---
tag: 0Y9D2FJRQ4
kind: definition
name: Second derivative
msc: [26A24]
defines: 0Q4QJWEHW3
uses: [084NGSTWM7]
sources:
  - url: https://en.wikipedia.org/wiki/Second_derivative
    locator: lead section
---

## 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|084NGSTWM7]] $f^{(2)}$. Another notation is $\frac{d^2 f}{dx^2}$.
If $f'' > 0$ on $I$, then $f$ is [[convex|0VV0HSW1ZS]] 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 $f'' < 0$.