x0=2;

Clear[f]

(*Método de Newton Para encontrar raiz de 2*)

f[x_]:=x^2-2

Newton[i_,x0_]:={x=x0;For[j=1,j≤i,j++,x=x-f[x]/f'[x];];x}[[1]]

x1=x0-f[x0]/f'[x0];

Manipulate[g=Plot[x^2-2,{x,1,9},AxesOrigin->{1,0}];
r1=Plot[f[it]+(x-it)*f'[it],{x,1,9},AxesOrigin->{1,0},PlotStyle->Red];
Show[g,r1],{it,{10->"(Subscript[x, 0])",Newton[1,10]->"(Subscript[x, 1])",Newton[2,10]->"(Subscript[x, 2])",Newton[3,10]->"(Subscript[x, 3])",Newton[4,10]->"(Subscript[x, 4])",Newton[5,10]->"(Subscript[x, 5])",Newton[6,10]->"(Subscript[x, 6])"}},ControlType->Setter]

Secante[i_,x0_,x1_]:={x=x0;y=x1;For[j=1,j≤i,j++,x=y-(x-y)*f[y]/(f[x]-f[y]);];x}[[1]]

(*Método da Secante para encontrar raiz de 2*)

Manipulate[g=Plot[x^2-2,{x,0,9},AxesOrigin->{1,0}];
it0=Secante[iter,9,10];
it=Secante[iter-1,9,10];
r1=Plot[f[it0]+(x-it0)*(((f[it]-f[it0]))/(it-it0)),{x,0,9},AxesOrigin->{1,0},PlotStyle->Red];
Show[g,r1],{iter,{1,2,3,4,5,6,7}},ControlType->Setter]

Or