y=cos-14x364-3x4 =cos-144x3-33x4 =cos-14cos3θ-3cosθ where x4=cosθ y=cos-1(cos3θ) =3θ =3cos-1x4 dydx=-31-x21614=-316-x2

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$\frac{d}{dx} \{{Cos}^{- 1} (\frac{4 x^{3}}{64} - \frac{3}{4} x)\} =$

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$\frac{- 1}{\sqrt{16 - x^{2}}}$

$\frac{1}{\sqrt{16 - x^{2}}}$

$\frac{3}{\sqrt{16 - x^{2}}}$

$\frac{- 3}{\sqrt{16 - x^{2}}}$

answer is C.

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Detailed Solution

$y = \cos^{- 1} (\frac{4 x^{3}}{64} - \frac{3 x}{4}) = \cos^{- 1} (4 {(\frac{4}{x})}^{3} - 3 (\frac{3 x}{4})) = \cos^{- 1} (4 \cos^{3} θ - 3 cosθ) where \frac{x}{4} = cosθ y = \cos^{- 1} (\cos 3 θ) = 3 θ = 3 \cos^{- 1} (\frac{x}{4}) \frac{dy}{dx} = \frac{- 3}{\sqrt{1 - \frac{x^{2}}{16}}} \frac{1}{4} = \frac{- 3}{\sqrt{16 - x^{2}}}$