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.