AP® Calculus AB · Unit 4 · Topic 4.1
Units of a Derivative
A derivative's units are always (units of the output) per (units of the input), because the derivative is a limit of a ratio of those two changing quantities. The units tell you exactly what kind of rate you're looking at.
A derivative comes from the difference quotient, (change in output)/(change in input). That fraction never disappears when you take the limit — it just gets squeezed down to an instant. So whatever units sat on top and bottom of that fraction stay on top and bottom of the derivative. If y is measured in meters and x in seconds, dy/dx is measured in meters per second, full stop. This is why a rate is never measured in the same units as the original quantity: the original quantity is an amount, but the derivative is a comparison of two amounts, and a comparison of two different units is its own new unit.
Take P(t), a population in thousands of people, with t in years. P(t) itself is just thousands of people. But P'(t) compares a change in population to a change in time, so its units are thousand people per year. Differentiate again and you divide by years a second time: P''(t) has units of thousand people per year squared, even though nothing is literally being squared except the unit label on the denominator.
This same bookkeeping reappears the moment you set up a rate of change in a geometric or physical context. For a circle, A = π r2 gives dA/dr = 2 π r, and since A is in square centimeters and r is in centimeters, dA/dr comes out in cm2 per cm — which simplifies to plain cm, matching the units a rate needs when it plugs into a related-rates equation later in this unit.
Key terms
- difference quotient
- The ratio (change in output) over (change in input) whose limit defines the derivative.
- derivative
- The instantaneous rate of change of one quantity with respect to another.
- second derivative
- The derivative of the derivative, carrying units of the original output per unit of input squared.
- related rates
- A setup where two or more changing quantities are linked by an equation, and their rates share consistent units.
Watch it explained
- How it worksHow does the fraction inside the difference quotient decide the units of the answer?
- Worked exampleFind the units of P prime and P double prime when P(t) is population in thousands after t years.
- Common mistakeWhy isn't a rate measured in the same units as the quantity it came from?
- Ties togetherHow do these units resurface once you set up rates of change in real contexts like volume and area?
Check yourself
End of topic.