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.