Question

If \[ y = \frac{1}{\sqrt{1 - 4 \sin^2 x \cos^2 x}}, \] then $\frac{dy}{dx}$ is:

• A. $2 \sec x \tan x$
• B. $\sin 2x$
• C. $2 \sec 2x \tan 2x$
• D. $\cos 2x$

Answer 1

The Correct Option is C

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:

\( 2 \sec 2x \tan 2x \).