---
tag: XXQ92H7T83
kind: concept
name: Delta derivative on a time scale
msc: ["26E70", 39A12]
sources:
  - url: https://en.wikipedia.org/wiki/Time-scale_calculus
---

## Summary

The delta derivative is the derivative of [[time-scale calculus|30G1M35TK1]], defined for functions on a time scale, that is, on a nonempty closed subset of $\R$.
At a point that is a limit from the right it is the usual derivative.
At a point isolated on the right it is a difference quotient across the gap.
On $\R$ it is the ordinary [[derivative|0V1EJCT8G6]], and on $\Z$ it is the forward difference, so it unifies differential and difference equations.

## Explanations

### L1 human

Picture a time line in which some stretches are continuous and some moments are separated by gaps.
At a moment followed by a gap, the delta derivative compares the value now with the value at the next moment, and divides by the length of the gap.
At a moment with no gap to its right, it is the usual slope.
For example, on the time scale $2\Z = \set{\dots, -2, 0, 2, 4, \dots}$, the delta derivative of $f(t) = t^2$ is $\frac{(t+2)^2 - t^2}{2} = 2t + 2$.