3.6 Higher-Order Derivatives

Second derivative

The second derivative f''(x) is the derivative of f'(x) — the rate at which the rate is changing. It relates to concavity of a graph and acceleration in motion.

Notation

Written f''(x), d²y/dx², or y''. Keep differentiating for third, fourth, etc. Example: f(x) = x⁴ ⇒ f'(x) = 4x³ ⇒ f''(x) = 12x².

Key terms

Second derivative

f'' = the derivative of f'.

Notation

f''(x), d²y/dx², y''.

Concavity

f'' describes how a graph curves.

Acceleration

The second derivative of position.

Example

f = x⁴ ⇒ f'' = 12x².

Try an AP-style question

For f(x) = x⁴ − 3x², what is f''(x)?

1. 4x³ − 6x
2. 12x² − 6
3. 12x² − 6x
4. 24x