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

Step-by-step solution:

1. Identify the outer and inner functions:

   • Outer function: $u^3$, where $u = 3x + 3$.
   • Inner function: $u = 3x + 3$.

2. Differentiate the outer function:

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

3. Differentiate the inner function:

   • The derivative of $3x + 3$ with respect to $x$ is $3$.

4. Apply the chain rule: Using the chain rule: $\frac{d}{dx} \left( (3x + 3)^3 \right) = 3(3x + 3)^2 \cdot 3$

5. Simplify: $\frac{d}{dx} \left( (3x + 3)^3 \right) = 9(3x + 3)^2$

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

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

Would you like further details or clarification on any step?

5 Related Questions:

1. How would you differentiate $f(x) = (5x - 4)^3$ using the chain rule?
2. Can you apply the chain rule to differentiate $f(x) = \sin(2x^2 + 1)$?
3. How does the chain rule work with higher powers, like $f(x) = (2x + 1)^5$?
4. What is the general formula for differentiating $f(x) = (ax + b)^n$ using the chain rule?
5. How would you handle the chain rule for trigonometric functions, e.g., $f(x) = \cos(3x^2)$?

Tip:

Always break down the differentiation process by clearly identifying the outer and inner functions before applying the chain rule to simplify the steps.