y=cos-1x2n-1x2n+1 =π-cos-11-x2n1+x2n Put xn=tanθ =π-cos-11-tan2θ1+tan2θ =π-cos-1(cos2θ) =π-2θ y=π-2 tan-1xn dydx=0-2 11+x2nnxn-1 =-2n·xn-11+x2n

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$\frac{d}{dx} [\cos^{- 1} (\frac{x^{2 n} - 1}{x^{2 n} + 1})] =$

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$\frac{- 2 {nx}^{n - 1}}{1 + x^{2 n}}$

$\frac{{nx}^{n - 1}}{1 + x^{2 n}}$

$\frac{- n . x^{n - 1}}{1 + x^{2 n}}$

${nx}^{n - 1}$

answer is A.

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Detailed Solution

$y = \cos^{- 1} (\frac{x^{2 n} - 1}{x^{2 n} + 1}) = π - \cos^{- 1} (\frac{1 - x^{2 n}}{1 + x^{2 n}}) Put x^{n} = tanθ = π - \cos^{- 1} (\frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}) = π - \cos^{- 1} (\cos 2 θ) = π - 2 θ y = π - 2 \tan^{- 1} (x^{n}) \frac{dy}{dx} = 0 - 2 \frac{1}{1 + x^{2 n}} {nx}^{n - 1} = \frac{- 2 n \cdot x^{n - 1}}{1 + x^{2 n}}$