I know the answer is $e^x$, but supposing I have no idea, what would be the steps required to get to that answer? Note that starting to test different functions is not a valid answer here as there are infinitely many.

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I started by doing 

$$
f(x)=\frac{d}{dx}f(x)
$$

so

$$
\int f(x)dx=f(x)
$$

and 

$$\int f(x)dx=\frac{d}{dx}f(x)$$
$$f(x)=\frac{d^2}{dx^2}f(x)$$

That leads us to

$$\frac{d^n}{dx^n}f(x)=\frac{d^m}{dx^m}f(x)$$

But that gives no new information...