TITLE: Are there any other functions that behave the same as $ce^x$ with respect to differentiation
QUESTION [8 upvotes]: $$\frac{d}{dx} ce^x = ce^x$$
Are there any other functions $f$ such that
$$\frac{d}{dx} f(x) = f(x)$$
or is $ f(x) = ce^x $ the only one?

REPLY [3 votes]: Consider the first order differential equation $$f'(x)=f(x)$$ which is separable. It's integration leads to $$f(x)=c e^x$$ and this is the only possible solution.