Differentiate `tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}` with respect to `cos^(-1)x^2`

Explanation

To differentiate the function \( u = \tan^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) \) with respect to \( v = \cos^{-1}(x^2) \), we will use the chain rule. Here are the steps to solve the problem:

### Step 1: Define the functions
Let:
- \( u = \tan^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) \)
- \( v = \cos^{-1}(x^2) \)

### Step 2: Differentiate \( u \) with respect to \( x \)
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