Question

The derivative of \( \sin^2 x \), w.r.t. \( x \) is:

• A. \( \cos 2x \)
• B. \( - \cos^2 x \)
• C. \( - \sin^2 x \)
• D. \( \sin 2x \)

Hint:

The derivative of \( \sin^2 x \) is \( 2 \sin x \cos x \), which simplifies to \( \sin 2x \) using a trigonometric identity.

Answer 1

The Correct Option is D

Step 1: Applying the chain rule.
To differentiate \( \sin^2 x \) with respect to \( x \), we use the chain rule. The chain rule states that the derivative of \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
Step 2: Differentiation of \( \sin^2 x \).
The derivative of \( \sin^2 x \) is: \[ \frac{d}{dx} \left( \sin^2 x \right) = 2 \sin x \cdot \cos x. \]
Step 3: Simplification.
Using the trigonometric identity \( \sin 2x = 2 \sin x \cos x \), we can rewrite the expression as: \[ \frac{d}{dx} \left( \sin^2 x \right) = \sin 2x. \]
Step 4: Conclusion.
Thus, the correct answer is (D) \( \sin 2x \). Final Answer:} \( \sin 2x \).