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On differentiating tan − 1 [ 1 + x 2 − 1 x ] with respect to x , the result would be

Options

  1. A1 2 . 1 1 + x 2
  2. B1 1 + x 2
  3. C2 1 + x 2
  4. D1 2 . 1 1 + 2 x

Correct answer

A. 1 2 . 1 1 + x 2

Step-by-step solution

Let us assume that, y = tan − 1 [ 1 + x 2 − 1 x ] Put : x = tan θ ∴ 1 + x 2 − 1 x = 1 + tan 2 θ − 1 tan θ = sec θ − 1 tan θ = 1 − cos θ sin θ = 2 sin 2 θ 2 2 sin θ 2 cos θ 2 = tan θ 2 On differentiating with respect to x : ∴ y = tan − 1 ( tan θ 2 ) = 1 2 θ = 1 2 tan − 1 x ⇒ d y d x = 1 2 . 1 1 + x 2 .

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