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NDA 2023 Maths · Mathematics · Derivatives · hard

The derivative of tan ^-1 ((1+x)/(1-x)) with respect to x is

Single correct option

The derivative of tan⁡−1 ⁣(1+x1−x)\tan^{-1}\!\left(\dfrac{1+x}{1-x}\right) with respect to xx is
  1. A. 21+x2\dfrac2{1+x^2}
  2. B. 11+x2\dfrac1{1+x^2}
  3. C. 11−x2\dfrac1{1-x^2}
  4. D. 11
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Answer: (B) 11+x2\dfrac1{1+x^2}

Since tan⁡−11+x1−x=tan⁡−11+tan⁡−1x=π4+tan⁡−1x\tan^{-1}\dfrac{1+x}{1-x}=\tan^{-1}1+\tan^{-1}x=\dfrac\pi4+\tan^{-1}x (for x<1x<1), the derivative is 11+x2\dfrac1{1+x^2}. Direct differentiation gives the same after simplification, and the constant π4\dfrac\pi4 vanishes.
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Original question written in the NDA 2023 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake