`tan^(-1)((x)/(sqrt(1-x^(2))))"w.r.t."sec^(-1)((1)/(2x^(2)-1))`

Explanation

To solve the problem of finding the derivative of \( u = \tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right) \) with respect to \( v = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \), we will use the chain rule and implicit differentiation.

### Step-by-Step Solution:

1. **Define the Functions**:
Let \( u = \tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right) \) and \( v = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right) \).

2. **Differentiate \( u \) with respect to \( x \)**:
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