Question

Find \( \frac{d}{dx} \left( \sec^{-1} x + \csc^{-1} x \right) \):

• A. 1
• B. 0
• C. 2
• D. -1

Hint:

Know the derivatives of inverse trigonometric functions: \[ \frac{d}{dx} \sec^{-1} x = \frac{1}{|x| \sqrt{x^2 - 1}}, \quad \frac{d}{dx} \csc^{-1} x = -\frac{1}{|x| \sqrt{x^2 - 1}} \]

Answer 1

The Correct Option is B

We differentiate term-by-term: \[ \frac{d}{dx} \left( \sec^{-1} x \right) = \frac{1}{|x| \sqrt{x^2 - 1}}, \quad x \in (-\infty, -1] \cup [1, \infty) \] \[ \frac{d}{dx} \left( \csc^{-1} x \right) = -\frac{1}{|x| \sqrt{x^2 - 1}}, \quad x \in (-\infty, -1] \cup [1, \infty) \] Adding both: \[ \frac{d}{dx} \left( \sec^{-1} x + \csc^{-1} x \right) = \frac{1}{|x| \sqrt{x^2 - 1}} - \frac{1}{|x| \sqrt{x^2 - 1}} = 0 \] \[ \therefore \frac{d}{dx} \left( \sec^{-1} x + \csc^{-1} x \right) = 0 \]