If \(y = 3\cos 2x - 5\sin 2x\), then \(\displaystyle\frac{d^{2}y}{dx^{2}}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-4y\)
By the chain rule, \(\frac{d}{dx}\cos 2x = -2\sin 2x\) and \(\frac{d}{dx}\sin 2x = 2\cos 2x\):
\( \displaystyle \frac{dy}{dx} = -6\sin 2x - 10\cos 2x \)
Differentiating again for the second derivative:
\( \displaystyle \frac{d^{2}y}{dx^{2}} = -12\cos 2x + 20\sin 2x \)
\( = -4(3\cos 2x - 5\sin 2x) \)
\( = -4y \)