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 $\frac{d}{dx} [\ln(x)] = \frac{1}{x}$.