Second derivative

Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be differentiable on $I$. If $f'$ is differentiable at $a \in I$, the second derivative of $f$ at $a$ is $f''(a) = (f')'(a)$.

Remarks

It is the higher derivative $f^{(2)}$. Another notation is $\frac{d^2 f}{dx^2}$. If $f'' > 0$ on $I$, then $f$ is convex on $I$. In mechanics, if $f(t)$ is the position at time $t$, then $f''(t)$ is the acceleration.

Explanations

L1 human

The second derivative is the rate of change of the slope. It tells how the graph bends: upward where $f'' > 0$ and downward where $f'' < 0$.