What does a function's smallest non-zero derivative say about the function? For example, say we have a function that looks like 6x^3, if you keep taking the derivative of this function until you get the smallest non-zero derivative, in this case 6x^3 -> 18x^2 -> 36x -> 36, what is the significance of the number 36 to the function 6x^3? I know that each function has a specific smallest non-zero derivative, however, each non-zero derivative can be characteristic of an infinite amount of functions in you keep integrating it. Is there anything to this thought, or am i just asking a pointless question? also, i was playing with numbers and derived that for nx^l, when l is a whole number, the smallest non zero derivative is (l-1)!*n*l.