The derivative of `sin^(-1)((sqrt(1+x)+sqrt(1-x))/(2))` with respect to x is

Options

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

Answer

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

### Step 1: Rewrite the function
Let:
\[ y = \sin^{-1}\left(\frac{\sqrt{1+x} + \sqrt{1-x}}{2}\right) \]

### Step 2: Substitute \( x \) with \( \cos(2\theta) \)
To simplify the expression, we can use the substitution:
\[ x = \cos(2\theta) \]
Then, we can express \( \sqrt{1+x} \) and \( \sqrt{1-x} \):
\[ \sqrt{1+x} = \sqrt{1 + \cos(2\theta)} = \sqrt{2\cos^2(\theta)} = \sqrt{2} \cos(\theta) \]
\[ \sqrt{1-x} = \sqrt{1 - \cos(2\theta)} = \sqrt{2\sin^2(\theta)} = \sqrt{2} \sin(\theta) \]

### Step 3: Substitute back into the function
Now substituting these into our expression for \( y \):
\[ y = \sin^{-1}\left(\frac{\sqrt{2}\cos(\theta) + \sqrt{2}\sin(\theta)}{2}\right) = \sin^{-1}\left(\frac{\sqrt{2}}{2}(\cos(\theta) + \sin(\theta))\right) \]

### Step 4: Simplify using trigonometric identities
Using the identity \( \sin(a + b) = \sin a \cos b + \cos a \sin b \):
\[ \cos(\theta) + \sin(\theta) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
Thus, we have:
\[ y = \sin^{-1}\left(\frac{1}{\sqrt{2}} \cdot \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right)\right) = \theta + \frac{\pi}{4} \]

### Step 5: Express \( \theta \) in terms of \( x \)
From our substitution \( x = \cos(2\theta) \), we can express \( \theta \) as:
\[ \theta = \frac{1}{2} \cos^{-1}(x) \]
So, substituting back:
\[ y = \frac{1}{2} \cos^{-1}(x) + \frac{\pi}{4} \]

### Step 6: Differentiate \( y \) with respect to \( x \)
Now, we differentiate:
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{d}{dx} \left(\cos^{-1}(x)\right) \]
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
The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{1}{2\sqrt{1-x^2}} \]