Higher-Order Derivatives

$f''(x) = \frac{d^2y}{dx^2}$ (second derivative).

$f'''(x) = \frac{d^3y}{dx^3}$ (third derivative).

• $f'(x)$: rate of change / velocity.
• $f''(x)$: rate of change of the rate / acceleration / concavity.

• $f''(x) > 0$: concave up (cup shape).
• $f''(x) < 0$: concave down (cap shape).
• $f''(x) = 0$: possible inflection point.

Where concavity changes: $f''(x) = 0$ AND $f''$ changes sign.

$f(x) = x^4 - 6x^2$. $f'(x) = 4x^3 - 12x$. $f''(x) = 12x^2 - 12$.

$f''(x) = 0$: $12(x^2 - 1) = 0$ → $x = \pm 1$.

$f''$ changes sign at both → inflection points at $x = -1$ and $x = 1$.