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

Answer

To find the derivative of the function \( f(x) = \frac{e^x}{1 + x^2} \), 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 \( f'(x) \) is given by:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

where \( u \) is the numerator and \( v \) is the denominator.

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = e^x \)
- \( v = 1 + x^2 \)

### Step 2: Differentiate \( u \) and \( v \)

Now, we need to find the derivatives \( u' \) and \( v' \):

- The derivative of \( u \):
\[ u' = \frac{d}{dx}(e^x) = e^x \]

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

### Step 3: Apply the Quotient Rule

Now, we can apply the quotient rule:

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

Substituting \( u \), \( u' \), \( v \), and \( v' \):

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

### Step 4: Simplify the Expression

Factor out \( e^x \) from the numerator:

\[ f'(x) = \frac{e^x \left(1 + x^2 - 2x\right)}{(1 + x^2)^2} \]

Now, simplify the expression inside the parentheses:

\[ 1 + x^2 - 2x = x^2 - 2x + 1 = (x - 1)^2 \]

Thus, we have:

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

### Final Answer

The derivative of \( f(x) = \frac{e^x}{1 + x^2} \) is:

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

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