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http://functions.wolfram.com/01.06.20.0005.02
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D[f[Sin[z]], {z, n}] == Sum[(Derivative[m][f][Sin[z]]/m!)
(D[(Sin[y] - Sin[z])^m, {y, n}] /. {y -> z}), {m, 0, 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["Sin", "[", "z", "]"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], RowBox[List["m", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["y", ",", "n"]], "}"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "y", "]"]], "-", RowBox[List["Sin", "[", "z", "]"]]]], ")"]], "m"]]], "/.", "\[InvisibleSpace]", RowBox[List["{", RowBox[List["y", "\[Rule]", "z"]], "}"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> sin </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> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <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> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> y </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> /. </mo> <mtext> ​ </mtext> <mrow> <mo> { </mo> <mrow> <mi> y </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> z </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </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> <sin /> <ci> z </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> f </ci> <apply> <sin /> <ci> z </ci> </apply> </apply> <list> <apply> <sin /> <ci> z </ci> </apply> <ci> m </ci> </list> </apply> <apply> <power /> <apply> <factorial /> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ReplaceAll </ci> <apply> <partialdiff /> <bvar> <ci> y </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <power /> <apply> <plus /> <apply> <sin /> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <ci> z </ci> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> <list> <apply> <ci> Rule </ci> <ci> y </ci> <ci> z </ci> </apply> </list> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["f", "[", RowBox[List["Sin", "[", "z_", "]"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["Sin", "[", "z", "]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["y", ",", "n"]], "}"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", "y", "]"]], "-", RowBox[List["Sin", "[", "z", "]"]]]], ")"]], "m"]]], "/.", "\[InvisibleSpace]", RowBox[List["{", RowBox[List["y", "\[Rule]", "z"]], "}"]]]], ")"]]]], RowBox[List["m", "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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