In[136]:= Table[
 With[{z = InverseFunction[f][x]},
  BellY[Table[{Pochhammer[n, j]/
       Derivative[1][f][
        z], -Derivative[j + 1][f][
         z]/((j + 1) Derivative[1][f][z]^(j + 1))}, {j, n - 1}]] -
    D[z, {x, n}] // Simplify], {n, 2, 6}]

Out[136]= {0, 0, 0, 0, 0}

In[137]:=
With[{p = 4, z = InverseFunction[f][x]},
  Sum[Derivative[m][g][z] BellY[p, m,
      Table[If[k == 1, 1/Derivative[1][f][z],
        BellY[Table[{Pochhammer[k, j]/
           Derivative[1][f][
            z], -Derivative[j + 1][f][
             z]/((j + 1) Derivative[1][f][z]^(j + 1))}, {j,
           k - 1}]]], {k, p}]], {m, 1, p}] - (D[
     g[z], {x, p}])] // Expand

Out[137]= 0