`x^(y) + y^(x) = 1`

If x ^(y) + y ^(x) = a ^(b) then show that (dy)/(dx) =- ((y x ^(y-1) + y^(x) log y)/( x ^(y) log x + x y ^(x -1)))

Solve for x,y. x^2 + y(x+1) = 17 and y^2 + x(y+1) = 13

If x^(y)=y^(x) show that (x/y)^(x/y)=x^((x)/(y)-1) if further x=2y then prove that y=2

{:("Column" A ,, "Column" B), (225x^(2) - 625 y^(2) = ,, (a) 25(x-2) (x-2)), (x^(2) - x - y - y^(2) = ,, (b) 25(3x- 5y) (3x + 5y)), (x^(2) - x - y^(2) + y = ,, (x + y) (x - y- 1)), (25x^(2) - 100 x + 100 = ,, (d) (x - y) (x + y -1)), (,,(e) (x + y) (x + y - 1)):}

The expression (x+y)^(-1).(x^(-1) + y^(-1)) is equivalent to :

The expression (x + y)^(-1).(x^(-1) + y^(-1)) is equivalent to :

If x^y+y^x= (x+y)^(x+y) , then prove that dy/dx= ((x+y)^(x+y) [1+log(x+y)]-yx^(y-1)-y^xlogy)/(x^ylogx+xy^(x-1) -(x+y)^(x+y) [1+log(x+y)]