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

### Function: 
\[ f(x) = x \ln x - x \]

### Step 1: Differentiate \( x \ln x \)
We apply the **product rule**, which states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative of their product is:
\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

Here, \( u(x) = x \) and \( v(x) = \ln x \).

- The derivative of \( u(x) = x \) is \( 1 \).
- The derivative of \( v(x) = \ln x \) is \( \frac{1}{x} \).

Now apply the product rule:
\[
\frac{d}{dx}(x \ln x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1
\]

### Step 2: Differentiate \( -x \)
The derivative of \( -x \) is simply \( -1 \).

### Step 3: Combine the results
Now, combining the two results:
\[
f'(x) = \ln x + 1 - 1 = \ln x
\]

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

Would you like further details or have any questions?

### Related Questions:
1. What are the critical points of \( f(x) = x \ln x - x \)?
2. What is the second derivative of \( f(x) = x \ln x - x \)?
3. How does the function \( f(x) = x \ln x - x \) behave at \( x = 1 \)?
4. What is the domain of the function \( f(x) = x \ln x - x \)?
5. How can the derivative \( f'(x) = \ln x \) be used to determine concavity?

### Tip:
Always remember to check the domain of the logarithmic function when differentiating expressions involving \( \ln x \), since \( \ln x \) is only defined for \( x > 0 \).

what is the derivative of f(x) = x ln x - x?

Learn how to find the derivative of the function f(x) = x ln x - x. This solution uses the product rule for differentiation, applying it to the logarithmic function and simplifying. Suitable for advanced high school or early college students.

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