Differentiate `sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-x^2)/(1+x^2))` with respect to `x ,if` `x in (0,1)` (ii) `x in (-1,0)` `x in (1,oo)` (iv) `x in (1,oo)`

Explanation

To differentiate the function \( f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \) with respect to \( x \), we can use a substitution method for easier differentiation. Let's go through the steps:

### Step-by-Step Solution

1. **Substitution**: Let \( x = \tan(\theta) \). Then, we have:
\[ \frac{2x}{1+x^2} = \frac{2\tan(\theta)}{1+\tan^2(\theta)} = \sin(2\theta) \]
...