Differentiating exponential functions - is base e the only situation?

Question

My maths book gives the example of;

Where $$ f = e^x $$ $$ f` = e^x $$

It only uses the example of base e in all of the questions so does that mean this is the only situation where the differential of an exponential function is the same as the original?

Answer 1

Indeed, the exponential function is the unique solution the the equation. Power series expansion can be used to verify.

Answer 2

Yes, since $$ Da^x=De^{x\ln a}=e^{x\ln a}\ln a=a^x\ln a, $$ and the only $a$ for which $\ln a=1$ is $a=e$.