Relative Rate of Change — Definition, Formula & Examples

Relative rate of change is the derivative of a function divided by the function's current value. It tells you the rate of change as a proportion of the quantity itself, rather than as an absolute amount.

For a differentiable function $f(t)$ , the relative rate of change at $t$ is defined as the ratio $\frac{f'(t)}{f(t)}$ , provided $f(t) \neq 0$ . This quantity equals the derivative of $\ln|f(t)|$ with respect to $t$ .

Key Formula

\text{Relative rate of change} = \frac{f'(t)}{f(t)} = \frac{d}{dt}\ln|f(t)|

Where:

$f(t)$ = The function value at time t
$f'(t)$ = The derivative (instantaneous rate of change) of f at time t

How It Works

To find the relative rate of change, first compute the derivative

f'(t)

, then divide by

f(t)

. The result is often expressed as a decimal or percentage. A relative rate of change of

0.05

means the function is growing at

5\%

per unit of time relative to its current size. This makes it especially useful for comparing growth across quantities of different magnitudes — a population of 1,000 growing by 50 per year and a population of 1,000,000 growing by 50,000 per year both have the same relative rate of change of

0.05

Worked Example

Problem: A company's revenue is modeled by

R(t) = 200e^{0.03t}

thousand dollars, where

t

is years since 2020. Find the relative rate of change of revenue at

t = 5

Find the derivative: Differentiate using the chain rule.

R'(t) = 200 \cdot 0.03 \cdot e^{0.03t} = 6e^{0.03t}

Compute the ratio: Divide the derivative by the original function.

\frac{R'(t)}{R(t)} = \frac{6e^{0.03t}}{200e^{0.03t}} = \frac{6}{200} = 0.03

Interpret the result: The relative rate of change is constant at 0.03, or 3% per year. This holds at

t = 5

and at every other time — a characteristic property of exponential functions.

Answer: The relative rate of change of revenue is

0.03

, or

3\%

per year.

Why It Matters

Economists use relative rate of change to express inflation rates and GDP growth as percentages, which are more meaningful than raw dollar amounts. In biology, relative growth rates let researchers compare organisms of vastly different sizes. Recognizing that the relative rate of change equals the derivative of

\ln|f(t)|

also simplifies many integration and differential equation problems in applied calculus.

Common Mistakes

Mistake: Confusing the (absolute) rate of change

f'(t)

with the relative rate of change

\frac{f'(t)}{f(t)}

Correction: The absolute rate tells you how many units the function changes per unit time. The relative rate tells you what fraction of the current value that change represents. Always divide by

f(t)