`y = sin^(-1) {(sqrt(1 +x) + sqrt(1 -x))/(2)} " then " (dy)/(dx) =` ?

Options

A. `(-1)/(2 sqrt(1 -x^(2)))`
B. `(1)/(2 sqrt(1 -x^(2)))`
C. `(1)/(2(1 +x^(2)))`
D. none of these

Explanation

To differentiate the function \( y = \sin^{-1} \left( \frac{\sqrt{1+x} + \sqrt{1-x}}{2} \right) \), we will follow these steps:

### Step 1: Substitute \( x \) with \( \cos(2\theta) \)

We know that:
\[ 1 + x = 1 + \cos(2\theta) = 2\cos^2(\theta) \]
\[ 1 - x = 1 - \cos(2\theta) = 2\sin^2(\theta) \]

Thus, we can rewrite the expression inside the inverse sine:
\[ \sqrt{1+x} = \sqrt{2\cos^2(\theta)} = \sqrt{2} \cos(\theta) \]
\[ \sqrt{1-x} = \sqrt{2\sin^2(\theta)} = \sqrt{2} \sin(\theta) \]

So, we have:
\[ y = \sin^{-1} \left( \frac{\sqrt{2} \cos(\theta) + \sqrt{2} \sin(\theta)}{2} \right) \]
This simplifies to:
\[ y = \sin^{-1} \left( \frac{\sqrt{2}}{2} (\cos(\theta) + \sin(\theta)) \right) \]

### Step 2: Simplify the expression

Using the identity \( \cos(\theta) + \sin(\theta) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \):
\[ y = \sin^{-1} \left( \frac{\sqrt{2}}{2} \cdot \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \right) \]
This simplifies to:
\[ y = \sin^{-1} \left( \sin\left(\theta + \frac{\pi}{4}\right) \right) \]
Thus, we have:
\[ y = \theta + \frac{\pi}{4} \]

### Step 3: Relate \( \theta \) back to \( x \)

Since \( x = \cos(2\theta) \), we have:
\[ \theta = \frac{1}{2} \cos^{-1}(x) \]
Substituting back, we get:
\[ y = \frac{1}{2} \cos^{-1}(x) + \frac{\pi}{4} \]

### Step 4: Differentiate \( y \)

Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{d}{dx} \left( \cos^{-1}(x) \right) \]
Using the derivative of \( \cos^{-1}(x) \):
\[ \frac{d}{dx} \left( \cos^{-1}(x) \right) = -\frac{1}{\sqrt{1-x^2}} \]
Thus:
\[ \frac{dy}{dx} = \frac{1}{2} \left( -\frac{1}{\sqrt{1-x^2}} \right) = -\frac{1}{2\sqrt{1-x^2}} \]

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

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