Delta derivative of reciprocal

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

• Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem 1.20 (iv)