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To find the slope of the tangent line to the function \( y = 3^x \) at \( x = 2 \), we need to calculate the derivative of \( y = 3^x \) and then evaluate it at \( x = 2 \).

### Steps:

1. **Differentiate the function:**
   The derivative of \( y = 3^x \) with respect to \( x \) is:
   \[
   \frac{dy}{dx} = 3^x \ln(3)
   \]
   This result comes from the fact that for exponential functions of the form \( a^x \), the derivative is \( a^x \ln(a) \), where \( a = 3 \) in this case.

2. **Evaluate the derivative at \( x = 2 \):**
   \[
   \frac{dy}{dx}\bigg|_{x=2} = 3^2 \ln(3) = 9 \ln(3)
   \]

Thus, the slope of the tangent line to the curve \( y = 3^x \) at \( x = 2 \) is \( 9 \ln(3) \).

Let me know if you'd like a more detailed explanation of any part!

### Here are some related questions:
1. What is the tangent line equation at \( x = 2 \) for \( y = 3^x \)?
2. How does the derivative of \( a^x \) change if \( a \) is a constant other than 3?
3. How do you find the tangent line at any point for \( y = a^x \)?
4. Can we use this method for other exponential functions, like \( y = e^x \)?
5. How do logarithmic functions relate to exponential functions in calculus?

**Tip:** Always remember that for an exponential function \( a^x \), the derivative involves multiplying by \( \ln(a) \), which is key for solving problems like this.

Find the slope for the line tangent to \( y = 3^x \) at \( x = 2 \).

Learn how to find the slope of the tangent line to the function \( y = 3^x \) at \( x = 2 \). This solution involves differentiation of exponential functions, a key concept in calculus. Suitable for students in Grades 10-12.

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