# Function which is never its own ($n^{th}$) derivative?

Is there any real-valued function $f(x)$ of a real variable $x$ with $n^{th}$ and $m^{th}$ derivatives never equal for nonequal nonnegative $m$ and $n$ and where the $n^{th}$ derivative of $f$ never has a vertical asymptote?

The idea is that f should be infinitely differentiable like $e^x$ or $sin(x)$ without being periodic by differentiation. In this case I would call $e^x$ 1-periodic by differentiation and $sin(x)$ 4-periodic by differentiation.

The asymptote requirement is because the solution $f(x)=\frac{1}{x}$ is obvious.

Also, is there a name for this class of functions?

$f(x)=\frac{1}{1+x^2}$ should do the trick.
I like $xe^x{}{}{}{}{}{}{}{}$.
$e^{2x}$ is not so bad either.
• I guess you like $e^{x^2}$ better. – LeoTheKub Feb 25 '14 at 3:50