---
tag: 027K9RMZ43
kind: definition
name: Derivative function
msc: [26A24]
defines: 0QKJ6NT0R9
uses: [0F1WR9C83K, 0A61B60B5R]
sources:
  - url: https://en.wikipedia.org/wiki/Derivative
    locator: Definition
---

## Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$ be [[differentiable|0A61B60B5R]] on $I$.
The derivative function of $f$ is the function $f' \colon I \to \mathbb{R}$ that maps each $x \in I$ to the [[derivative|0F1WR9C83K]] $f'(x)$ of $f$ at $x$.

## Remarks

Other notations are $\frac{df}{dx}$ and $D f$.
For example, the derivative function of $x^3$ is $3 x^2$.
The derivative function need not be continuous. See [[x^2 sin(1/x)|0WDAKKN4NC]].
Mathlib defines `deriv f` on all of $\mathbb{R}$, with value $0$ at points where $f$ is not differentiable.

## Explanations

### L1 human

The derivative function collects the slopes of $f$ at all points into one new function.
We can then differentiate it again.