f = Cos[x];

d = (∂_x) f

da1[h_] := ((Cos[x + h] - Cos[x])/h)

Series[da1[h] - d, {h, 0, 3}]

L10[t_] := 
        If[Abs[t] < (1/(10^20)), 20, N[-Log[10, Abs[t]]]];

err[k_] := 
      L10[d - da1[(10^(-k))] /. x -> 1.1]

Do[Print[N[(10^(-k))], "   ", err[k]], {k, 1, 15, 2}]

Plot[err[k], {k, 0, 15}];

da2[h_] := ((Cos[x + h] - Cos[x - h])/(2 h));

err2[h_] := 
        L10[d - da2[(10^(-h))] /. x -> 
              1.1];

Series[da2[h] - d, {h, 0, 5}]

Plot[{err[k], err2[k]}, {k, 0, 15}];

da4[h_] := -da2[h]/3 + (4/3) da2[(h/2)];

err4[h_] := 
        L10[d - da4[(10^(-h))] /. x -> 
              1.1];

Series[da4[h] - d, {h, 0, 5}]

Plot[{err[k], err2[k], err4[k]}, {k, 0, 15}];