# Delta derivative

Let $\mathbb{T}$ be a time scale and let $f \colon \mathbb{T} \rightarrow \mathbb{R}$ and let $t \in \mathbb{T}^{\kappa}$. We define the $\Delta$-derivative of $f$ at $t$ to be the number $f^{\Delta}(t)$ (if it exists) so that there exists a $\delta >0$ so that for all $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$, $$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$ We sometimes use the notation $\dfrac{\Delta}{\Delta t} f(t)$ or $\dfrac{\Delta f}{\Delta t}$ for $f^{\Delta}(t)$.

## Properties of the $\Delta$-derivative

Delta derivative of constant

Delta differentiable implies continuous

Delta derivative at right-scattered

Delta derivative at right-dense

Delta simple useful formula

Delta derivative of sum

Delta derivative of constant multiple

Delta derivative of product (1)

Delta derivative of product (2)

Delta derivative of reciprocal

Delta derivative of quotient

Delta derivative of squaring function

Delta derivative of classical polynomial

Delta derivative of reciprocal of classical polynomial

Relationship between nabla derivative and delta derivative

Relationship between delta derivative and nabla derivative

Delta mean value theorem

## References

- Martin Bohner and Allan Peterson:
*Dynamic Equations on Time Scales*(2001)... (previous)... (next): Definition 1.10 - Dorota Mozyrska and Ewa Pawluszewicz:
*Functional series on time scales*(2008)... (previous)... (next): Definition 2.1