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Delta derivative of constant multiple - Revision history
2024-03-28T08:18:54Z
Revision history for this page on the wiki
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http://timescalewiki.org/index.php?title=Delta_derivative_of_constant_multiple&diff=1299&oldid=prev
Tom at 05:45, 10 June 2016
2016-06-10T05:45:23Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 05:45, 10 June 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (ii)</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (ii)</div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_derivative_of_constant_multiple&diff=1289&oldid=prev
Tom at 05:36, 10 June 2016
2016-06-10T05:36:46Z
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (<del class="diffchange diffchange-inline">iii</del>)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (<ins class="diffchange diffchange-inline">ii</ins>)</div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_derivative_of_constant_multiple&diff=1288&oldid=prev
Tom at 05:36, 10 June 2016
2016-06-10T05:36:17Z
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 05:36, 10 June 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Theorem==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Theorem==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Let $\mathbb{T}$ be a [[time scale]] and $f<del class="diffchange diffchange-inline">,g </del>\colon \mathbb{T} \rightarrow \mathbb{R}$ [[delta derivative|delta differentiable]]. Then the <del class="diffchange diffchange-inline">product </del>function $<del class="diffchange diffchange-inline">fg</del>$ is delta differentiable with</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Let $\mathbb{T}$ be a [[time scale]]<ins class="diffchange diffchange-inline">, $\alpha \in \mathbb{R}$, </ins>and $f \colon \mathbb{T} \rightarrow \mathbb{R}$ [[delta derivative|delta differentiable]]. Then the function $<ins class="diffchange diffchange-inline">\alpha f</ins>$ is delta differentiable with</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$$(<del class="diffchange diffchange-inline">fg</del>)^{\Delta}(t)=<del class="diffchange diffchange-inline">f^{</del>\<del class="diffchange diffchange-inline">Delta}(t)g(t)+</del>f<del class="diffchange diffchange-inline">(\sigma(t))g</del>^{\Delta}(t)<del class="diffchange diffchange-inline">,$$</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$$(<ins class="diffchange diffchange-inline">\alpha f</ins>)^{\Delta}(t)=\<ins class="diffchange diffchange-inline">alpha </ins>f^{\Delta}(t)<ins class="diffchange diffchange-inline">.</ins>$$</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">where </del>$<del class="diffchange diffchange-inline">\sigma</del>$ <del class="diffchange diffchange-inline">denotes the [[forward jump]].</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof==</div></td></tr>
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Tom
http://timescalewiki.org/index.php?title=Delta_derivative_of_constant_multiple&diff=1285&oldid=prev
Tom: Created page with "==Theorem== Let $\mathbb{T}$ be a time scale and $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ delta differentiable. Then the product function $fg$ i..."
2016-06-10T05:34:04Z
<p>Created page with "==Theorem== Let $\mathbb{T}$ be a <a href="/index.php/Time_scale" title="Time scale">time scale</a> and $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ <a href="/index.php/Delta_derivative" title="Delta derivative">delta differentiable</a>. Then the product function $fg$ i..."</p>
<p><b>New page</b></p><div>==Theorem==<br />
Let $\mathbb{T}$ be a [[time scale]] and $f,g \colon \mathbb{T} \rightarrow \mathbb{R}$ [[delta derivative|delta differentiable]]. Then the product function $fg$ is delta differentiable with<br />
$$(fg)^{\Delta}(t)=f^{\Delta}(t)g(t)+f(\sigma(t))g^{\Delta}(t),$$<br />
where $\sigma$ denotes the [[forward jump]].<br />
<br />
==Proof==<br />
<br />
==References==<br />
* {{BookReference|Dynamic Equations on Time Scales|2001|Martin Bohner|author2=Allan Peterson|prev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (iii)</div>
Tom