Is it possible to derive the number e, starting with the assumption that there exists a base number for an exponential function that is equal to its own derivative?

I've seen proofs that [e^x]' = e^x, using the definition of e. What I'm looking for is a way to start with the assumption that the above is true, and get to the limit definition of e.

I searched around a bit, without any luck, so I thought I'd ask here.