The derivative of sin−1x with respect to cos−11−x2 is

y=sin−1xPut x=sinθy=sin−1(sinθ)=θdydθ=1u=cos−11−x2=cos−11−sin2θu=cos−1(cosθ)=θdudθ=1dydu=1

The derivative of sin−1x with respect to cos−11−x2 is

y=sin−1xPut x=sinθy=sin−1(sinθ)=θdydθ=1u=cos−11−x2=cos−11−sin2θu=cos−1(cosθ)=θdudθ=1dydu=1

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The derivative of $\sin^{- 1} x$ with respect to $\cos^{- 1} \sqrt{1 - x^{2}}$ is

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$\cos^{- 1} x$

$\frac{1}{\sqrt{1 - x^{2}}}$

answer is C.

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Detailed Solution

$y = \sin^{- 1} x$

Put $x = \sin θ$

$y = \sin^{- 1} (\sin θ) = θ$

$\frac{d y}{d θ} = 1$

$u = \cos^{- 1} \sqrt{1 - x^{2}} = \cos^{- 1} \sqrt{1 - \sin^{2} θ}$

$u = \cos^{- 1} (\cos θ) = θ$

$\frac{d u}{d θ} = 1$

$\frac{d y}{d u} = 1$

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The derivative of $\sin^{- 1} x$ with respect to $\cos^{- 1} \sqrt{1 - x^{2}}$ is