W|A and Mathematica agree here. Note that these two functions are not the same; the input in the question is f1[x].

f1[x_] := (x^2)^(1/3)
f2[x_] := (x^(1/3))^2

So the derivatives do not agree. Both are correct though.

{D[f1[x], x], D[f2[x], x]}

(* Out[256]= {(2 x)/(3 (x^2)^(2/3)), 2/(3 x^(1/3))} *)

(I'll make this a Community wiki and also vote to close since it reduces to a misunderstanding of the definition, not to mention off-topic if W|A is the issue.)