Algorithms: Big-O notation and calculation

I was wondering if there is any calculation relation implicit in the Big-O notation.

For example, a linear algorithm according to the Big-O notation reduces the size of the problem by a constant amount at each step, and also involves observing each part of the input a constant number of times. The derivative of a linear expression is a constant expression, so there is some indication of a pattern. However, I have not been able to discover how to generalize these facts to other kinds of Big-O algorithms.

Do derivatives / antiderivatives help match the Big-O description of an algorithm with its behavior?