Question

If \(f(x) = x^{\sec^{-1} x}\), then find \(f'(2)\).

• A. \(\frac{2^{\pi/3}}{6} \left(\pi - \sqrt{3} \log 2\right)\)
• B. \(\frac{2^{\pi/6}}{6} \left(\pi + \sqrt{3} \log 2\right)\)
• C. \(\frac{2^{\pi/3}}{6} \left(\pi + \sqrt{3} \log 2\right)\)
• D. \(\frac{2^{\pi/6}}{6} \left(\pi - \sqrt{3} \log 2\right)\)

Hint:

Use logarithmic differentiation and derivatives of inverse secant function carefully.

Answer 1

The Correct Option is C

Given \(f(x) = x^{\sec^{-1} x}\), write as \[ f(x) = e^{\sec^{-1} x \log x}. \] Differentiate using product and chain rules: \[ f'(x) = f(x) \left( \frac{d}{dx} (\sec^{-1} x) \log x + \frac{\sec^{-1} x}{x} \right). \] Use \[ \frac{d}{dx} \sec^{-1} x = \frac{1}{|x| \sqrt{x^2 -1}}. \] At \(x=2\), \[ \sec^{-1} 2 = \frac{\pi}{3},
f(2) = 2^{\pi/3}. \] Calculate derivatives and substitute values to find \[ f'(2) = \frac{2^{\pi/3}}{6} \left(\pi + \sqrt{3} \log 2\right). \]