You have the function \( f(x) = \frac{\ln(x) + a}{x} \), where \( a \) is a constant.

Here are some things we can explore with this function:

1. **Domain**: The natural logarithm function \( \ln(x) \) is defined for \( x > 0 \), so the domain of \( f(x) \) is \( x > 0 \).

2. **Derivative** (to find critical points, slope, or concavity): We can apply the quotient rule or rewrite the function as \( f(x) = \ln(x) \cdot x^{-1} + a \cdot x^{-1} \).

3. **Asymptotic behavior**: As \( x \to 0^+ \) or \( x \to \infty \), we can analyze how the function behaves, especially near these boundaries.

4. **Critical points and extrema**: Using the first derivative, we can find where the function has local maxima or minima.

5. **Second derivative**: To examine concavity or inflection points.

Would you like to explore the derivative, find specific points, or focus on another aspect of the function?

### Follow-up Questions:
1. What is the first derivative of \( f(x) \) using the quotient rule?
2. How does \( f(x) \) behave as \( x \to \infty \)?
3. Are there any critical points for \( f(x) \)?
4. Can you analyze the concavity of the function using the second derivative?
5. What is the effect of changing the constant \( a \) on the function's shape?

**Tip**: The derivative can provide insights into the increasing or decreasing nature of the function and help identify local extrema!

Explore the behavior of the function f(x) = (ln x + a)/x, including domain, derivatives, critical points, and asymptotic behavior. Learn how the quotient rule is applied to find the first derivative, and discover how changing the constant 'a' affects the function's shape. Ideal for undergraduate calculus students.

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