TITLE: Infinite number of Derivatives
QUESTION [6 upvotes]: Is there a kind of function (other than trigonometric) that you can take infinite amount of derivatives without it ever becoming 0. Algebraic functions now matter how long, or how many powers  it has it can eventually be derived to 0. I am not including trigonometric functions because they are circular in nature. I mean a function that will not go on a circle like trigonometric do; But will have infinite derivatives.

REPLY [3 votes]: Further to other answers here, only polynomials of positive integer order eventually run to a zero derivative.  The repeated derivative of any polynomial $x^{y}$ where $y$ is non-integer or where $y<0$ will never become zero.  (The logarithm is a special starting point for the latter case.)