In[8]:= f[u_] := D[g[u], u]

In[9]:= f[u]^2 == 4 g[u]^3 - a*g[u] - b

Out[9]= Derivative[1][g][u]^2 == -b - a g[u] + 4 g[u]^3

In[10]:= FullSimplify[f'''[u]/f'[u] - 1.5 f''[u]^2/f'[u]^2]

(
!(*SuperscriptBox[(g),
TagBox[
RowBox[{"(", "4", ")"}],
Derivative],
MultilineFunction->None])[u] (g^[Prime][Prime])[u] - 1.5
!(*SuperscriptBox[(g),
TagBox[
RowBox[{"(", "3", ")"}],
Derivative],
MultilineFunction->None])[u]^2)/(g^[Prime][Prime])[u]^2