If `y=x^(1/3)-x^(-1/3)`, prove that `y^(3)+3y=x-(1)/(x)`

If y^(1/3) + y^(-1/3) = 2x, prove that (x^2 - 1)y_2 + xy_1 = 9y .

If x^(3)=y^(4) prove that (x/y)^(4/3)+(y/x)^(3/4)=x^(1/3)+y^(-1/4)

If x+y=t+(1)/(t) and x^(3)+y^(3)=t^(3)+(1)/(t^(3)) then prove that (dy)/(dx)=-(1)/(x^(2))

If x+y=t+(1)/(t) and x^(3)+y^(3)=t^(3)+(1)/(t^(3)) then prove that (dy)/(dx)=-(1)/(x^(2))

If x^2=y^3 , then prove that (x/y)^(3/2)+(y/x)^-(2/3)=x^(1/2)+y^(1/3) .

If x+y+z=xyz , prove that: a) (3x-x^(3))/(1-3x^(2))+(3y-y^(3))/(1-3y^(2))+(3z-z^(3))/(1-3z^(2))= (3x-x^(3))/(1-3x^(2)).(3y-y^(3))/(1-3y^(2)).(3z-z^(3))/(1-3z^(2)) b) (x+y)/(1-xy) + (y+z)/(1-yz)+(z+x)/(1-zx)= (x+y)/(1-xy) .(y+z)/(1-yz).(z+x)/(1-zx)

If the join of (x_(1),y_(1)) and (x_(2),y_(2)) makes on obtuse angle at (x_(3),y_(3)), then prove than (x_(3)-x_(1))(x_(3)-x_(2))+(y_(3)-y_(1))(y_(3)-y_(2))<0

If sqrt(1-x^6)+sqrt(1-y^6)=a^3(x^3-y^3) , prove that : dy/dx = x^2/y^2sqrt((1-y^6)/(1-x^6)) .

If sqrt(1 - x^6) + sqrt(1 - y^6) = a^3 (x^3 - y^3) prove that (dy)/(dx) = (x^2 sqrt(1 - y^6))/(y^2 sqrt(1 - x^6))

If x + y + z = xyz , prove that (3x -x^(3))/ (1-3x^(2)) + (3y -y^(3))/(1- 3y^(2)) +(3z -z^(3))/(1- 3z^(2)) = (3x -x^(3))/(1-3x)^(2) * (3y- y^(3))/(1-3x)^(2)* (3z- z^(3))/(1-3z)^(2) .