MHT CET · Maths · Differentiation
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 \(\frac{3}{4}\)
Answer & Solution
Correct Answer
(D) \(\frac{3}{4}\)
Step-by-step Solution
Detailed explanation
\(f(x) = \cos^{-1}\left(\sin \sqrt{\frac{1+x}{2}}\right)+x^x = \frac{\pi}{2} - \sqrt{\frac{1+x}{2}} + x^x\) \(f'(x) = \frac{d}{dx}\left(\frac{\pi}{2}\right) - \frac{d}{dx}\left(\left(\frac{1+x}{2}\right)^{1/2}\right) + \frac{d}{dx}(x^x)\)
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