What's the derivative of moment-generating function?

Question

If $X$ follows a distribution with p.d.f. $f(x)$ and its moment-generating function is $$M_{X}(t) = E(e^{tX}).$$ What should the derivative of $M_{X}(t)$ , $\frac{d}{dt}M_{X}(t)$ be?

Thanks for your time.

Answer 1

You have $$\mathbb{E}(e^{tX}) = \int e^{tx}f(x) ~\mathrm{d}x$$ so we obtain $$\frac{d}{dt} \mathbb{E}(e^{tX}) = \int xe^{tx}f(x)~\mathrm{d}x = \mathbb{E}(Xe^{tX})$$ at least if this integral exists finitely for all $s$ in some neighborhood of $t$.