AP Calculus AB and BC glossary

Second derivative

The second derivative is what you get by differentiating the derivative. It measures how the rate of change is itself changing, it determines concavity, and in motion problems it is acceleration.

f''(x) = \frac{d}{dx}\left[f'(x)\right] = \frac{d^{2}y}{dx^{2}}

A positive second derivative means the graph is concave up and the first derivative is increasing. That is a stronger statement than the function increasing, and the exam tests the difference constantly.

The mistake

Reading a positive second derivative as the function increasing. It says the SLOPE is increasing; a function can be decreasing and concave up at the same time.

Appears in: Unit 3: Chain Rule, Implicit, and Inverses, Unit 5: Analytical Applications