d/dx f(x) = f(x)

Are there any other functions from R to R whose derivative everywhere equals itself, beside x -> a*e^x for constant a in R?

It is very simple. If f(x)=f'(x), then f'(x)/f(x)=1. Since f'(x)/f(x)-1=0, then d(ln(f(x))-x)/dx must also equal to 0 since it is equal to f'(x)/f(x)-1. Since it can easily be proven that only constant functions have 0 derivatives everywhere, then ln(f(x))-x must be a constant function C. The rest easily follows.