Differentiate the following w.r.t. x :
`sin^(-1)((2x)/(1+x^(2))),-1ltxlt1`

Answer

To differentiate the function \( y = \sin^{-1}\left(\frac{2x}{1+x^2}\right) \) with respect to \( x \), we will use the chain rule and the quotient rule.

### Step-by-Step Solution:

1. **Identify the function**:
Let \( u = \frac{2x}{1+x^2} \). Then, we can rewrite the function as:
\[ y = \sin^{-1}(u) \]

2. **Differentiate \( y \) with respect to \( u \)**:
The derivative of \( \sin^{-1}(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \frac{1}{\sqrt{1-u^2}} \]

3. **Differentiate \( u \) with respect to \( x \)**:
We need to differentiate \( u = \frac{2x}{1+x^2} \) using the quotient rule. The quotient rule states that if \( u = \frac{f(x)}{g(x)} \), then:
\[ \frac{du}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \]
Here, \( f(x) = 2x \) and \( g(x) = 1+x^2 \).

- Calculate \( f'(x) = 2 \)
- Calculate \( g'(x) = 2x \)

Now applying the quotient rule:
\[ \frac{du}{dx} = \frac{(2)(1+x^2) - (2x)(2x)}{(1+x^2)^2} = \frac{2 + 2x^2 - 4x^2}{(1+x^2)^2} = \frac{2 - 2x^2}{(1+x^2)^2} = \frac{2(1-x^2)}{(1+x^2)^2} \]

4. **Combine using the chain rule**:
Now, using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1-u^2}} \cdot \frac{2(1-x^2)}{(1+x^2)^2} \]

5. **Substitute back for \( u \)**:
We need to find \( 1 - u^2 \):
\[ u^2 = \left(\frac{2x}{1+x^2}\right)^2 = \frac{4x^2}{(1+x^2)^2} \]
Therefore:
\[ 1 - u^2 = 1 - \frac{4x^2}{(1+x^2)^2} = \frac{(1+x^2)^2 - 4x^2}{(1+x^2)^2} = \frac{1 + 2x^2 + x^4 - 4x^2}{(1+x^2)^2} = \frac{1 - 2x^2 + x^4}{(1+x^2)^2} \]

6. **Final expression**:
Now substituting \( 1 - u^2 \) into the derivative:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\frac{1 - 2x^2 + x^4}{(1+x^2)^2}}} \cdot \frac{2(1-x^2)}{(1+x^2)^2} \]
Simplifying:
\[ \frac{dy}{dx} = \frac{2(1-x^2)}{(1+x^2)^2} \cdot \frac{\sqrt{(1+x^2)^2}}{\sqrt{1 - 2x^2 + x^4}} = \frac{2(1-x^2)}{(1 - 2x^2 + x^4)} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{2(1-x^2)}{1 - 2x^2 + x^4} \]