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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?

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marked as duplicate by Aryabhata, The Chaz 2.0, Qiaochu Yuan Mar 22 '12 at 21:22

This question was marked as an exact duplicate of an existing question.

Well, that depends. What definition of $e^x$ are you using? – Qiaochu Yuan Mar 22 '12 at 21:15
up vote 0 down vote accepted

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.

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