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We are given the following:

- \( f(2) = 4 \)
- \( f'(x) = 3^x \)
- We need to find the equation of the tangent line to \( f(x) \) at \( x = 2 \).

### Steps to Solve:
1. **Slope of the tangent line**:
   The slope of the tangent line at \( x = 2 \) is given by \( f'(2) \), which we can calculate by substituting \( x = 2 \) into the derivative \( f'(x) = 3^x \):
   \[
   f'(2) = 3^2 = 9
   \]
   So, the slope of the tangent line is \( 9 \).

2. **Point on the curve**:
   The point at which the tangent line touches the curve is \( (2, f(2)) \). From the problem, we are given \( f(2) = 4 \), so the point is \( (2, 4) \).

3. **Equation of the tangent line**:
   The equation of a line is given by the point-slope form:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( m \) is the slope and \( (x_1, y_1) \) is the point on the line. Substituting \( m = 9 \) and \( (x_1, y_1) = (2, 4) \):
   \[
   y - 4 = 9(x - 2)
   \]
   Simplifying:
   \[
   y - 4 = 9x - 18
   \]
   \[
   y = 9x - 14
   \]

### Final Answer:
The equation of the tangent line to \( f(x) \) at \( x = 2 \) is:
\[
y = 9x - 14
\]

Would you like more details on any step or have any other questions?

Here are 5 related questions to explore:
1. How would the tangent line change if \( f'(x) \) was a different exponential function?
2. What is the geometric meaning of the tangent line?
3. How do you determine the slope of a tangent line for functions with higher powers of \( x \)?
4. How can the equation of the normal line be derived using the same information?
5. What role does the derivative play in understanding the behavior of functions?

**Tip**: Remember, the derivative at a point gives the slope of the tangent line at that point.

Suppose that f(2) = 4 and f'(x) = 3^x. What is the equation of the tangent line to f(x) at x = 2?

Discover how to find the equation of the tangent line to f(x) at x = 2 where f(2) = 4 and f'(x) = 3^x. This involves using the point-slope form of a line and calculating the derivative of an exponential function. Suitable for high school or early college students.

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