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Let's differentiate the function \( f(x) = e^x \cdot x^2 \cdot \ln(x) \) with respect to \( x \).

To differentiate this product of three functions, we will use the product rule. The product rule for three functions \( u(x) \), \( v(x) \), and \( w(x) \) is given by:

\[
\frac{d}{dx}[u(x) \cdot v(x) \cdot w(x)] = u'(x) \cdot v(x) \cdot w(x) + u(x) \cdot v'(x) \cdot w(x) + u(x) \cdot v(x) \cdot w'(x)
\]

In this case:
- \( u(x) = e^x \)
- \( v(x) = x^2 \)
- \( w(x) = \ln(x) \)

Let's differentiate each function:

1. \( u'(x) = \frac{d}{dx} [e^x] = e^x \)
2. \( v'(x) = \frac{d}{dx} [x^2] = 2x \)
3. \( w'(x) = \frac{d}{dx} [\ln(x)] = \frac{1}{x} \)

Now, applying the product rule:

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

Simplify the expression:

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

Finally, combine like terms:

\[
f'(x) = e^x \left( x \ln(x) + 2x \ln(x) + x^2 \right)
\]

This simplifies to:

\[
f'(x) = e^x \left( 3x \ln(x) + x^2 \right)
\]

So the derivative of the function is:

\[
f'(x) = e^x \cdot \left( x^2 + 3x \ln(x) \right)
\]

Would you like any further details or have any questions?

### Further Questions:
1. Would you like to find the critical points of \( f(x) \) using the derivative?
2. Should we examine the second derivative to determine concavity or inflection points?
3. Do you need to evaluate \( f'(x) \) at specific values of \( x \)?
4. Are there any points of interest (like maximums or minimums) in \( f(x) \) you'd like to explore?
5. Would you like to see a plot of the function and its derivative?

**Tip:** Understanding how to differentiate functions with multiple products is crucial in calculus, especially when dealing with complex expressions.

Learn how to differentiate the function f(x) = e^x * x^2 * ln(x) using the product rule for differentiation. This problem involves calculus concepts, including differentiation of exponential, polynomial, and logarithmic functions. Suitable for college-level calculus students.

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