Difference between revisions of "Delta derivative of quotient"
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==Theorem== | ==Theorem== | ||
| − | Let $\mathbb{T}$ be a [[time scale]], $t \in \mathbb{T}^{\kappa}$, $f \colon \mathbb{T} \rightarrow \mathbb{R}$ [[delta derivative|delta differentiable]], and $ | + | Let $\mathbb{T}$ be a [[time scale]], $t \in \mathbb{T}^{\kappa}$, $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ [[delta derivative|delta differentiable]], and $g(t)g(\sigma(t)) \neq 0$. Then $\dfrac{f}{g}$ is delta differentiable and |
| − | $$\left( \dfrac{ | + | $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{g(t)f^{\Delta}(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))},$$ |
where $\sigma$ denotes the [[forward jump]]. | where $\sigma$ denotes the [[forward jump]]. | ||
| + | |||
| + | ==Proof== | ||
==References== | ==References== | ||
| − | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of product (2)|next=}}: Theorem 1.20 ( | + | * {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of product (2)|next=Delta derivative of classical polynomial}}: Theorem 1.20 (v) |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 05:44, 10 June 2016
Theorem
Let $\mathbb{T}$ be a time scale, $t \in \mathbb{T}^{\kappa}$, $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ delta differentiable, and $g(t)g(\sigma(t)) \neq 0$. Then $\dfrac{f}{g}$ is delta differentiable and $$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{g(t)f^{\Delta}(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))},$$ where $\sigma$ denotes the forward jump.
Proof
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous)... (next): Theorem 1.20 (v)
