Question Detail
Question
If \( f(x) = \sqrt{4\sin 2x + 2} \), find \( f'(0) \).
Correct Answer
Option D is the correct answer.
Detailed Explanation
Let ( u = 4\sin 2x + 2 ). Then ( f(x) = u^{1/2} ). By chain rule: ( f'(x) = \frac{1}{2}u^{-1/2} \cdot \frac{du}{dx} ). Here ( \frac{du}{dx} = 8\cos 2x ). At ( x=0 ), ( u=2 ) and ( \frac{du}{dx}=8 ). So ( f'(0) = \frac{1}{2}(2)^{-1/2}(8) = \frac{4}{\sqrt{2}} = 2\sqrt{2} ).
Hint
Use the chain rule and substitute \( x = 0 \) after differentiating.