TITLE: How can I prove that there is a function that is its own derivative?
QUESTION [20 upvotes]: How can I prove that a function that is its own derivative exists? And how can I prove that this function is of the form $a(b^x)$?

REPLY [42 votes]: $f(x) = 0$ is trivially its own derivative, and is of the form $a(b^x)$ for $a=0$ and any positive $b$. That's all we need to solve the problem posed.