How to calculate what function coincides with the equation $f(x)=\dfrac{d}{dx}f(x)$?

Question

Possible Duplicate:
Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$

I know that that the equation that coincides with $f(x)=\dfrac{d}{dx}f(x)$ is the function $f(x)=e^x$

But how can it be calculated? What is the prove of that?

Answer 1

Of course the easiest way is just trying. But you can also notice that the equation implies that $f$ is infinitely differentiable. Whit a leap of faith you may assume it is analytic, and write its series expansion, plug it in the equation, and find its coefficients.