AP Calculus AB and BC glossary

Units of a Derivative

Also called: Units of dy/dx, Units of a rate of change

The units of a derivative are the units of the output divided by the units of the input. If a tank holds W gallons after t minutes, W prime is measured in gallons per minute, and a second derivative divides by the input unit once more, giving gallons per minute per minute.

Read the derivative as a fraction and the units follow. If $W(t)$ is water in a tank in gallons and $t$ is in minutes, $W'(t)$ is in gallons per minute. Differentiate a second time and the input unit appears twice, so $W''(t)$ is in gallons per minute per minute. Position in metres against seconds gives velocity in metres per second and acceleration in metres per second squared.

The units are what turn a number into a sentence. $W'(7) = -3$ gallons per minute says the water is leaving at $3$ gallons a minute at $t = 7$ , not that the tank holds $-3$ gallons. The same division sits inside the difference quotient, where a change in gallons is divided by a change in minutes, so an average rate of change over an interval carries exactly the units the derivative at a point does. Leibniz notation reads them off directly: $\frac{dW}{dt}$ is gallons over minutes.

The mistake

Carrying a rate's units into its integral. If $R(t)$ is in litres per hour, then $\int_1^4 R(t)\,dt$ is in litres, and the sentence should say how much the amount of liquid changed between hour $1$ and hour $4$ ; how much arrived is only right when $R$ stays non-negative across the whole interval. The reverse slip costs the same point: reporting a derivative in the units of the original function, as though $W'$ were still measured in gallons.

Appears in: Unit 2: Defining the Derivative, Unit 4: Contextual Applications