Rate (mathematics)

A rate is a comparison of two measurements, typically expressed as a quotient with different units, such as miles per hour or dollars per kilogram. The average rate of change of a function f over an interval [a, b] is given by (f(b) − f(a))/(b − a), which represents the slope of the secant line connecting the two points. Introducing the limit as the interval shrinks to zero yields the instantaneous rate of change, or derivative, f′(x) = lim_h→0 (f(x+h) − f(x))/h. This concept is the foundation of differential calculus and allows the precise description of continuously varying phenomena.¹

Rates are ubiquitous in the sciences. In physics, velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity. Chemical reaction rates measure how quickly reactants are consumed or products formed. In economics, interest rates represent the cost of borrowing money, and exchange rates link the value of one currency to another. Demography uses birth and death rates per capita to model population dynamics. Engineering relies on flow rates, thermal transfer rates, and strain rates to design systems. Every such application uses the core mathematical idea of a ratio of changes.¹²

Rates can be classified in several ways. A constant rate occurs when the change is uniform over time, as in linear motion; a variable rate requires calculus to describe. In finance, the nominal interest rate does not account for compounding, whereas the effective annual rate does. Reaction rates in chemistry are often expressed as instantaneous rates, determined from the slope of concentration vs. time curves at a given point. Growth rates, such as in population or economic output, may be simple or compound, and continuous compounding leads to the exponential function. The distinction between average and instantaneous rates is crucial in all fields.²³

Although the modern concept of a rate is tied to the derivative, ancient civilizations used rates implicitly. Babylonian astronomers computed the velocity of celestial bodies using ratios of distance and time. The word “rate” entered English from Latin rata (fixed amount) and originally referred to a proportional share. In the 17th century, Newton and Leibniz independently formalized instantaneous rates, leading to the development of calculus. A less commonly known fact is that rates can be dimensionless when the two quantities have the same unit, such as a strain rate in materials science (change in length per unit length per time). Zero rates and infinite rates (e.g., at a vertical tangent) represent edge cases that challenge intuition.¹⁴

Rate

A ratio comparing two quantities, typically with different units, expressing how one changes with respect to the other.

Derivative

The instantaneous rate of change of a function, defined as the limit of the average rate of change as the interval approaches zero.

Average rate of change

The change in a function over an interval divided by the length of the interval; the slope of the secant line.

Instantaneous rate of change

The rate of change at a specific point, given by the derivative.

Slope

The steepness of a line, measured as the ratio of vertical change to horizontal change; equivalent to a rate of change.

Velocity

The rate of change of position with respect to time; a vector quantity.

Interest rate

The proportion of a loan charged as interest to the borrower, typically expressed as an annual percentage of the principal.