To find the derivative of $f(x) = e^x(x^2 + 2)(x + 2)$, we'll apply the product rule and chain rule.

Steps:

1. Define the function as a product of three parts: $f(x) = g(x) \cdot h(x) \cdot i(x)$ where:

   • $g(x) = e^x$
   • $h(x) = x^2 + 2$
   • $i(x) = x + 2$

2. Use the product rule: [ \frac{d}{dx}[g(x) \cdot h(x) \cdot i(x)] = \frac{d}{dx}[g(x)] \cdot h(x) \cdot i(x) + g(x) \cdot \frac{d}{dx}[h(x)] \cdot i(x) + g(x) \cdot h(x) \cdot \frac{d}{dx}[i(x)] ]

3. Calculate the derivatives of each part:

   • $\frac{d}{dx}[g(x)] = \frac{d}{dx}[e^x] = e^x$
   • $\frac{d}{dx}[h(x)] = \frac{d}{dx}[x^2 + 2] = 2x$
   • $\frac{d}{dx}[i(x)] = \frac{d}{dx}[x + 2] = 1$

4. Substitute these values into the product rule expression: $\frac{d}{dx}[f(x)] = e^x \cdot (x^2 + 2) \cdot (x + 2) + e^x \cdot 2x \cdot (x + 2) + e^x \cdot (x^2 + 2) \cdot 1$

5. Factor out $e^x$ from the entire expression: $f'(x) = e^x \left[ (x^2 + 2)(x + 2) + 2x(x + 2) + (x^2 + 2) \right]$

6. Now simplify the terms inside the brackets:

   • $(x^2 + 2)(x + 2) = x^3 + 2x^2 + 2x + 4$
   • $2x(x + 2) = 2x^2 + 4x$
   • $x^2 + 2 = x^2 + 2$

7. Combine all terms: $f'(x) = e^x \left[ x^3 + 2x^2 + 2x + 4 + 2x^2 + 4x + x^2 + 2 \right]$ $f'(x) = e^x \left[ x^3 + 5x^2 + 6x + 6 \right]$

Thus, the derivative of $f(x) = e^x(x^2 + 2)(x + 2)$ is: $f'(x) = e^x(x^3 + 5x^2 + 6x + 6)$

Would you like any further details or clarification?

Here are five related questions:

1. What is the derivative of $f(x) = e^x(x^3 + x^2)$?
2. Can you explain the product rule in more detail?
3. How do we apply the chain rule in more complex functions?
4. What are some common mistakes when differentiating products of functions?
5. How does factoring out common terms help simplify derivatives?

Tip: Always double-check each term when applying the product rule to avoid missing any part of the function.