Difference between revisions of "Delta derivative"

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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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
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Let $\mathbb{T}$ be a [[time_scale | 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 $s \in (t-\delta,t+\delta) \bigcap \mathbb{T}$,
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
 
$$|[f(\sigma(t))-f(s)]-f^{\Delta}(t)[\sigma(t)-s]| \leq \epsilon |\sigma(t)-s|.$$
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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==
 
==Properties of the $\Delta$-derivative==
*$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t)$
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[[Delta derivative of constant]]<br />
*Sum rule:
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[[Delta differentiable implies continuous]]<br />
$$(f+g)^{\Delta}(t)=f^{\Delta}(t)+g^{\Delta}(t)$$
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[[Delta derivative at right-scattered]]<br />
*Constant rule:if $\alpha$ is constant with respect to $t$, then
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[[Delta derivative at right-dense]]<br />
$$(\alpha f)^{\Delta}(t) = \alpha f^{\Delta}(t)$$
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[[Delta simple useful formula]]<br />
*Product Rule I
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[[Delta derivative of sum]]<br />
$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t))$$
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[[Delta derivative of constant multiple]]<br />
*Product Rule II
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[[Delta derivative of product (1)]]<br />
$$(fg)^{\Delta}(t) = f(t)g^{\Delta}(t)+ f^{\Delta}(t)g(\sigma(t))$$
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[[Delta derivative of product (2)]]<br />
*Quotient Rule:
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[[Delta derivative of reciprocal]]<br />
$$\left( \dfrac{f}{g} \right)^{\Delta}(t) = \dfrac{f^{\Delta}(t)g(t)-f(t)g^{\Delta}(t)}{g(t)g(\sigma(t))}$$
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[[Delta derivative of quotient]]<br />
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[[Delta derivative of squaring function]]<br />
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[[Delta derivative of classical polynomial]]<br />
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[[Delta derivative of reciprocal of classical polynomial]]<br />
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[[Relationship between nabla derivative and delta derivative]]<br />
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[[Relationship between delta derivative and nabla derivative]]<br />
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[[Delta mean value theorem]]<br />
  
==Interesting Examples==
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== References ==
The jump operator $\sigma$ is not $\Delta$-differentiable on all time scales. Consider $\mathbb{T}=
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* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Induction on time scales|next=Delta differentiable implies continuous}}: Definition 1.10
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* {{PaperReference|Functional series on time scales|2008|Dorota Mozyrska|author2=Ewa Pawluszewicz|prev=Left dense|next=Regulated}}: Definition 2.1

Revision as of 14:54, 21 October 2017

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