The derivative of the given function \( f(x) = \ln(x) \) with respect to \( x \) can be found using the chain rule.

To differentiate \( \ln(x) \), the result is:

\[
\frac{d}{dx} \left( \ln(x) \right) = \frac{1}{x}
\]

Thus:

\[
\frac{d}{dx} \left( \ln(x) \right) = \frac{1}{x}
\]

If you were asking for the second derivative (denoted as \( \frac{d^2}{dx^2} \left( \ln(x) \right) \)), it would be:

\[
\frac{d^2}{dx^2} \left( \ln(x) \right) = \frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}
\]

Would you like more details or further clarification?

### Relative questions:
1. How do you differentiate other logarithmic functions like \( \ln(f(x)) \)?
2. What is the derivative of \( \ln(ax + b) \)?
3. How do you find the derivative of \( e^{\ln(x)} \)?
4. Can you explain the chain rule used in differentiating \( \ln(x) \)?
5. What is the general rule for differentiating natural logarithms?

### Tip:
Always remember that the derivative of \( \ln(x) \) is only valid for \( x > 0 \) because \( \ln(x) \) is undefined for \( x \leq 0 \).

Explore how to differentiate the natural logarithmic function ln(x) using the chain rule. The solution provides step-by-step differentiation, including the first and second derivatives. This is ideal for students in Grades 11-12 studying advanced calculus topics.

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