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http://functions.wolfram.com/01.19.20.0004.01
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D[f[Sinh[z]], {z, n}] ==
(-1)^n Sum[(1/m!) Sum[Binomial[m, j]
Sum[((-1)^(j - l) 2^(j - m) Sinh[z]^j (2 l + j - m)^n
Binomial[m - j, l] Derivative[m][f][Sinh[z]])/E^((j + 2 l - m) z),
{l, 0, m - j}], {j, 0, m - 1}], {m, 1, n}] /;
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["f", "[", RowBox[List["Sinh", "[", "z", "]"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], RowBox[List[FractionBox["1", RowBox[List["m", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], RowBox[List["m", "-", "j"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "l"]]], SuperscriptBox["2", RowBox[List["j", "-", "m"]]], " ", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "l"]], "+", "j", "-", "m"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["j", "+", RowBox[List["2", " ", "l"]], "-", "m"]], ")"]]]], " ", "z"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", "j"]], ",", "l"]], "]"]], " ", RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["Sinh", "[", "z", "]"]], "]"]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> j </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mi> l </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mi> j </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> l </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> m </mi> <mo> - </mo> <mi> j </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> l </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", "j"]], Identity, Rule[Editable, True]]], List[TagBox["l", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msup> <mi> f </mi> <semantics> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "m", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> f </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> l </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> l </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <ci> l </ci> </apply> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <sinh /> <ci> z </ci> </apply> </apply> <list> <apply> <sinh /> <ci> z </ci> </apply> <ci> m </ci> </list> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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