Question

Derivative of \( \left( \sin^{-1}x + \cos^{-1}x \right) \) with respect to \( x \) is:

• A. -1
• B. 0
• C. \( \frac{\pi}{2} \)
• D. 2

Hint:

The derivative of \( \sin^{-1}x + \cos^{-1}x \) is always 0 because their sum is constant and equal to \( \frac{\pi}{2} \).

Answer 1

The Correct Option is B

Step 1: Using the identity of inverse trigonometric functions. 
We know that: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] This identity holds for all values of \( x \) in the domain of the functions. 
Step 2: Taking the derivative. 
Differentiating both sides with respect to \( x \): \[ \frac{d}{dx} \left( \sin^{-1}x + \cos^{-1}x \right) = \frac{d}{dx} \left( \frac{\pi}{2} \right) \] Since \( \frac{\pi}{2} \) is a constant, its derivative is 0. Thus, the derivative is: \[ \frac{d}{dx} \left( \sin^{-1}x + \cos^{-1}x \right) = 0 \] Step 3: Conclusion. 
Therefore, the correct answer is 0, corresponding to option (b).