D[f[Cos[z]], {z, n}] == Sum[(Derivative[m][f][Cos[z]]/m!) (D[(Cos[y] - Cos[z])^m, {y, n}] /. {y -> z}), {m, 0, n}] /; Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["f", "[", RowBox[List["Cos", "[", "z", "]"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "m", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["Cos", "[", "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["Cos", "[", "y", "]"]], "-", RowBox[List["Cos", "[", "z", "]"]]]], ")"]], "m"]]], "/.", "\[InvisibleSpace]", RowBox[List["{", RowBox[List["y", "\[Rule]", "z"]], "}"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<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> &#8706; </mo> <mi> n </mi> </msup> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </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[&quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> y </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> /. </mo> <mtext> &#8203; </mtext> <mrow> <mo> { </mo> <mrow> <mi> y </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> z </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </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> <cos /> <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> <cos /> <ci> z </ci> </apply> </apply> <list> <apply> <cos /> <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> <cos /> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <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> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["f", "[", RowBox[List["Cos", "[", "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["Cos", "[", "z", "]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["y", ",", "n"]], "}"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", "y", "]"]], "-", RowBox[List["Cos", "[", "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"]]]]]]]]]]

2001-10-29