To differentiate the function \( f(x) = x\ln(x) - x \), we will apply the rules of differentiation.

### Step-by-step differentiation:

1. **Differentiate \( x\ln(x) \):**
   - Use the product rule: If \( u(x) = x \) and \( v(x) = \ln(x) \), then \( (uv)' = u'v + uv' \).
   - \( u'(x) = 1 \) and \( v'(x) = \frac{1}{x} \).
   - So, \( \frac{d}{dx} [x\ln(x)] = 1\cdot\ln(x) + x\cdot\frac{1}{x} = \ln(x) + 1 \).

2. **Differentiate \( -x \):**
   - The derivative of \( -x \) is \( -1 \).

Now, combining the two results:
\[
f'(x) = (\ln(x) + 1) - 1 = \ln(x)
\]

### Final Answer:
\[
f'(x) = \ln(x)
\]

Would you like further details on this process or have any questions?

Here are 5 related questions you might find interesting:
1. How do you apply the product rule in other functions?
2. What is the derivative of \( x^2 \ln(x) \)?
3. How do you differentiate logarithmic functions involving higher powers of \( x \)?
4. Can you explain the chain rule with examples?
5. What is the derivative of \( \ln(f(x)) \) where \( f(x) \) is a complex function?

**Tip**: When differentiating logarithmic functions, remember that \( \frac{d}{dx} [\ln(x)] = \frac{1}{x} \).

Learn how to differentiate the function f(x) = xln(x) - x step-by-step using the product rule. This guide explains key concepts like logarithmic functions and basic differentiation. Suitable for students in Grades 11-12 or college-level calculus.

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