Answer is: option4
\( \frac{2\sqrt{x} + 1}{4\sqrt{x^2 + x\sqrt{x}}} \)Solution:
\( f(x) = \sqrt{x + \sqrt{x}} \)
\( f'(x) = \frac{1}{2 \sqrt{x + \sqrt{x}}} \cdot \left(1 + \frac{1}{2 \sqrt{x}} \right) \)
\( f'(x) = \frac{1}{2 \sqrt{x + \sqrt{x}}} \cdot \left(\frac{2 \sqrt{x} + 1}{2 \sqrt{x}} \right) \)
\( f'(x) = \frac{2 \sqrt{x} + 1}{4 \sqrt{x} \sqrt{x + \sqrt{x}}} \)
\( f'(x) = \frac{2 \sqrt{x} + 1}{4 \sqrt{x^2 + x \sqrt{x}}} \)Ans