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Let's solve the given problem by finding the derivative of the function \( f(x) = x^2 \cdot \ln(x) - \frac{x^2}{2} \).

### Step 1: Differentiate the first term \( x^2 \cdot \ln(x) \)
We will apply the product rule for differentiation, which is:
\[
(fg)' = f'g + fg'
\]
Here, \( f(x) = x^2 \) and \( g(x) = \ln(x) \).

- The derivative of \( x^2 \) is \( 2x \).
- The derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Now applying the product rule:
\[
\frac{d}{dx}(x^2 \cdot \ln(x)) = 2x \cdot \ln(x) + x^2 \cdot \frac{1}{x} = 2x \cdot \ln(x) + x
\]

### Step 2: Differentiate the second term \( \frac{x^2}{2} \)
The derivative of \( \frac{x^2}{2} \) is:
\[
\frac{d}{dx}\left(\frac{x^2}{2}\right) = \frac{2x}{2} = x
\]

### Step 3: Combine the results
Now we combine the derivatives of both terms:
\[
f'(x) = \left(2x \ln(x) + x\right) - x
\]

Simplifying:
\[
f'(x) = 2x \ln(x)
\]

Thus, the derivative of the function \( f(x) = x^2 \cdot \ln(x) - \frac{x^2}{2} \) is:
\[
f'(x) = 2x \ln(x)
\]

Would you like more details or have any questions?

### Related questions:
1. How do you apply the product rule in calculus for more complex functions?
2. What is the derivative of \( x^n \cdot \ln(x) \) for different values of \( n \)?
3. How does the chain rule work when differentiating functions involving logarithms?
4. Can we use implicit differentiation for logarithmic functions?
5. How would the derivative change if the function was \( f(x) = x^3 \ln(x) - \frac{x^3}{2} \)?

### Tip:
When differentiating products or quotients of functions, always consider using the product or quotient rule, respectively, and simplify your result to avoid errors.

Find the derivative of the function f given by f(x) = x^2 * ln(x) - x^2/2.

Learn how to find the derivative of the function f(x) = x^2 * ln(x) - x^2/2 by applying the product rule and logarithmic differentiation. This problem covers key calculus concepts and is suitable for college-level students.

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