Question

`d/(dx)(tan^-1  (sqrt(1 + x^2) - 1)/x)` is equal to:

Options

• `1/(1 + x^2)`

• `x^2/(2sqrt(1 + x^2) (sqrt(1 + x^2) - 1)`

• `2/(1 + x^2)`

• `1/(2(1 + x^2))`

Answer 1

`1/(2(1 + x^2))`

Explanation:

We have given `d/(dx) (tan^-1   (sqrt(1 + x^2) - 1)/x)`

Let x = tan θ

⇒ θ = tan^–1x

Given expression = `d/(dx) (tan^-1  (sqrt(1 + tan^2 theta) - 1)/tan theta)`

= `d/(dx)[tan^-1  (2sin^2  theta/2)/(2sin  theta/2  cos  theta/2)]`

= `d/(dx) (tan^-1  tan  theta/2)`

= `d/(dx)(theta/2)`

= `1/2 d/(dx) (tan^-1 x)`

= `1/(2(1 + x^2)`