`(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 solve the problem \(\frac{d}{dx} \cos^{-1}(\sqrt{\cos x})\) for \(0 < x < \frac{\pi}{2}\), we will follow these steps:

### Step 1: Differentiate the function
We start by applying the chain rule for differentiation. The derivative of \(\cos^{-1}(u)\) is given by \(-\frac{1}{\sqrt{1-u^2}}\) where \(u = \sqrt{\cos x}\).

So, we have:
\[
\frac{d}{dx} \cos^{-1}(\sqrt{\cos x}) = -\frac{1}{\sqrt{1 - (\sqrt{\cos x})^2}} \cdot \frac{d}{dx}(\sqrt{\cos x})
...