Prove that:`|(1,2y),(x,0)|=|(y,5x),(y,3x)|`

Prove that: 4|(x,y),(3x,0)|=3|(x,0),(5x,-4y)|

By using properties of determinant prove that |(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0

If (x^y)(y^x)=1 , prove that (dy)/(dx)=-(y(y+xlogy))/(x(ylogx+x))

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

If xsqrt(1 + y) + ysqrt(1 + x) = 0 x != y prove that (dy)/(dx) = (-1)/((1 + x)^(2))

Prove that (frac(cos(7x-5y)+cos(7y-5x))(sin(7x-5y)+sin(7y-5x)))=cot(x+y)

If [(1,2,-3),(0,4,5),(0,0,1)][(x),(y),(z)]=[(1),(1),(1)] , then (x, y, z) is equal to

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