FullSimplify[(y (x^y y - x y^x Log[y]))/(x (x y^x - x^y y Log[x])) == 
                 (x y Log[y] - y^2)/(x y Log[x] - x^2), 

                 x^y == y^x]

will return `True`.

If you want _Mathematica_ to give you the same answer as the hand-calculated one, you can 

    Solve[Dt[x^y == y^x, x], Dt[y, x]] /. Rule @@ (x^y == y^x) // Simplify

or

    Solve[Dt[Log /@ (x^y == y^x), x], Dt[y, x]]