D[Root[Function[z, Sum[Subscript[a, j] z^j, {j, 0, n}]], k], Subscript[a, k]] == -(R^k/(k Subscript[a, k] R^(k - 1) + Sum[(1 - KroneckerDelta[k, j]) j Subscript[a, j] R^(j - 1), {j, 1, n}])) /; R == Root[Function[z, Sum[Subscript[a, j] z^j, {j, 0, n}]], k]

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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> <mrow> <mrow> <mfrac> <mo> &#8706; </mo> <mrow> <mo> &#8706; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> </mfrac> <mo> &#8290; </mo> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mi> R </mi> <mi> k </mi> </msup> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> R </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> k </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> j </mi> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mi> R </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> R </mi> <mo> &#10869; </mo> <msubsup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8706; 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</ms> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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2001-10-29