Find the derivative of `(e^(x))/(1 + x^(2))`

Answer

To find the derivative of the function \( y = \frac{e^x}{1 + x^2} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the 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 1: Identify \( u \) and \( v \)
Let:
- \( u = e^x \)
- \( v = 1 + x^2 \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
- The derivative of \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^x) = e^x \]

- The derivative of \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(1 + x^2) = 0 + 2x = 2x \]

### Step 3: Apply the Quotient Rule
Now, substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

\[ \frac{dy}{dx} = \frac{(1 + x^2)(e^x) - (e^x)(2x)}{(1 + x^2)^2} \]

### Step 4: Simplify the expression
Factor out \( e^x \) from the numerator:

\[ \frac{dy}{dx} = \frac{e^x(1 + x^2 - 2x)}{(1 + x^2)^2} \]

### Final Answer
Thus, the derivative of \( y = \frac{e^x}{1 + x^2} \) is:

\[ \frac{dy}{dx} = \frac{e^x(1 + x^2 - 2x)}{(1 + x^2)^2} \]