What is the relationship between higher-order derivatives and their functions? When I say higher order I mean derivatives above the second derivative.

For example,

f(x) = x[sup]4[/sup]

f’(x) = 4x[sup]3[/sup] finds the slope of f(x) at x; it evaluates the rate of change of f. As f increases, the slope increases.

f’’(x) = 12x[sup]2[/sup] finds the slope of f’(x); it evaluates the rate of change of the slope of f. The slope changes more over shorter intervals and progressively slows down until x=0, where it changes progressively faster.

So, what is the relevance of f’’’(x) to f? f[sup]4/sup? f[sup]infinity/sup? In a similar note, what does the rate of change of acceleration called, and what does it measure?