Ross(Probability, 10th Edition) says in Section 7.7 on moment generating functions that they are defined and differentiated as $$M(t)=E[e^{tx}]\Rightarrow M'(t) = \frac{d}{dt} E[e^{tx}]$$
As explained in the textbook, we can "almost always" interchange the differentiation operators to get: $$=E \left[\frac{d}{dt}(e^{tx}) \right ]$$
- What is the math/method to justify this.
- What are the exceptions? Where can this not be justified.