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 \( \frac{1}{|x|\sqrt{x^2 - 1}} \).

Answer 1

The Correct Option is A

Step 1: Differentiate the provided expression with respect to \( x \)
Let \( y = \tan^{-1}(\cos\sqrt{x}) + \sec^{-1}(e^x) \). We aim to compute \( \frac{dy}{dx} \). Applying the chain rule yields:
\[ \frac{dy}{dx} = \frac{d}{dx} \tan^{-1}(\cos\sqrt{x}) + \frac{d}{dx} \sec^{-1}(e^x) \]
\[ = \frac{1}{1 + (\cos\sqrt{x})^2} \cdot \frac{d}{dx} (\cos\sqrt{x}) + \frac{1}{|e^x|\sqrt{e^{2x} - 1}} \cdot \frac{d}{dx} (e^x) \]
\[ = \frac{1}{1 + \cos^2\sqrt{x}} \cdot (-\sin\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} + \frac{e^x}{e^x\sqrt{e^{2x} - 1}} \]
\[ = -\frac{\sin\sqrt{x}}{2\sqrt{x}(1 + \cos^2\sqrt{x})} + \frac{1}{\sqrt{e^{2x} - 1}} \]

Step 2: Evaluate the derivative at \( x = \frac{\pi^2}{4} \)
Substitute \( x = \frac{\pi^2}{4} \) into the computed derivative:
\[ \frac{dy}{dx} \Bigg|_{x = \frac{\pi^2}{4}} = -\frac{\sin\sqrt{\frac{\pi^2}{4}}}{2\sqrt{\frac{\pi^2}{4}}(1 + \cos^2\sqrt{\frac{\pi^2}{4}})} + \frac{1}{\sqrt{e^{2(\frac{\pi^2}{4})} - 1}} \]
\[ = -\frac{\sin\frac{\pi}{2}}{2(\frac{\pi}{2})(1 + \cos^2\frac{\pi}{2})} + \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} \]
\[ = -\frac{1}{\pi(1 + 0)} + \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} \]
\[ = -\frac{1}{\pi} + \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} \]
\[ = \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} - \frac{1}{\pi} \]

The value of the given expression at \( x = \frac{\pi^2}{4} \) is \( \frac{1}{\sqrt{e^{\frac{\pi^2}{2}} - 1}} - \frac{1}{\pi} \). This corresponds to option (A).