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.