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To find the equation of the tangent line to the curve \( y = \frac{1 + x}{1 + e^x} \) at the specified point \( (0, \frac{1}{2}) \), we need to follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)
To find the slope of the tangent line, we’ll first find the derivative \( \frac{dy}{dx} \).

Given:
\[
y = \frac{1 + x}{1 + e^x}
\]
we can use the quotient rule for differentiation, which states:
\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}
\]
where \( u = 1 + x \) and \( v = 1 + e^x \).

1. **Differentiate \( u \):**
   \[
   u' = \frac{d}{dx}(1 + x) = 1
   \]

2. **Differentiate \( v \):**
   \[
   v' = \frac{d}{dx}(1 + e^x) = e^x
   \]

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

Simplifying the numerator:
\[
= \frac{1 + e^x - e^x - x e^x}{(1 + e^x)^2} = \frac{1 - x e^x}{(1 + e^x)^2}
\]

### Step 2: Evaluate the Derivative at \( x = 0 \)
Now, substitute \( x = 0 \) into \( \frac{dy}{dx} \) to find the slope at the point \( (0, \frac{1}{2}) \).

1. **Substitute \( x = 0 \):**
   \[
   \frac{dy}{dx} \Bigg|_{x=0} = \frac{1 - (0) \cdot e^0}{(1 + e^0)^2} = \frac{1}{(1 + 1)^2} = \frac{1}{4}
   \]

Thus, the slope of the tangent line at \( (0, \frac{1}{2}) \) is \( \frac{1}{4} \).

### Step 3: Write the Equation of the Tangent Line
Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m = \frac{1}{4} \), \( x_1 = 0 \), and \( y_1 = \frac{1}{2} \), we get:

\[
y - \frac{1}{2} = \frac{1}{4}(x - 0)
\]

Simplifying, we find:
\[
y = \frac{1}{4}x + \frac{1}{2}
\]

### Final Answer
The equation of the tangent line to the curve at the point \( (0, \frac{1}{2}) \) is:
\[
y = \frac{1}{4}x + \frac{1}{2}
\]

Would you like further details on any part of this solution?

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Here are some related questions you might find helpful:
1. How do we use the chain rule in differentiation?
2. What other methods could we use to find the tangent line?
3. How does the quotient rule apply to functions with exponents?
4. What is the significance of the slope of the tangent line at a point?
5. How can we verify that \( (0, \frac{1}{2}) \) is on the original curve?

**Tip:** Always recheck differentiation steps, especially with complex functions, to ensure accuracy.

Find the equation of the tangent line to the curve y = (1 + x) / (1 + e^x) at the point (0, 1/2).

Step-by-step solution to finding the tangent line to the curve y = (1 + x) / (1 + e^x) at the point (0, 1/2). Includes differentiation using the quotient rule and application of the point-slope formula.

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