`y=log[(x+sqrt(x^2+25))/(sqrt(x^2+25)-x)],f i n d(dy)/(dx)`

Options

A. `(1)/(sqrt(x^(2)+25))`
B. `(2)/(sqrt(x^(2)+25))`
C. `(-1)/(sqrt(x^(2)+25))`
D. `(-2)/(sqrtx^(2)+25)`

Answer

B

Explanation

We have, `y=log[(x+sqrt(x^(2)+25))/(sqrt(x^(2)+25)-x)]`
`impliesy=log[((x+sqrt(x^(2)+25))^(2))/(x^(2)+25-x^(2))]`
`impliesy=log[((x+sqrt(x^(2)+25))^(2))/(25)]`
`impliesy=2log(x+sqrt(x^(2)+25)-log25`
On differentiating both sides w.r.t. x, we get
`(dy)/(dx)=(2)/(x+sqrt(x^(2))+25))(1+(1)/(2sqrt(x^(2)+25))(2x))`
`=(2)/(x+sqrt(x^(2)+25))(sqrt(x^(2)+25)+x)/(sqrt(x^(2)+25)))=(2)/(sqrt(x^(2)+25))`