It recently occurred to me that calculus is really about finding good local polynomial approximations of functions. The derivative is often first introduced as finding the slope of a function "at a point". What this of course really means is we want to find the slope of the line which best locally approximates the function within a sufficiently small region of that point. When we learn about second derivatives we are told that we are simply taking the derivative of the first derivative which is true enough. But what we are really doing is we are finding the coefficient of the x^2 term in the best local quadratic approximator of the function. In general taking derivatives of a function just means finding the best local polynomial approximators and then looking at the appropriate coefficients for the order of derivative you want.
Of course in practice it goes exactly the other way. If you want a good polynomial approximimation of a function around a point you use the taylor series expansion of the function which you build up from its derivatives.