Question

At \( x = \frac{\pi^2}{4} \), \( \frac{d}{dx} \left( \tan^{-1}(\cos\sqrt{x}) + \sec^{-1}(e^x) \right) = \)

• A. \( \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} - \frac{1}{\pi} \)
• B. \( \frac{\pi}{4} + \frac{1}{\sqrt{e^{\pi^2} + e^{\frac{\pi^2}{2}}}} \)
• C. \( \frac{1}{\sqrt{e^{\pi^2} + e^{\frac{\pi^2}{2}}}} + \frac{2}{\pi} \cot \left( \frac{\pi}{2} \right) \)
• D. \( \frac{1}{\sqrt{e^{\pi}}} + \frac{1}{\pi} \)

Hint:

Remember the derivatives of inverse trigonometric functions and use the chain rule appropriately. Also, remember that \( \sec^{-1}(x) \) is defined for \( |x| \ge 1 \), and its derivative is given by \( \frac{1}{|x|\sqrt{x^2 - 1}} \).