Question

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

• A. $9y$
• B. $-9y$
• C. $3\surd (1–y^2)$
• D. $-3\surd (1–y^2)$

Answer 1

The correct option is B

$-9y$

Explanation for the correct option:

Step 1: Find the value of $\frac{dy}{dx}$

Given: $y={\cos }^2\left(\frac{3x}{2}\right)-{\sin }^2\left(\frac{3x}{2}\right)$

We know that ${\cos }^{2}x–{\sin }^{2}x=\cos 2x$

So,

$$\begin{gathered} y=\cos 2\left(\frac{3x}{2}\right) \\ y=\cos 3x \end{gathered}$$

Now, differentiate with respect to $x$

$\frac{dy}{dx}=-3sin3x\mathbf{[}\mathbf{\because }\mathbf{}\frac{\mathbf{d}}{\mathbf{dx}}\mathbf{\left(}\mathbf{cosx}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{sinx}\mathbf{]}$

Step 2: Find the value of $\frac{d^{2}y}{d{x}^2}$

Again differentiate again with respect to $x$

Then,

$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=-9cos3x\left[\because \frac{d\left(\sin x\right)}{dx}=\cos x\right] \\ \frac{d^{2}y}{d{x}^2}=-9y \end{gathered}$$

Hence, the correct option is B.