If \(y = \displaystyle\frac{1}{\sqrt{1 - 4\sin^2 x\cos^2 x}}\), then \(\displaystyle\frac{dy}{dx} =\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2\sec 2x\tan 2x\)
By the double-angle formula \(\sin 2x = 2\sin x\cos x\):
\(4\sin^2 x\cos^2 x = \sin^2 2x\)
\(1 - \sin^2 2x = \cos^2 2x\)
\( \displaystyle y = \frac{1}{\sqrt{\cos^2 2x}} = \frac{1}{|\cos 2x|} \)
Where \(\cos 2x \gt 0\), the case the options take, this is \(y = \sec 2x\). (Where \(\cos 2x \lt 0\), y = −sec 2x and the derivative only changes sign.) By the chain rule:
\( \displaystyle \frac{dy}{dx} = \sec 2x\tan 2x\times\frac{d}{dx}(2x) \)
\( \displaystyle = 2\sec 2x\tan 2x \)