D[Integrate[f[t], {t, low[x], upper[x]}], x]
(*Out[]=-f[low[x]] (low^[Prime])[x]+f[upper[x]] (upper^[Prime])[x]*)

Clear["Global`*"]
(*a=110.;b=55.;d=1.;m1=18.;m2=42.;m=m2/m1;
*)
w[l_, e_] := (-((m1*a)/2) Log[1 - (l^(-4) + 2*l^2 - 3)/a] - (m2*b)/
      2 Log[1 - (l^-4*e^4 + 2 l^2*e^-2 - 3)/b])/m1
dw[l_, e_] := D[w[l, e], l]
f[l_, e_] := dw[l, e]/(1 - l^3)
sup[x_] := ((d + x^3)/(1 + d))^(1/3)
intf[x_, e_] := Integrate[f[l, e], {l, x, sup[x]},
                Assumptions-> a > 0 && b > 0 && d > 0 && m1 > 0 && m2 > 0]
D[intf[x, e], x]