`(d)/(dx)cos^(-1)sqrt(cosx),0ltxlt(pi)/(2)` is equal to

Options

A. `(1)/(2)sqrt(1+sec x)`
B. `sqrt(1+sec x)`
C. `-(1)/(2)sqrt(1+sec x)`
D. `-sqrt(1+sec x)`

Explanation

To find the derivative of \( \cos^{-1}(\sqrt{\cos x}) \) with respect to \( x \), we will apply the chain rule and the formula for the derivative of the inverse cosine function.

### Step-by-step Solution:

1. **Identify the Function**:
We need to differentiate \( y = \cos^{-1}(\sqrt{\cos x}) \).

2. **Apply the Derivative of Inverse Cosine**:
...