If `y=cos^(-1) sqrt((1+cos x)//(2)),"then " dy/dx=`

Options

A. ` (1)/(2) `
B. `(-1)/(2) `
C. ` 1`
D. `-1`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \cos^{-1} \left( \sqrt{\frac{1 + \cos x}{2}} \right) \), we can follow these steps:

### Step 1: Simplify the expression inside the inverse cosine
We know that:
\[ \sqrt{\frac{1 + \cos x}{2}} = \cos\left(\frac{x}{2}\right) \]
This is derived from the double angle formula for cosine, where:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Setting \( 2\theta = x \) gives us \( \theta = \frac{x}{2} \). Thus, we can rewrite \( y \) as:
\[ y = \cos^{-1} \left( \cos\left(\frac{x}{2}\right) \right) \]

### Step 2: Apply the property of inverse functions
Using the property of inverse functions, we have:
\[ y = \frac{x}{2} \quad \text{(for } 0 \leq x \leq \pi \text{)} \]

### Step 3: Differentiate \( y \) with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \]

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