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: $y'(x) = 3^{9x} (9 \ln 3)$.

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.