The problem presented is asking for the derivative of $\log_x(e)$ with respect to $x$, where $x > 0$.

The function is $\log_x(e)$, which represents the logarithm of $e$ (Euler's number) with base $x$.

Step-by-step solution:

1. Rewriting the logarithm:
   Recall that $\log_x(e)$ can be rewritten using the change of base formula: $\log_x(e) = \frac{\ln(e)}{\ln(x)}$ where $\ln$ represents the natural logarithm (logarithm base $e$).

2. Simplifying:
   Since $\ln(e) = 1$, this simplifies to: $\log_x(e) = \frac{1}{\ln(x)}$

3. Taking the derivative:
   Now we need to differentiate $\frac{1}{\ln(x)}$ with respect to $x$. Using the chain rule, the derivative of $\frac{1}{\ln(x)}$ is: $\frac{d}{dx} \left( \frac{1}{\ln(x)} \right) = -\frac{1}{(\ln(x))^2} \cdot \frac{1}{x}$

4. Final result:
   So, the derivative is: $\frac{d}{dx} \log_x(e) = -\frac{1}{x (\ln(x))^2}$

This is the required derivative of $\log_x(e)$ with respect to $x$.

Would you like further clarification or additional details on any step?

Here are five questions related to this topic:

1. How do you differentiate logarithmic functions with different bases?
2. What is the change of base formula and why is it important in logarithms?
3. What role does the chain rule play in differentiation of logarithmic functions?
4. How does natural logarithm $\ln$ simplify differentiation in base $e$?
5. What is the significance of Euler’s number $e$ in calculus and logarithms?

Tip: When differentiating complex expressions, always look for ways to simplify the function first, such as applying logarithmic identities.