To begin, restate the expression for \( y \) using trigonometric identities:
\( y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}} \).
Employ the double angle identity \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \) to rewrite \( y \):
\( y = \frac{1}{\sqrt{1 - \sin^2(2x)}} \).
The expression \( 1 - \sin^2(2x) \) simplifies to \( \cos^2(2x) \). Consequently:
\( y = \frac{1}{\cos(2x)} = \sec(2x) \).
Differentiate \( y \) with respect to \( x \):
\( \frac{dy}{dx} = \frac{d}{dx} (\sec(2x)) \).
The derivative of \( \sec(2x) \) is:
\( \frac{dy}{dx} = \sec(2x) \tan(2x) \times 2 \).
Thus:
\( \frac{dy}{dx} = 2 \sec^2(2x) \tan(2x) \).
The final result is:
\( 2 \sec 2x \tan 2x \).