D[Mod[m, x], x] == Sign[x] (IntegerPart[m/x] - (m/x^2) Sum[KroneckerDelta[k, 0] DiracDelta[m/x - k], {k, -Infinity, Infinity}])

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`x </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> x </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> int </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> x </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mtext> </mtext> </mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> k </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mfrac> <mi> m </mi> <mi> x </mi> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> x </ci> </bvar> <apply> <rem /> <ci> $CellContext`m </ci> <ci> $CellContext`x </ci> </apply> </apply> <apply> <times /> <apply> <ci> Sign </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <ci> int </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> DiracDelta </ci> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29