AP Calculus AB and BC glossary

Relative rate of change

Also called: Logarithmic derivative, Percentage rate of change

The relative rate of change of a positive quantity is its derivative divided by its current value, so growth is measured as a fraction of size rather than in absolute units. It is usually reported as a percent per unit time, and it equals the derivative of the natural logarithm of the quantity.

Dividing by $f(x)$ cancels the units of the quantity itself and leaves a rate measured in reciprocal time, which is what makes the number comparable across quantities of very different size. Multiplying by 100 turns it into a percent per unit time. The identity below is why logarithmic differentiation and relative rate are the same computation.

\big(\ln f(x)\big)' = \frac{f'(x)}{f(x)}, \qquad f(x) > 0

A constant relative rate is exactly what exponential behavior means. For $f(t) = Ce^{kt}$ with $C \neq 0$ , the derivative is $f'(t) = kCe^{kt}$ , so the relative rate is $k$ at every instant. Read in reverse, a quantity whose relative rate holds at $0.03$ per year is $Ce^{0.03t}$ , which multiplies by $e^{0.03} \approx 1.0305$ each year, a genuine yearly increase of about 3.05 percent rather than 3.

The mistake

Ranking growth by the derivative alone, or reporting $f'(x)$ itself as a percent. A city of $50{,}000$ gaining $500$ people a year has relative rate $\frac{500}{50000} = 0.01$ , one percent per year, while a town of $200$ gaining $20$ a year has relative rate $\frac{20}{200} = 0.10$ , ten percent per year. The smaller derivative is the faster relative growth.

Economics builds its single most-used number out of two of these. Price elasticity of demand is the relative rate of change of quantity divided by the relative rate of change of price, which is why it comes out as a pure number with no units attached: price elasticity of demand.

Appears in: Unit 4: Contextual Applications