If you point out the dependency on e to Mathematica like this
In[1]:= Z?[b_, e_] := 0.5 (Exp[-2 b e] + Exp[-b e] + 1)^2
In[2]:= L[b_, e_] := -Log[Z?[b, e]]
In[3]:= ?prom[b_, e_] := D[L[b, e], b]
In[4]:= ?prom[b, e]
Out[4]= -((2. (-2 e E^(-2 b e) - e E^(-b e)))/(1 + E^(-2 b e) + E^(-b e)))
then does everything work?