If `u=log((x^2+y^2)/(x+y))`, prove that `x(delu)/(delx)+y(delu)/(dely)=1`

If u=log(x^(2)+y^(2)+z^(2)) prove that (del^(2)u)/(del x^(2))+(del^(2)u)/(del y^(2))+(del^(2)u)/(del z^(2) = 2/(x^(2)+y^(2)+z^(2)

If u = log ((x ^ (2) + y ^ (2)) / (x + y)), prove that x (u) / (x) + y (u) / (y) = 1

If u = (x^2+y^2+z^2)^(-1/2) then prove that (del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2) =0

if y=log x^(x) prove that (dy)/(dx)=1+log x

If y=log(x+(1)/(x)), prove that (dy)/(dx)=(x-1)/(2x(x+1))

If y^(x)=e^(y-x), prove that (dy)/(dx)=((1+log y)^(2))/(log y)

If y^(x)=e^(y-x), prove that (dy)/(dx)=((1+log y)^(2))/(log y)

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

If u = x ^ (2) tan ^ (- 1) ((y) / (x)) - y ^ (2) tan ^ (- 1) ((x) / (y)), prove that (del ^ (2) u) / (del x del y) = (x ^ (2) -y ^ (2)) / (x ^ (2) + y ^ (2))

If tan^(-1) (y/x) = log sqrt(x^(2) + y^(2)) , prove that dy/dx = (x+y)/(x-y)