Question

If $y=\sin x+\cos 2x$, then $\frac{d^{2}y}{d{x}^2}$ equals

• A. $\sin x+4\sin 2x$
• B. $-\sin x+4\cos 2x$
• C. $-(\sin x+4\cos 2x)$
• D. None of these

Answer 1

Answer: (C)

$-(\sin x+4\cos 2x)$

Given $y=\sin x+\cos 2x$

Differentiating w.r.t x, we get

$\frac{dy}{dx}=\frac{d(\sin x+\cos 2x)}{dx}$

$\frac{dy}{dx}=\frac{d\left(\sin x\right)}{dx}+\frac{d\left(\cos 2x\right)}{dx}$

$\frac{dy}{dx}=\cos x+(-2\sin 2x)$

$\frac{dy}{dx}=\cos x-2\sin 2x$

Again differentiating w.r.t x, we get

$\frac{d^{2}y}{d{x}^2}=\frac{d(\cos x-2\sin 2x)}{dx}$

$\frac{d^{2}y}{d{x}^2}=\frac{d\left(\cos x\right)}{dx}-\frac{\left(2\sin 2x\right)}{dx}$

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

$\frac{d^{2}y}{d{x}^2}=-\sin x-4\cos 2x$

$\frac{d^{2}y}{d{x}^2}=-(\sin x+4\cos 2x)$