`y=tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))),

Explanation

To solve the problem step by step, we will differentiate the given function \( y = \tan^{-1} \left( \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} - \sqrt{1-x^2}} \right) \).

### Step 1: Simplifying the Argument of the Inverse Tangent
We start by letting \( x^2 = \cos(2\theta) \). Then we can express \( \sqrt{1+x^2} \) and \( \sqrt{1-x^2} \) in terms of \( \theta \):

\[ \sqrt{1+x^2} = \sqrt{1+\cos(2\theta)} = \sqrt{2\cos^2(\theta)} = \sqrt{2} \cos(\theta) \]
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