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The problem involves finding the derivative of the function \( f(x) = 2^x \cdot \tan(3x) \) at \( x = 0 \). Here's the step-by-step solution:

### Step 1: Define the Function
The function is given by:

\[
f(x) = 2^x \cdot \tan(3x)
\]

### Step 2: Use the Product Rule
To find the derivative \( f'(x) \), we use the product rule:

\[
f'(x) = \frac{d}{dx} (2^x) \cdot \tan(3x) + 2^x \cdot \frac{d}{dx} (\tan(3x))
\]

### Step 3: Differentiate Each Part
1. Derivative of \( 2^x \):
   
   The derivative of \( 2^x \) with respect to \( x \) is:

   \[
   \frac{d}{dx}(2^x) = 2^x \ln(2)
   \]

2. Derivative of \( \tan(3x) \):

   Using the chain rule, the derivative of \( \tan(3x) \) is:

   \[
   \frac{d}{dx}(\tan(3x)) = 3 \sec^2(3x)
   \]

### Step 4: Combine the Derivatives
Substituting these into the product rule formula, we get:

\[
f'(x) = (2^x \ln(2)) \cdot \tan(3x) + 2^x \cdot (3 \sec^2(3x))
\]

### Step 5: Evaluate at \( x = 0 \)
To find \( f'(0) \), substitute \( x = 0 \):

1. For the first term:

   \[
   (2^0 \ln(2)) \cdot \tan(3 \times 0) = 1 \cdot \ln(2) \cdot 0 = 0
   \]

2. For the second term:

   \[
   2^0 \cdot 3 \sec^2(3 \times 0) = 1 \cdot 3 \cdot 1 = 3
   \]

Thus:

\[
f'(0) = 0 + 3 = 3
\]

### Conclusion
The correct answer is:

\[
\boxed{3}
\]

Let me know if you need further details or have questions! Here are some related questions for further exploration:

1. How does the chain rule apply when differentiating composite functions?
2. What is the significance of the natural logarithm in exponential function derivatives?
3. How do you apply the product rule to more than two functions?
4. Can you explore the behavior of \( \tan(x) \) near zero?
5. What happens if you change the base of the exponential function?

**Tip:** Always evaluate each part of a function separately when using the product rule to ensure accuracy!

If f : (-π/6, π/6) → ℝ is defined by f(x) = 2^x tan(3x) for every x ∈ (-π/6, π/6), then f'(0) is?

This problem explores how to find the derivative of the function f(x) = 2^x tan(3x) at x = 0 using the product rule, chain rule, and knowledge of trigonometric and exponential function derivatives. Suitable for advanced math students in Grades 11-12.

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