If ` y = tan^(-1)( sqrt((x+1)/(x-1))) " for " |x| gt 1 " then " (dy)/(dx)` =

Options

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

Answer

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

### Step 1: Rewrite the expression
Let \( y = \tan^{-1}\left(\sqrt{\frac{x+1}{x-1}}\right) \).

### Step 2: Use a trigonometric substitution
To simplify the expression, we can let \( x = \sec(\theta) \). Then, we have:
\[ \sqrt{\frac{x+1}{x-1}} = \sqrt{\frac{\sec(\theta) + 1}{\sec(\theta) - 1}} \]

### Step 3: Simplify the square root
Using the identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \):
\[ \sqrt{\frac{\sec(\theta) + 1}{\sec(\theta) - 1}} = \sqrt{\frac{\frac{1}{\cos(\theta)} + 1}{\frac{1}{\cos(\theta)} - 1}} = \sqrt{\frac{1 + \cos(\theta)}{1 - \cos(\theta)}} \]

### Step 4: Use trigonometric identities
Using the identities \( 1 + \cos(\theta) = 2\cos^2\left(\frac{\theta}{2}\right) \) and \( 1 - \cos(\theta) = 2\sin^2\left(\frac{\theta}{2}\right) \):
\[ \sqrt{\frac{1 + \cos(\theta)}{1 - \cos(\theta)}} = \sqrt{\frac{2\cos^2\left(\frac{\theta}{2}\right)}{2\sin^2\left(\frac{\theta}{2}\right)}} = \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)} = \cot\left(\frac{\theta}{2}\right) \]

### Step 5: Substitute back into \( y \)
Now we can rewrite \( y \):
\[ y = \tan^{-1}\left(\cot\left(\frac{\theta}{2}\right)\right) = \frac{\pi}{2} - \frac{\theta}{2} \]

### Step 6: Express \( \theta \) in terms of \( x \)
Since \( x = \sec(\theta) \), we have \( \theta = \sec^{-1}(x) \). Therefore:
\[ y = \frac{\pi}{2} - \frac{1}{2} \sec^{-1}(x) \]

### Step 7: Differentiate \( y \) with respect to \( x \)
Now we differentiate \( y \):
\[ \frac{dy}{dx} = 0 - \frac{1}{2} \cdot \frac{d}{dx} \sec^{-1}(x) \]
Using the derivative of \( \sec^{-1}(x) \):
\[ \frac{d}{dx} \sec^{-1}(x) = \frac{1}{|x| \sqrt{x^2 - 1}} \]
Thus,
\[ \frac{dy}{dx} = -\frac{1}{2} \cdot \frac{1}{|x| \sqrt{x^2 - 1}} = -\frac{1}{2 |x| \sqrt{x^2 - 1}} \]

### Final Result
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{1}{2 |x| \sqrt{x^2 - 1}} \]