AP EAMCET · Maths · Differentiation
If \(f(x)=x^{\operatorname{Sec}^{-1} x}\), then \(f^{\prime}(2)=\)
- A \(\frac{2^{\pi / 3}}{6}(\pi-\sqrt{3} \log 2)\)
- B \(\frac{2^{\pi / 6}}{6}(\pi+\sqrt{3} \log 2)\)
- C \(\frac{2^{\pi / 3}}{6}(\pi+\sqrt{3} \log 2)\)
- D \(\frac{2^{\pi / 6}}{6}(\pi-\sqrt{3} \log 2)\)
Answer & Solution
Correct Answer
(C) \(\frac{2^{\pi / 3}}{6}(\pi+\sqrt{3} \log 2)\)
Step-by-step Solution
Detailed explanation
\(\log f(x) = \operatorname{Sec}^{-1} x \log x\) \(\frac{f'(x)}{f(x)} = \frac{1}{x\sqrt{x^2-1}}\log x + \frac{1}{x}\operatorname{Sec}^{-1} x\) \(f'(x) = x^{\operatorname{Sec}^{-1} x} \left( \frac{\log x}{x\sqrt{x^2-1}} + \frac{\operatorname{Sec}^{-1} x}{x} \right)\)…
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