https://file.aiquickdraw.com/mathbot/file/1729296266890pj5n9e0w.jpg

We are given \( y(x) = 3^{9x} \) and need to find \( y'(x) \), the derivative of the function.

### Steps:
1. **Use the chain rule**: 
   \[
   y(x) = 3^{9x}
   \]
   To differentiate this, recall that the derivative of an exponential function \( a^x \) with respect to \( x \) is:
   \[
   \frac{d}{dx} a^x = a^x \ln(a)
   \]
   Applying this to \( 3^{9x} \), we need to handle the \( 9x \) in the exponent. 

2. **Differentiate \( 3^{9x} \)**:
   Using the chain rule, the derivative of \( 3^{9x} \) is:
   \[
   y'(x) = 3^{9x} \cdot \ln(3) \cdot \frac{d}{dx}(9x)
   \]
   The derivative of \( 9x \) is 9, so:
   \[
   y'(x) = 3^{9x} \cdot \ln(3) \cdot 9
   \]
   \[
   y'(x) = 9 \cdot 3^{9x} \cdot \ln(3)
   \]

### Correct Answer:
Looking at the options, the correct one is:
\[
y'(x) = 3^{9x} \cdot (9 \ln 3)
\]

This matches the last option in the image: \( y'(x) = 3^{9x} (9 \ln 3) \).

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Do you want further details or have any questions?  
Here are five related questions to deepen your understanding:
1. How does the chain rule work for composite functions like this?
2. Can we generalize the derivative rule for \( a^{kx} \) (where \( k \) is a constant)?
3. Why do we multiply by \( \ln(3) \) when differentiating \( 3^x \)?
4. What would the derivative of a function like \( e^{9x} \) be, and how is it different from \( 3^{9x} \)?
5. How does changing the base of the exponential function affect the derivative?

**Tip:** Always remember that when differentiating an exponential function with a constant base (e.g., \( 3^{f(x)} \)), you need to use the chain rule and include the logarithm of the base in the derivative.

Find the derivative of the exponential function y(x) = 3^(9x) using the chain rule. This problem involves understanding the differentiation of exponential functions and applying the chain rule to solve for y'(x). Suitable for advanced high school students in Grades 11-12.

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