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 $e^x$.