If `y=(x^2)/2+(xsqrt(x^2+1))/2+logsqrt(x+sqrt(x^2+1))` prove that `2y=x(dy)/(dx)+log((dy)/(dx)).`

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

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

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

If y=log(sqrt(x)+(1)/(sqrt(x))). Prove that (dy)/(dx)=(x-1)/(2x(x+1))

If y sqrt(x^(2)+1)= log (sqrt(x^(2)+1)-x) , prove that (x^(2)+1)(dy)/(dx) +xy+1=0 .

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

If y = sqrt(x) + (1)/(sqrt(x)) prove that 2x(dy)/(dx) + y = 2sqrt(x)

If y= ((sqrt(x)+1)(x^(2)-sqrt(x)))/(xsqrt(x)+x+sqrt(x)), Prove that (dy)/(dx) = 1 .

If y=log\ [x+sqrt(x^2+1)] , prove that (x^2+1)(d^2\ y)/(dx^2)+x(dy)/(dx)=0

If y=log\ [x+sqrt(x^2+1)] , prove that (x^2+1)(d^2\ y)/(dx^2)+x(dy)/(dx)=0