`d/dx cos^-1 sqrtcos x` 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 \( y = \cos^{-1}(\sqrt{\cos x}) \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let \( u = \sqrt{\cos x} \). Then, we can rewrite the function as:
\[ y = \cos^{-1}(u) \]

### Step 2: Differentiate the outer function
The derivative of \( y = \cos^{-1}(u) \) with respect to \( u \) is given by:
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