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NDA 2018 Maths · Mathematics · Derivatives · hard

For |x|<1/2, the derivative of sin ^-1(3x-4x^3 with respect to x is

Single correct option

For ∣x∣<12|x|<\frac12, the derivative of sin⁡−1(3x−4x3)\sin^{-1}(3x-4x^3) with respect to xx is
  1. A. 11−x2\frac{1}{\sqrt{1-x^2}}
  2. B. 31−x2\frac{3}{\sqrt{1-x^2}}
  3. C. 31−x2\frac{3}{1-x^2}
  4. D. −31−x2-\frac{3}{\sqrt{1-x^2}}
Show answer and solution

Answer: (B) 31−x2\frac{3}{\sqrt{1-x^2}}

Put x=sin⁡θx=\sin\theta, so 3x−4x3=sin⁡3θ3x-4x^3=\sin3\theta. For ∣x∣<12|x|<\frac12 we have ∣3θ∣<π2|3\theta|<\frac\pi2, hence the function equals 3θ=3sin⁡−1x3\theta=3\sin^{-1}x, with derivative 31−x2\frac{3}{\sqrt{1-x^2}}. Outside this range the identity changes by a constant and sign, which is why the restriction is given.
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Original question written in the NDA 2018 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