To find the derivative of the function \( f'(x) = x^2 \ln x \), we will use the **product rule** of differentiation. The product rule states that if you have a function \( f(x) = u(x) \cdot v(x) \), then its derivative is:

\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]

In this case:

- \( u(x) = x^2 \) and \( v(x) = \ln x \).

Let's compute the derivatives of \( u(x) \) and \( v(x) \):

1. \( u'(x) = \frac{d}{dx}(x^2) = 2x \)
2. \( v'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} \)

Now, apply the product rule:

\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]

Substituting the values:

\[
f'(x) = (2x) \cdot \ln x + x^2 \cdot \frac{1}{x}
\]

Simplifying further:

\[
f'(x) = 2x \ln x + x
\]

So, the derivative of \( f(x) = x^2 \ln x \) is:

\[
\boxed{f'(x) = 2x \ln x + x}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the derivative of \( f(x) = x^2 e^x \)?
2. How would you differentiate \( f(x) = x^2 \sin(x) \)?
3. What is the second derivative \( f''(x) \) of \( f(x) = x^2 \ln x \)?
4. How does the product rule apply to three functions multiplied together?
5. How would you find the derivative of \( f(x) = x^2 \ln(x^2) \)?

**Tip:** Remember that logarithmic differentiation can be useful when differentiating products, quotients, or powers where the exponents are functions of the variable.

Explore how to differentiate the function f(x) = x^2 ln x using the product rule of differentiation. Step-by-step solution with formulas and explanations. Suitable for advanced high school students and anyone studying calculus.

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