Delta derivative of reciprocal

From timescalewiki
Revision as of 15:19, 21 January 2023 by Tom (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Theorem

Let $\mathbb{T}$ be a time scale, $t \in \mathbb{T}^{\kappa}$, $f \colon \mathbb{T} \rightarrow \mathbb{R}$ delta differentiable, and $f(t)f(\sigma(t)) \neq 0$. Then $\dfrac{1}{f}$ is delta differentiable and $$\left( \dfrac{1}{f} \right)^{\Delta}(t) = -\dfrac{f^{\Delta}(t)}{f(t)f(\sigma(t))},$$ where $\sigma$ denotes the forward jump.

References