I noticed the following difference
Assuming[x ∈ Reals, FullSimplify[D[Abs[2 x], x]]] (* 2 Sign[x] *) Assuming[x ∈ Reals, FullSimplify[Abs'[2 x]]] (* Sign[x] *)
While it is clear that the first line of code above returns the correct result, I wonder why the call of
Derivative fails? A little further investigation shows that
f[x_] := Abs[2 x] f' (* 2 Derivative[Abs][#1] & *)
which is correct and implies that the difference might come from with or without
SetDelayed. But I still do not fully understand why the second code failed.