Question

The first derivative of the function \( \left( \cos^{-1}\left(\sin\sqrt{\frac{1+x}{2}}\right) + x^x \right) \) with respect to \( x \) at \( x = 1 \) is

• A. \( \frac{1}{4} \)
• B. \( \frac{5}{4} \)
• C. \( \frac{-1}{2} \)
• D. \( \frac{3}{4} \)

Hint:

Remember: \(\frac{d}{dx}(x^x) = x^x(1 + \log x)\).

Answer 1

The Correct Option is D

Step 1: Concept Use the identity \(\cos^{-1}(\sin \theta) = \frac{\pi}{2} - \theta\) and logarithmic differentiation for \(x^x\).

Step 2: Meaning Let \(y = f(x) + g(x)\). \(f(x) = \frac{\pi}{2} - \sqrt{\frac{1+x}{2}}\). \(g(x) = x^x\).

Step 3: Analysis \(f'(x) = -\frac{1}{2\sqrt{\frac{1+x}{2}}} \cdot \frac{1}{2} = -\frac{1}{4\sqrt{\frac{1+x}{2}}}\). At \(x=1\), \(f'(1) = -\frac{1}{4}\). \(g'(x) = x^x(1 + \log x)\). At \(x=1\), \(g'(1) = 1^1(1 + 0) = 1\). Total derivative = \(-\frac{1}{4} + 1 = \frac{3}{4}\).

Step 4: Conclusion The derivative at \(x=1\) is \(\frac{3}{4}\). Final Answer: (D)