TITLE: n-th derivative where $n$ is a real number?
QUESTION [5 upvotes]: We know that 
$$\frac{d^n}{dt^n} e^{at}= a^n e^{at}; \, n\in \mathbb N.$$
I want to know if the result is true if $n$ is a real number, i.e., $n\in \mathbb R$ ?

REPLY [3 votes]: It's one possible definition. You know the result holds true for any non-negative integer $n$, so you define it to be $a^ne^{at}$ for fractional $n$. As others have pointed out this is one of the starting points of fractional calculus.