I think the answers so far do not really take account of Kevin's remark:

I should remark that this is a list of different ways of thinking about the derivative, which isn't the same thing as a list of different formal definitions of the derivative.

We could just copy from http://en.wikipedia.org/wiki/Derivative_%28generalizations%29 or cite Fermat categories .... Perhaps this is an alternative way:

Smoothing: Continuity in $p$ asserts roughly that you can draw the graph in one line. Differentiabilty is the next step towards smoothness: The line has no zig-zags / edges. The derivative is then the direction of the line.