CellPrint[TextCell[Quiet[cdf =Manipulate[Column[{ "Fórmula usada:\n"<>ToString@ TraditionalForm[(("f")^("("<>ToString[u]<>")"))[x0]]<>"≈"TraditionalForm@formulas[[u]], Dynamic@SuFunc}],Style[proposito,Bold,10],Spacer[{100,3}],
 (*Inicio contadores *)
 {{u,1,"Elija el orden de la derivada para aproximar: "},1,4,1,ControlType->SetterBar},
{{ff,Cos[x],"f(x) = "},ControlType->InputField},
{{punto,2,"x0 = "},ControlType->InputField},
{{hache,0.1,"h = "},ControlType->InputField},
Button["Calcular", Dynamic@ SuFunc]
(* Fin contadores *)
,Alignment->Center,Deployed->True,AppearanceElements->"ResetButton",LocalizeVariables->True,ContinuousAction->False,FrameLabel->{{None,None},{Column[{,Row[{Style["Autores: Juan Félix Ávila Herrera y Enrique Vílchez Quesada",Bold,10,RGBColor[0,0,0.5]]}],Row[{Style["Escuela de Informática | Universidad Nacional de Costa Rica",Bold,10,RGBColor[0.5,0.1,0]]}]},Center],Style["Aproximación numérica de las primeras cuatro derivadas ", Bold,14, RGBColor[0,0.33,0.33]]}},
Initialization:>(
F[x0_, h_, f_] := ((-f[x0-h]+f[h+x0])/(2h));
formulas ={g1[x0_] = Together@F[x0, h, f],
g2[x0_]=Together@F[x0, h, g1] ,
g3[x0_]=Together@F[x0, h, g2] ,
g4[x0_]=Together@F[x0, h, g3]};
 
Clear[f];

(*($PreRead=(#/. s_String/;StringMatchQ[s,NumberString]&&Precision@ToExpression@s==MachinePrecision:>s<>"`12."&));*)
proposito ="Propósito: Aproximar las primeras cuatro derivadas y usarlas para aproximar el polinomio de Taylor.";
SuFunc := Module[{aux, resp, pt, coef},
resp[x_] := If[x==Round[x], x,ToString[x, InputForm]<> "≈"<>ToString[Sign[x]AccountingForm[Abs@N@x]]];
f[t_] := ff/.x->t;
aux =formulas[[u]]/.{x0-> punto, h->hache, f->g };
aux = ToString@ TraditionalForm[(("f")^("("<>ToString[u]<>")"))[punto]]<>" ≈ "<>ToString[Sign[aux]AccountingForm[Abs@aux]]<>"\nUsando Mathematica: "<> ToString@ TraditionalForm[(("f")^("("<>ToString[u]<>")"))[punto]]<>" = "<>ToString[resp[D[f[x], {x, u}]/.x->punto]];
pt =f[punto]+(∑_(k=1)^4)(coef =formulas[[k]]/.{x0-> punto, h->hache, f->g };
 (coef/(k!))*(((x-punto))^k));
aux = aux <>"\n----------------------------------------------\nPolinomio de Taylor de orden 4 (usando las fórmulas):\n(Subscript[P, 4])(x) ="<>ToString[N@Expand@pt, InputForm];
Clear[f];
aux
];
Dynamic@SuFunc
)]],"Text"]]

CloudDeploy[cdf,Permissions->"Public"]

Table[(1/(h^m))(∑_(k=0)^m)(((-1))^(k+m))Binomial[m, k] f[x0+k*h], {m, 1, 5}]

formulas =Table[If[m==1,g[x0_]:=Together[((-f[x0-h]+f[h+x0])/(2h))]; g[x0] , 
 g[x0_]=Together[((-g[x0-h]+g[h+x0])/(2h))]
], {m, 1, 4}]

g1[x0_] = ((-f[x0-h]+f[h+x0])/(2h))
g2[x0_]=Together@F[x0, h, g1] 
g3[x0_]=Together@F[x0, h, g2] 
g4[x0_]=Together@F[x0, h, g3]

{g1[x0_] = ((-f[x0-h]+f[h+x0])/(2h)),
g2[x0_]=Together@F[x0, h, g1] ,
g3[x0_]=Together@F[x0, h, g2] ,
g4[x0_]=Together@F[x0, h, g3]}