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

Given, \(y=\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\)
\[
\begin{aligned}
&\Rightarrow \quad y=\cos ^{2} \frac{3 x}{2}-\left(1-\cos ^{2} \frac{3 x}{2}\right) \\
&\Rightarrow \quad y=2 \cos ^{2} \frac{3 x}{2}-1
\end{aligned}
\]
\[
\begin{aligned}
&\Rightarrow \frac{d y}{d x}=2-2 \cos \frac{3 x}{2}\left(-\sin \frac{3 x}{2}\right)\left(\frac{3}{2}\right) \\
&\Rightarrow \frac{d y}{d x}=-6 \cos \frac{3 x}{2} \sin \frac{3 x}{2} \\
&\Rightarrow \frac{d^{2} y}{d x^{2}}=-6\left[\cos \frac{3 x}{2}\left(\cos \frac{3 x}{2}\right) \cdot \frac{3}{2}-\sin \frac{3 x}{2}\right. \\
&\Rightarrow \frac{d^{2} y}{d x^{2}}=-9\left[\cos ^{2} \frac{3 x}{2}-\sin ^{2} \frac{3 x}{2}\right] \\
&\Rightarrow \frac{d^{2} y}{d x^{2}}=-9 y
\end{aligned}
\]