Differentiate the function f(x) = x ln x - x

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

1. The function is given as: 
\[ f(x) = x \ln x - x \]

2. Differentiate each term separately:
   - For the term \( x \ln x \), we will use the product rule, which states that \(\frac{d}{dx}[u \cdot v] = u'v + uv'\).
     - Let \( u = x \) and \( v = \ln x \).
     - Then, \( u' = 1 \) and \( v' = \frac{1}{x} \).
     - Applying the product rule: 
     \[ \frac{d}{dx}(x \ln x) = x \cdot \frac{1}{x} + \ln x \cdot 1 = 1 + \ln x \]

   - For the term \( -x \), the derivative is straightforward:
     \[ \frac{d}{dx}(-x) = -1 \]

3. Combine the results:
\[ f'(x) = 1 + \ln x - 1 = \ln x \]

So, the derivative of the function \( f(x) = x \ln x - x \) is:
\[ f'(x) = \ln x \]

Would you like more details or have any questions? 

Here are 5 questions you might ask next:
1. How do you apply the product rule in differentiation?
2. What are the basic rules of differentiation?
3. Can you differentiate \( x^2 \ln x \) for me?
4. How do you find the derivative of \( \ln x \)?
5. How do you integrate the function \( \ln x \)?

**Tip:** When differentiating a product of two functions, remember to use the product rule, which ensures each part of the product is accounted for correctly.

Learn how to differentiate the function f(x) = x ln x - x with a step-by-step guide. This problem covers differentiation using the product rule and involves natural logarithms. Suitable for advanced high school students and anyone studying calculus.

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