Functions that are always less than their derivatives I was wondering if there are functions for which $$f'(x) > f(x)$$ for all $x$. Only examples I could think of were $e^x - c$ and simply $- c$ in which $c > 0$. Also, is there any significance in a function that is always less than its derivative?

Edit: Thank you very much for all the replies. It seems almost all functions that apply are exponential by nature... 
Are there more examples like - 1/x?
Again are there any applications/physical manifestations of these functions? [for example an object with a velocity that is always greater than its position/acceleration  is always greater than its velocity]

A very simple example is $f(x) = -1 < 0 = f'(x)$.  Relevant to your edit:  this isn't exponential at all.
Other examples that aren't immediately exponential:


*

*$\frac{-\pi}{2} + \arctan x$ is everywhere negative and everywhere strictly monotonically increasing, so is everywhere less than its derivative.

*$-1 + \mathrm{erf}(x)$ is also everywhere negative and everywhere strictly monotonically increasing.  (These are very similar, since they are shifted copies of the CDFs of the (standard/normalized) Cauchy and Gaussian distributions.)

*$\frac{1}{2}\left( x - \sqrt{x^2 + 4} \right)$ is the lower branch of a hyperbola having the $x$-axis and the line $y = x$ as asymptotes.  It is everywhere negative and everywhere strictly monotonically increasing.

