If `x^y=y^x`, then prove that `(x/y)^(x/y)=(x)^((x)/(y)-1)`.

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

If x^(y)=y^(x) , prove that (dy)/(dx)=((y)/(x)-log y)/((x)/(y)-log x)

If x^(y) + y^(x) = a^(b) then prove that (dy)/(dx)=-[(yx^(y-1)+y^(x)Logy)/(x^(y)Logx+xy^(x-1))] .

If ye^y=x prove that (dy)/(dx)= y/(x(1+y))

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)]

If ye^(y)=x, prove that,(dy)/(dx)=(y)/(x(1+y))

if x+y+z=0,then Prove that (xyz)/((x+y)(y+z)(z+x))=-1 where(x!=-y,y!=-z,z!=-x)

If x^(x)+y^(x)=1, prove that (dy)/(dx)=-{(x^(x)(1+log x)+y^(x)log y)/(xy^((x-1)))}

If y=x^(x^(x^(...oo))) , then prove that, (dy)/(dx)=(y^(2))/(x(1-y log x)) .

If (sinx)^y=x+y , prove that (dy)/(dx)=(1-(x+y)ycotx)/((x+y)logsinx-1)