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`

Options

A. `y'=x+sqrt(x^(2)+1)`
B. `y'=(1)/(x+sqrt(x^(2)+1))`
C. `2y=xy'+log_(e)y'`
D. `2y=xy'-log_(e)y'`

Explanation

`y=(x^(2))/(2)+(1)/(2)xsqrt(x^(2)+1)+(1)/(2)log_(e)(x+sqrt(x^(2)+1))`
`rArr" "y'=x+(1)/(2)[(x^(2))/(sqrt(x^(2)+1))+sqrt(x^(2)+1)]+(1)/(2sqrt(x^(2)+1))`
`=x+(1)/(2)[(x^(2)+1)/(sqrt(x^(2)+1))]+(sqrt(x^(2)+1))/(2)`
`=x+sqrt(x^(2)+1)`
`"Also, "2y=x^(2)+xsqrt(x^(2)+1)+log_(e)(x+sqrt(x^(2)+1))`
`=x(x+sqrt(x^(2)+1))+log_(e)(x+sqrt(x^(2)+1))`
`=xy'+log_(e)y'`