Question

If y = log `[(x + sqrt(x^2 + 25))/(sqrt(x^2 + 25) - x)]` then `"dy"/"dx"` = ______.

Options

• `1/sqrt(x^2 + 25)`

• `2/sqrt(x^2 + 25)`

• `(- 1)/sqrt(x^2 + 25)`

• `(- 2)/sqrt(x^2 + 25)`

Answer 1

If y = log `[(x + sqrt(x^2 + 25))/(sqrt(x^2 + 25) - x)]` then `"dy"/"dx"` = `underline(2/sqrt(x^2 + 25))`.

Explanation:

We have, y = log`[(x + sqrt(x^2 + 25))/(sqrt(x^2 + 25) - x)]`

`=> "y" = log[(x + sqrt((x^2 + 25))^2)/(x^2 + 25 - x^2)]`

`=> "y" = log[(x + sqrt((x^2 + 25))^2)/25]`

`=> "y" = 2 log(x + sqrt(x^2 + 25)) - log 25`

On differentiating both sides w.r.t. x, we get

`"dy"/"dx" = 2/(x + sqrt(x^2 + 25)) (1 + 1/(2 sqrt(x^2 + 25)) (2x))`

`= 2/(x + sqrt(x^2 + 25)) ((sqrt(x^2 + 25) + x)/sqrt(x^2 + 25))`

`= 2/sqrt(x^2 + 25)`