with respect to the exponent of distribution

Question

I want to see how the following expression increases or decreases with $n$.

$a-b(n)+\int Y(x)dF(x)^{n}$

Actually, $n$ is an integer, so that $F(x)^n$ actually means the distribution of

$max(X_1,X_2,...,X_n)$

But anyways, as a first step, I am assuming $n$ is a real number to take derivative. But I don't know how to take derivative with respect to $n$

Does it look like something like this

$-b'(n)+?$

But how should I differentiate that?