It seems like you're asking for the derivative of the expression \(\frac{dx}{1 + e^x}\). Let's interpret this and proceed accordingly.

Assuming you meant the derivative of \( \frac{1}{1 + e^x} \), we can apply the chain rule.

Let’s solve the derivative of \( f(x) = \frac{1}{1 + e^x} \):

1. Rewrite the function as:  
   \[
   f(x) = (1 + e^x)^{-1}
   \]

2. Differentiate using the chain rule:
   \[
   f'(x) = - (1 + e^x)^{-2} \cdot e^x
   \]

So, the derivative is:
\[
f'(x) = \frac{-e^x}{(1 + e^x)^2}
\]

Would you like more details on how the chain rule was applied, or any other clarifications?

Here are five related questions to consider:
1. What is the derivative of \( e^x \)?
2. How does the chain rule work in calculus?
3. What is the second derivative of \( \frac{1}{1 + e^x} \)?
4. Can you apply the quotient rule to \( \frac{1}{1 + e^x} \)?
5. What is the integral of \( \frac{1}{1 + e^x} \)?

**Tip:** When differentiating expressions with exponential functions, remember that the derivative of \( e^x \) is always \( e^x \).

Explore how to find the derivative of the function 1/(1 + e^x) using the chain rule. This solution includes step-by-step differentiation, applying the chain rule, and covers the key calculus concept of exponential functions. Suitable for college-level calculus students.

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