Have you read this thread? Is that close to what you are asking?
If not, please try to explain exactly what you mean.
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.
I've attached a picture which should clarfy. What I mean is if there is a way to start with the assumption that there exisits a number^x which is its own derivative, and solve for that number to get to the last expression in my picture.