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

Step 1: Identify the Outer and Inner Functions

• The function $f(x)$ is a composition of two functions:
  • Outer function: $u^3$ where $u = 3x + 3$.
  • Inner function: $u = 3x + 3$.

Step 2: Differentiate the Outer Function

• Differentiate $u^3$ 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

• According to the chain rule: $f'(x) = \frac{d}{dx} \left( (3x + 3)^3 \right) = \frac{d}{du}(u^3) \cdot \frac{du}{dx}$ Substitute $\frac{d}{du}(u^3) = 3u^2$ and $\frac{du}{dx} = 3$: $f'(x) = 3(3x + 3)^2 \cdot 3$

Step 5: Simplify the Result

$f'(x) = 9(3x + 3)^2$

So, the derivative is:

$f'(x) = 9(3x + 3)^2$

Would you like more details, or have any questions? Here are five related questions to explore this concept further:

1. How would you differentiate $f(x) = (2x + 5)^4$ using the chain rule?
2. Can you find the second derivative of $f(x) = (3x + 3)^3$?
3. How would you differentiate $f(x) = (x^2 + 1)^5$ using the chain rule?
4. What happens when you differentiate $f(x) = (2x + 1)^{-2}$?
5. Can you differentiate $g(x) = \sin(3x + 2)$ using the chain rule?

Tip:

When using the chain rule, always identify the inner and outer functions clearly to avoid errors in differentiation.