`tan^(-1) "" ( sqrt"" (1+x) - sqrt"" (1-x))/(sqrt (1+x)+sqrt""(1-x)),(dy)/(dx)=`

Options

A. `(1)/(sqrt""(1-x2))`
B. `(1)/(2 sqrt"" (1-x^2))`
C. `(x)/( sqrt""(1-x^2))`
D. None

Answer

To solve the problem \( y = \tan^{-1} \left( \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \right) \) and find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Simplify the Expression
We start with the expression for \( y \):
\[ y = \tan^{-1} \left( \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \right) \]
Let’s denote:
\[ u = \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}} \]

### Step 2: Use Substitution
We can use the substitution \( x = \cos(2\theta) \). Then:
\[ 1 + x = 1 + \cos(2\theta) = 2\cos^2(\theta) \]
\[ 1 - x = 1 - \cos(2\theta) = 2\sin^2(\theta) \]
Substituting these into \( u \):
\[ u = \frac{\sqrt{2\cos^2(\theta)} - \sqrt{2\sin^2(\theta)}}{\sqrt{2\cos^2(\theta)} + \sqrt{2\sin^2(\theta)}} \]
This simplifies to:
\[ u = \frac{\sqrt{2}(\cos(\theta) - \sin(\theta))}{\sqrt{2}(\cos(\theta) + \sin(\theta))} = \frac{\cos(\theta) - \sin(\theta)}{\cos(\theta) + \sin(\theta)} \]

### Step 3: Further Simplification
Using the tangent subtraction formula:
\[ u = \tan\left(\frac{\pi}{4} - \theta\right) \]
Thus, we have:
\[ y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \theta\right)\right) = \frac{\pi}{4} - \theta \]

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

### Step 5: Differentiate \( y \) with respect to \( x \)
Now we differentiate \( y \):
\[ \frac{dy}{dx} = 0 - \frac{1}{2} \cdot \frac{d}{dx}(\cos^{-1}(x)) \]
Using the derivative of \( \cos^{-1}(x) \):
\[ \frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^2}} \]
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 \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{1-x^2}} \]