Interpreting the Derivative in Context: Units and Rates from Tables

The sentence that earns the point

A derivative in context is the rate of change of one quantity with respect to another. If $W(t)$ is the water temperature in °C at time $t$ minutes, then $W'(4)=1.5$ means: "At $t=4$ minutes, the temperature of the water is increasing at a rate of $1.5$ °C per minute." Graders award the interpretation point only when the sentence has all four ingredients: what is changing, increasing or decreasing, the value with units, and when.

Units are always output units per input unit: $\dfrac{dV}{dt}$ in litres per hour, $\dfrac{dC}{dx}$ in dollars per item; a second derivative doubles the denominator, so $W''(t)$ is in °C per minute per minute.

Estimating a rate from a table

With only a table, estimate $f'(a)$ by the difference quotient over the smallest interval containing $a$ and show the arithmetic. Example: $W(4)=61$, $W(7)=70$ give $W'(5)\approx\dfrac{70-61}{7-4}=3$ °C per minute — "at $t=5$ minutes the temperature is increasing at approximately $3$ °C per minute."

Rates outside motion

The same reading covers marginal cost $C'(x)$, population growth $P'(t)$, or the rate $R(t)$ at which water enters a tank. When $R$ is itself a rate, $R'(t)\lt0$ says the inflow is slowing, not that the amount is decreasing — two claims AP distractors deliberately swap.