`(e^(x))/((1+x^(2)))`

Answer

To differentiate the function \( y = \frac{e^x}{1 + x^2} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then its derivative is given by:

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

where \( u = e^x \) and \( v = 1 + x^2 \).

### Step-by-Step Solution:

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

2. **Differentiate \( u \) and \( v \)**:
- The derivative of \( u \) is:
\[ \frac{du}{dx} = e^x \]
- The derivative of \( v \) is:
\[ \frac{dv}{dx} = 0 + 2x = 2x \]

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

4. **Simplify the numerator**:
- The numerator becomes:
\[ (1 + x^2)e^x - 2xe^x = e^x(1 + x^2 - 2x) \]
- Thus, we can rewrite the derivative as:
\[ \frac{dy}{dx} = \frac{e^x(1 + x^2 - 2x)}{(1 + x^2)^2} \]

5. **Final Result**:
- Therefore, the derivative of \( y = \frac{e^x}{1 + x^2} \) is:
\[ \frac{dy}{dx} = \frac{e^x(1 + x^2 - 2x)}{(1 + x^2)^2} \]