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

Options

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

Answer

To solve the problem, we need to differentiate the function given by \( y = \cos^{-1} \left( \sqrt{\frac{1+x}{2}} \right) \) with respect to \( x \).

### Step-by-Step Solution:

1. **Identify the Function**:
\[ y = \cos^{-1} \left( \sqrt{\frac{1+x}{2}} \right) \]

2. **Differentiate Using the Chain Rule**:
The derivative of \( \cos^{-1}(u) \) where \( u = \sqrt{\frac{1+x}{2}} \) is given by:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]

3. **Calculate \( u^2 \)**:
\[ u^2 = \frac{1+x}{2} \]
Therefore,
\[ 1 - u^2 = 1 - \frac{1+x}{2} = \frac{2 - (1+x)}{2} = \frac{1 - x}{2} \]

4. **Find \( \frac{du}{dx} \)**:
Since \( u = \sqrt{\frac{1+x}{2}} \),
\[ \frac{du}{dx} = \frac{1}{2\sqrt{\frac{1+x}{2}}} \cdot \frac{d}{dx} \left( \frac{1+x}{2} \right) = \frac{1}{2\sqrt{\frac{1+x}{2}}} \cdot \frac{1}{2} = \frac{1}{4\sqrt{\frac{1+x}{2}}} \]

5. **Substituting into the Derivative**:
Now substituting \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{\frac{1 - x}{2}}} \cdot \frac{1}{4\sqrt{\frac{1+x}{2}}} \]

6. **Simplifying the Expression**:
We can simplify this further:
\[ \frac{dy}{dx} = -\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{1 - x}{2}} \cdot \sqrt{\frac{1+x}{2}}} = -\frac{1}{4} \cdot \frac{1}{\sqrt{\frac{(1-x)(1+x)}{4}}} = -\frac{1}{4} \cdot \frac{2}{\sqrt{1-x^2}} = -\frac{1}{2\sqrt{1-x^2}} \]

### Final Result:
\[ \frac{dy}{dx} = -\frac{1}{2\sqrt{1-x^2}} \]