ddxsin−15x+121−x213=

y=sin−1⁡5x+121−x213 Let x=sin⁡θ ∴y=sin−1⁡5sin⁡θ+12cos⁡θ13=sin−1⁡513sin⁡θ+1213cos⁡θ ∴y=sin−1⁡(sin⁡θcos⁡α+sin⁡αcos⁡θ) ∴dydx=11−x2

ddxsin−15x+121−x213=

y=sin−1⁡5x+121−x213 Let x=sin⁡θ ∴y=sin−1⁡5sin⁡θ+12cos⁡θ13=sin−1⁡513sin⁡θ+1213cos⁡θ ∴y=sin−1⁡(sin⁡θcos⁡α+sin⁡αcos⁡θ) ∴dydx=11−x2

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$\frac{d}{d x} \{\sin^{- 1} (\frac{5 x + 12 \sqrt{1 - x^{2}}}{13})\} =$

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

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

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

answer is A.

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Detailed Solution

$y = \sin^{- 1} (\frac{5 x + 12 \sqrt{1 - x^{2}}}{13}) Let x = \sin θ ∴ y = \sin^{- 1} (\frac{5 \sin θ + 12 \cos θ}{13}) = \sin^{- 1} (\frac{5}{13} \sin θ + \frac{12}{13} \cos θ) ∴ y = \sin^{- 1} (\sin θ \cos α + \sin α \cos θ) ∴ \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^{2}}}$