(*            Formula di Taylor        -1-    *)

f[x_]:=Sin[x]

?D

D[f[x],x]

D[ArcTan[x],x]

D[x^3ArcCos[x],{x,3}]

?Simplify

Simplify[%5]

?Collect

Collect[%,x]

?Together

Together[%8]

?Expand

Expand[%11]

Together[D[ArcCos[x],{x,3}]]

Together[D[ArcCos[x],{x,3}]/.x->x0]

D[ArcCos[x],{x,3}]/.x->(1/2)

D[ArcCos[x],x]/.x->x0

D[ArcCos[x],x]/.x->(1/2)

?Sum

Sum[k,{k,n}]

Sum[g[k],{k,n}]

Clear[f]

D[f[x],{x,k}]/.x->x0

?Factorial

Factorial[3]

Factorial[5]

taylor[f_,x0_,npas_][x_]:=f[x0]+Sum[(D[f[x],{x,k}]/.x->x0)/Factorial[k]*(x-x0)^k,{k,1,npas}]

taylor[f,0,1][x]

taylor[f,0,5][x]

taylor[Sin,0,5][x]