rate of change

A rate of change tells you how much the output moves for each step of the input. Driving 120 miles in 2 hours is an average rate of 60 miles per hour: that single number folds the whole trip into one summary of how fast position changed with time. Speed, growth, cooling, interest — almost every changing quantity is described by some rate.

For a function f, the average rate of change between inputs a and b is the change in output divided by the change in input, (f(b) - f(a)) / (b - a). Geometrically this is the slope of the secant line — the straight line drawn through the two points (a, f(a)) and (b, f(b)) on the graph. Rise over run: how much the function climbed, shared out evenly across the interval.

This is the seed from which the derivative grows. The average rate smooths over a whole stretch, so it can hide what happens moment to moment — a car averaging 60 mph might have been parked, then sped to 90. To capture the rate at a single instant rather than over an interval, you shrink b toward a and watch the secant slope approach a limiting value; that limit is the instantaneous rate of change, the derivative.