If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:
- $2 \sec x \tan x$
- $\sin 2x$
- $2 \sec 2x \tan 2x$
- $\cos 2x$
The Correct Option is C
Solution and Explanation
Begin by rewriting the expression for \( y \) in terms of trigonometric identities:
\( y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}} \).
Using the double angle identity \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \), rewrite \( y \):
\( y = \frac{1}{\sqrt{1 - \sin^2(2x)}} \).
The term \( 1 - \sin^2(2x) \) simplifies to \( \cos^2(2x) \). Thus:
\( y = \frac{1}{\cos(2x)} = \sec(2x) \).
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:
\( \frac{dy}{dx} = 2 \sec^2(2x) \tan(2x) \).
Therefore, the correct answer is:
\( 2 \sec 2x \tan 2x \).
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