D[ProductLog[z], z] == ProductLog[z]/(z (1 + ProductLog[z]))

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["ProductLog", "[", "z", "]"]]]], "\[Equal]", FractionBox[RowBox[List["ProductLog", "[", "z", "]"]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ProductLog", "[", "z", "]"]]]], ")"]]]]]]]]]

<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> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mfrac> <mrow> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> W </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> W </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> W </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <ci> W </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["ProductLog", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["ProductLog", "[", "z", "]"]], RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ProductLog", "[", "z", "]"]]]], ")"]]]]]]]]]

2001-10-29