Derivative of a moment generating function

Question

Okay so I am assuming that you know what is the expectation of a random variable so let me define the Moment generating function of a random variable
Let $X$ be a random variable whose probability density function is $f(x)$ then a real-valued function $M:\mathbb{R}\to\mathbb{R}$ defined by $M(t)=E(e^{tX})$,when $E(X)$ is the expectation of the random variable $X$.Now I have to find the derivative of the random variable at $t=0$
$\frac{d}{dt}M(t)=\frac{d}{dt}E(e^{tX})$ = $E(\frac{d}{dt}e^{tX})$ ,my question how we are taking the derivative inside $E$?

Answer 1

The expectation is $$\int \exp(t X) f(x) d x.$$ Differentiate under the integral sign, to get $$\int X \exp(t X) f(x) d x.$$