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.

Appears in: Unit 4: Contextual Applications