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

Options

A. `(1)/(sqrt(1-x^(4)))`
B. `-(1)/(sqrt(1-x^(4)))`
C. `(x)/(sqrt(1-x^(4)))`
D. `-(x)/(sqrt(1-x^(4)))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function

\[ y = \tan^{-1} \left( \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}} \right), \]

we will follow these steps:

### Step 1: Simplify the expression inside the arctangent

Let \( u = \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}} \).

### Step 2: Use trigonometric identities

We can rewrite \( \sqrt{1+x^2} \) and \( \sqrt{1-x^2} \) in terms of trigonometric functions.

Let \( x^2 = \cos(2\theta) \). Then:
- \( 1 + x^2 = 1 + \cos(2\theta) = 2\cos^2(\theta) \)
- \( 1 - x^2 = 1 - \cos(2\theta) = 2\sin^2(\theta) \)

Substituting these into \( u \):

\[ u = \frac{\sqrt{2\cos^2(\theta)} + \sqrt{2\sin^2(\theta)}}{\sqrt{2\cos^2(\theta)} - \sqrt{2\sin^2(\theta)}} \]

### Step 3: Factor out the square root of 2

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 4: Use the tangent addition formula

Using the tangent addition formula:

\[ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \]

We can express \( u \) as:

\[ u = \tan\left(\frac{\pi}{4} + \theta\right) \]

### Step 5: Substitute back into \( y \)

Thus, we have:

\[ y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} + \theta\right)\right) \]

### Step 6: Simplify \( y \)

Since \( \frac{\pi}{4} + \theta \) is in the range of the arctangent function, we can write:

\[ y = \frac{\pi}{4} + \theta \]

### Step 7: Relate \( \theta \) back to \( x \)

Recall that \( \theta = \frac{1}{2} \cos^{-1}(x^2) \). Therefore:

\[ y = \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x^2) \]

### Step 8: Differentiate \( y \)

Now, differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = 0 + \frac{1}{2} \cdot \frac{d}{dx} \left( \cos^{-1}(x^2) \right) \]

Using the derivative of \( \cos^{-1}(x) \):

\[ \frac{d}{dx} \left( \cos^{-1}(x^2) \right) = -\frac{1}{\sqrt{1 - (x^2)^2}} \cdot 2x = -\frac{2x}{\sqrt{1 - x^4}} \]

Thus:

\[ \frac{dy}{dx} = \frac{1}{2} \left( -\frac{2x}{\sqrt{1 - x^4}} \right) = -\frac{x}{\sqrt{1 - x^4}} \]

### Final Result

So, the derivative is:

\[ \frac{dy}{dx} = -\frac{x}{\sqrt{1 - x^4}} \]