rateofchange

Rate of change is a measure of how a quantity changes as another quantity changes. In calculus, it is formalized as the derivative of a function with respect to a variable. There are instantaneous rates of change, given by the derivative at a point, and average rates of change over an interval, given by the slope of the secant line between two points.

The instantaneous rate of change of a function f with respect to x is defined as the limit as h approaches zero of [f(x+h)−f(x)]/h. This limit, when it exists, is denoted f′(x) or df/dx. The average rate of change over an interval [a,b] is [f(b)−f(a)]/(b−a).

Example: for f(x)=x^2, f′(x)=2x, so the instantaneous rate at x=3 is 6. The average rate of change over [2,5] is (25−4)/(5−2)=21/3=7.

In physics, velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity. In economics and biology, rates of change describe growth, decay, or marginal effects. Rates of change are central to modeling with differential equations, where dy/dt = g(y,t) specifies how a quantity changes over time, and to numerical settings where discrete rates are approximated by difference quotients.