Infinite number of Derivatives 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. 
 A: In addition to what everyone else is saying: rational functions that aren't polynomials, such as $\dfrac1x$, $\dfrac1{1-x}$, $\dfrac x{x^2+2x+2}$, etc.
Also things like $x^x$, which is halfway between a power (e.g. $x^n$) and an exponential (e.g. $n^x$). It grows faster than exponentials, by the way. You might not have learned how to differentiate this, yet.
Honestly, anything other than polynomials.
A: 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.)
A: $\log{x}$ and its various derivatives and antiderivatives.
Really, polynomials are the exception here: it is easy enough to prove by induction that any smooth function $f$ for which $d^nf/dx^n=0$ is a polynomial of degree less than $n$.
A: Let's suppose a function $f$ is given by a power series which converges within a open set $D$ (for simplicity, let's assume $0 \in D$).  Then, we can write
$$f(x) = \sum_{n=0}^\infty a_n x^n$$
provided that infinitely many $a_n$'s are nonzero, $f^{(k)}(0) = k!a_k$ will be nonzero for infinitely many $k$.  For example, $f(x)=e^x$ can be written as
$$e^x = \sum_{n=0}^\infty \frac{1}{n!}x^n$$
and the derivatives of $e^x$ never vanish. 
A: The only smooth functions for which a derivative of some order is identically zero are polynomials.
This is essentially just computing $\int^{(n)}0\;dx$ for arbitrary $n$ (don't forget the additive constant at each stage).
A: Certainly, $f(x) = Ce^{\pm x} \neq 0$ for a fixed non-zero $C$.
A: Of course there are, a simple example is $f(x)=e^x$. Since $\frac{d}{dx}e^x=e^x$, the $n$th derivative of $f(x)$ will also be $f(x)$, which will never be 0.
A: Many special functions, such as $\text{erf} (x),\Gamma(x), J_0 (x)$ etc have derivatives such that it will never equal 0 after $n$ derivatives.
