D[BesselK[0, z], z] == -BesselK[1, z]

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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> <msub> <mi> K </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <msub> <mi> K </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> BesselK </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> BesselK </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["BesselK", "[", RowBox[List["0", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", RowBox[List["BesselK", "[", RowBox[List["1", ",", "z"]], "]"]]]]]]]]

2001-10-29