4.1 Interpreting the Derivative in Context

Rate with units

In context, the derivative is a rate of change with units of (output units) per (input unit). If V(t) is volume in liters and t is seconds, then V'(t) is in liters per second.

Sign and meaning

A positive derivative means the quantity is increasing; a negative derivative means it is decreasing. So f'(3) = 5 means at x = 3, f is increasing at 5 output units per input unit.

Key terms

Derivative in context

A rate of change with real-world units.

Units

Output units per input unit.

Positive derivative

The quantity is increasing.

Negative derivative

The quantity is decreasing.

Interpreting f'(a)

The instantaneous rate of change at x = a.

Try an AP-style question

P(t) is a town's population in thousands, t years after 2020. What does P'(5) = 1.2 mean?

1. In 2025, the population is growing at about 1,200 people per year
2. In 2025, the population is 1.2 thousand people
3. In 2025, the population is 1,200 people larger than in 2020
4. Over five years, the population grew by 1.2 thousand