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]]

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]]