AP Calculus AB and BC glossary

Higher-order derivative

Also called: Second derivative

A higher-order derivative is what you get by differentiating repeatedly. The second derivative is the derivative of the derivative and measures how the rate of change is itself changing, which is what determines concavity and acceleration.

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

In motion problems the chain of meanings is concrete: position, then velocity, then acceleration. In graph analysis, the first derivative controls increasing and decreasing behavior while the second controls concavity.

Notation varies by order. Primes are standard through the third derivative, after which a parenthesized superscript takes over, as in $f^{(4)}(x)$ . In Leibniz form the second derivative is $\frac{d^2y}{dx^2}$ .

The mistake

Reading $\frac{d^2y}{dx^2}$ as something squared. It is a single symbol for differentiating twice, not a power of anything.

Appears in: Unit 2: Defining the Derivative, Unit 5: Analytical Applications