Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$

Question

Let $n \geqslant 1$, Show that the function $g(u)=\int_{-\infty}^{\infty} \frac{x^n e^{ux}}{e^x+1}dx$ is differentiable in $(0,1)$, where $u \in (0,1)$. What I did is just use the definition of derivative, but not sure whether I can switch the limit and the integral.