To find the derivative of \( f(x) = e^{3x} \cdot \cos(2x) \), we will use the **product rule** and **chain rule**.

The product rule states:
\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]
Here:
- \( f(x) = e^{3x} \)
- \( g(x) = \cos(2x) \)

### Step 1: Differentiate \( e^{3x} \)
Using the chain rule:
\[
f'(x) = \frac{d}{dx}(e^{3x}) = 3e^{3x}
\]

### Step 2: Differentiate \( \cos(2x) \)
Again using the chain rule:
\[
g'(x) = \frac{d}{dx}(\cos(2x)) = -2\sin(2x)
\]

### Step 3: Apply the product rule
Now applying the product rule:
\[
f'(x) = (3e^{3x}) \cdot \cos(2x) + e^{3x} \cdot (-2\sin(2x))
\]

### Final Answer:
\[
f'(x) = 3e^{3x} \cos(2x) - 2e^{3x} \sin(2x)
\]

Would you like further explanation on any steps or a breakdown of the product and chain rules?

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Here are five related questions to deepen your understanding:
1. How would you apply the product rule to \( f(x) = x^2 \cdot e^{x} \)?
2. What is the derivative of \( \sin(3x) \cdot e^{x} \)?
3. How would the second derivative \( f''(x) \) be computed for this function?
4. What is the chain rule and how does it apply to exponential functions like \( e^{kx} \)?
5. How do you handle the differentiation of trigonometric functions like \( \cos(ax) \) or \( \sin(bx) \)?

**Tip:** When dealing with derivatives involving products of functions, always check if the product or chain rule applies, as they are crucial for handling such cases efficiently.

Learn how to find the derivative of f(x) = e^(3x) * cos(2x) using the product rule and chain rule. This problem involves exponential and trigonometric functions and demonstrates step-by-step differentiation. Suitable for advanced high school or college students.

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