Find a function $f(x)$ so that $f(x)=f'(x)$ 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.

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...

You can solve this using differential equations.
You have:
$$\frac{df}{dx}=f$$
This is a separable ODE, so if $f\neq 0$:
$$\int \frac{1}{f}~df=\int dx$$
Integrate both sides, and you should get the set of solutions.
