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NDA 2019 Maths · Mathematics · Derivatives · hard

If y=sin ^-1x+x√(1-x^2 for |x|<1, then dy/dx at x=1/2 is

Single correct option

If y=sin⁡−1x+x1−x2y=\sin^{-1}x+x\sqrt{1-x^2} for ∣x∣<1|x|<1, then dydx\dfrac{dy}{dx} at x=12x=\dfrac12 is
  1. A. 32\frac{\sqrt3}{2}
  2. B. 3\sqrt3
  3. C. 23\frac{2}{\sqrt3}
  4. D. 22
Show answer and solution

Answer: (B) 3\sqrt3

Differentiate: 11−x2+1−x2−x21−x2=1−x21−x2+1−x2=21−x2\frac{1}{\sqrt{1-x^2}}+\sqrt{1-x^2}-\frac{x^2}{\sqrt{1-x^2}}=\frac{1-x^2}{\sqrt{1-x^2}}+\sqrt{1-x^2}=2\sqrt{1-x^2}. At x=12x=\frac12 this is 2⋅32=32\cdot\frac{\sqrt3}{2}=\sqrt3. The value 23\frac{2}{\sqrt3} is 11−x2\frac{1}{\sqrt{1-x^2}} alone.
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Original question written in the NDA 2019 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