The function given is:

\[
f(x) = \ln(x) - x + 2
\]

To find its derivative \( f'(x) \), we differentiate each term with respect to \( x \):

1. The derivative of \( \ln(x) \) is \( \frac{1}{x} \).
2. The derivative of \( -x \) is \( -1 \).
3. The derivative of the constant \( 2 \) is \( 0 \).

Thus, the derivative is:

\[
f'(x) = \frac{1}{x} - 1
\]

This matches the derivative you provided: 

\[
f'(x) = \frac{1}{x} - 1
\]

Would you like more details on finding the derivative or anything else about the function?

Here are some related questions to expand on this topic:

1. What is the critical point(s) of the function \( f(x) = \ln(x) - x + 2 \)?
2. How would you determine if the function \( f(x) = \ln(x) - x + 2 \) has any local extrema?
3. Can you compute the second derivative of \( f(x) \) and analyze its concavity?
4. What is the domain and range of \( f(x) = \ln(x) - x + 2 \)?
5. How does the function behave as \( x \to \infty \) or \( x \to 0^+ \)?

**Tip:** To analyze the behavior of a function, always look at its critical points and inflection points by finding the first and second derivatives.

Explore the differentiation of the function f(x) = ln(x) - x + 2, with detailed steps on how to find f'(x) = 1/x - 1. This problem covers logarithmic functions, basic derivative rules, and their applications. Ideal for students in Grades 11-12 studying calculus.

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