---
tag: AZBDB5F4M4
kind: concept
name: Difference quotient
msc: [26A24, 26A06]
sources:
  - url: https://en.wikipedia.org/wiki/Difference_quotient
same_as:
  wikidata: Q1224446
  wikipedia: Difference quotient
---

## Summary

The difference quotient of a function between two points is the change of the value divided by the change of the input.
It is the slope of the secant line through two points of the graph.
The [[derivative|0F1WR9C83K]] is the limit of the difference quotient as the two points merge.
Difference quotients also appear on their own in the mean value theorem and in numerical approximation of derivatives.

## Explanations

### L1 human

If you drive $120$ kilometres in $2$ hours, your average speed is $60$ kilometres per hour.
That average speed is a difference quotient: change of position over change of time.
On a graph, it is the slope of the straight line joining two points of the curve.
For $f(x) = x^2$ between $x = 1$ and $x = 3$, the difference quotient is $(9 - 1)/(3 - 1) = 4$.