(ii) Let `y= (x)/(a^(2) +x^(2))` then find dy/dx

Answer

To find the derivative of the function \( y = \frac{x}{a^2 + x^2} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form of \( \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step-by-Step Solution:

1. **Identify \( u \) and \( v \)**:
- Let \( u = x \)
- Let \( v = a^2 + x^2 \)

2. **Find \( \frac{du}{dx} \)**:
- The derivative of \( u \) with respect to \( x \) is:
\[ \frac{du}{dx} = 1 \]

3. **Find \( \frac{dv}{dx} \)**:
- The derivative of \( v \) with respect to \( x \) is:
\[ \frac{dv}{dx} = 0 + 2x = 2x \]

4. **Apply the quotient rule**:
- Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{(a^2 + x^2)(1) - (x)(2x)}{(a^2 + x^2)^2} \]

5. **Simplify the numerator**:
- Calculate the numerator:
\[ (a^2 + x^2) - 2x^2 = a^2 + x^2 - 2x^2 = a^2 - x^2 \]

6. **Write the final expression for \( \frac{dy}{dx} \)**:
- Thus, the derivative is:
\[ \frac{dy}{dx} = \frac{a^2 - x^2}{(a^2 + x^2)^2} \]

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