To differentiate $f(x) = (3x + 3)^3$ using the chain rule, follow these steps:

1. Outer function: Recognize that the outer function is $u^3$, where $u = 3x + 3$.

   • The derivative of $u^3$ with respect to $u$ is $3u^2$.

2. Inner function: The inner function is $u = 3x + 3$.

   • The derivative of $u$ with respect to $x$ is $\frac{d}{dx}(3x + 3) = 3$.

3. Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. $\frac{d}{dx}(3x + 3)^3 = 3(3x + 3)^2 \cdot 3 = 9(3x + 3)^2$

Thus, the derivative of $f(x) = (3x + 3)^3$ is: $f'(x) = 9(3x + 3)^2$

Would you like more details on any step? Here are five related questions to expand your understanding:

1. How does the chain rule apply to other functions, such as trigonometric functions?
2. What is the difference between the chain rule and the product rule in differentiation?
3. Can you explain how to differentiate $f(x) = \sin(2x^2 + 1)$ using the chain rule?
4. How does implicit differentiation relate to the chain rule?
5. What is the geometric interpretation of the chain rule?

Tip: When applying the chain rule, always identify the outer and inner functions clearly to avoid confusion.