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 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 classical polynomial
Delta derivative of reciprocal of classical polynomial
Relationship between nabla derivative and delta derivative
Relationship between delta derivative and nabla derivative

References

• Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Definition 1.10