---
tag: 4W31MKE01V
kind: concept
name: Approximate derivative
msc: [26A24]
sources:
  - url: https://en.wikipedia.org/wiki/Approximate_limit
---

## Summary

The approximate derivative of a real function at a point is a limit of its difference quotients.
In this limit one may ignore a set that is very thin near the point.
The ignored set must have density $0$ at the point, in the sense of [[Lebesgue's density theorem|WSVXYSNHNX]].
Approximate derivatives exist in many cases where ordinary derivatives do not, and they are the derivatives used in the [[Khinchin integral|Q1TV4F608D]].

## Explanations

### L1 human

An ordinary derivative looks at all points near $x$.
An approximate derivative is allowed to skip a set of points, as long as that set takes up a smaller and smaller share of each small interval around $x$.
For example, change the zero function to the value $1$ at the points $1, \frac12, \frac13, \dots$.
The new function has no derivative at $0$, but its approximate derivative at $0$ is $0$, since the changed points form a countable set, which is invisible to measure.