if $y = \cos^{2} (\frac{3 x}{2}) - \sin^{2} (\frac{3 x}{2}), then \frac{d^{2} y}{d x^{2}}$ is

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-9y

$3 \sqrt{1 - y^{2}}$

$- 3 \sqrt{1 - y^{2}}$

answer is C.

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Detailed Solution

Given $y = \cos^{2} (\frac{3 x}{2}) - \sin^{2} (\frac{3 x}{2})$
$y = \cos 3 x \Rightarrow \frac{d y}{d x} = - 3 \sin 3 x$
and $\frac{d^{2} y}{d x^{2}} = - 3 \times 3 \cos 3 x = - 9 \cos 3 x = - 9 y$

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