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

Step 1: Set the inner function

Let: $u = 3x + 3$ So now the function becomes: $f(x) = u^3$

Step 2: Differentiate the outer function

Now differentiate $f(x) = u^3$ with respect to $u$: $\frac{d}{du}(u^3) = 3u^2$

Step 3: Differentiate the inner function

Differentiate $u = 3x + 3$ with respect to $x$: $\frac{du}{dx} = 3$

Step 4: Apply the chain rule

Using the chain rule, $\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx}$. Substituting the values: $\frac{df}{dx} = 3u^2 \cdot 3 = 9u^2$ Now substitute back $u = 3x + 3$: $\frac{df}{dx} = 9(3x + 3)^2$

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

Would you like further clarification or a breakdown of any of the steps?

Here are five related questions to consider:

1. How does the chain rule apply when differentiating composite functions?
2. Can you differentiate $f(x) = (5x - 2)^4$ using the same technique?
3. What is the derivative of $f(x) = (3x + 3)^2$ using the chain rule?
4. How would the result change if the inner function was $u = x^2 + 1$ instead?
5. How does the chain rule compare to the product and quotient rules?

Tip: Always differentiate the outer function first and then multiply by the derivative of the inner function when using the chain rule!