If `y=(x^(2))/(2)+(1)/(2)xsqrt(x^(2)+1)+lnsqrt(x+sqrt(x^(2)+1))` then the value of `xy'+logy'` is

Options

A. y
B. 2y
C. 0
D. `-2y`

Answer

B

Explanation

`y=(x^(2))/(2)+(1)/(2)xsqrt(x^(2)+1)+(1)/(2)log(x+sqrt((x^(2)+1)))`
`rArr" "y'=x+(1)/(2)sqrt(x^(2)+1)+(1)/(2)x(x)/(sqrt(x^(2)+1))+(1)/(2(x+sqrt(x^(2)+1)))xx(1+(x)/(sqrt(x^(2)+1)))`
`=x+((x^(2)+1+x^(2)))/(2sqrt(x^(2)+1))+(1)/(2sqrt(x^(2)+1))`
`=x+(2x^(2)+1)/(2sqrt(x^(2)+1))+(1)/(2sqrt(x^(2)+1))`
`=x+(2x^(2)+1+1)/(2sqrt(x^(2)+1))`
`rArr" "y'=x+sqrt(x^(2)+1)`
`rArr" "xy'+logy'=x(x+sqrt(x^(2)+1)+1)+log(x+sqrt(x^(2)+1))`
`=x^(2)+xsqrt(x^(2)+1)+log(x+sqrt(x^(2)+1))`
`=2y`