If `y=(x^2)/2+1/2xsqrt(x^2+1)+(log)_esqrt(x+sqrt(x^2+1))` , prove that `2y=x y^(prime)+(log)_e y^(prime),w h e r ey '` denotes the derivative w.r.t `xdot`

If quad (x^(2))/(2)+(1)/(2)x sqrt(x^(2)+1)+log_(e)sqrt(x+sqrt(x^(2)+1))y=(x^(2))/(2)+(1)/(2)x sqrt(x^(2)+1)+log_(e)sqrt(x+sqrt(x^(2)+1)) prove that 2y=xy'+log_(e)y', where y' denotes the derivative w.r.x.

"If "y=(x^(2))/(2)+(1)/(2)xsqrt(x^(2)+1)+log_(e)sqrt(x+sqrt(x^(2)+1)), then

y=(x^(2))/(2)+(1)/(2)x sqrt(x^(2)+1)+ln sqrt(x+sqrt(x^(2)+1)) prove that 2y=xy'+ln y'

If y=x-x^(2), then the derivative of y^(2) w.r.t x^(2) is

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

"If "y^(1//m)=(x+sqrt(1+x^(2)))," then "(1+x^(2))y_(2)+xy_(1) is (where y_(r) represents the rth derivative of y w.r.t. x)

The derivative of y=x^(2^(x)) w.r.t. x is

y=[log(x+sqrt(x^(2)+1))]^(2) then prove that (x^(2)+1)y_(2)+xy_(1)=2

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)