Differentiate f(x) = (3x + 3)3 using the chain rule

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.

Differentiate f(x) = (3x + 3)^3 using the chain rule

Learn how to differentiate f(x) = (3x + 3)^3 using the chain rule. This detailed step-by-step solution explains key calculus concepts like the chain rule and power rule, providing the final result f'(x) = 9(3x + 3)^2. Suitable for students in Grades 11-12 or introductory college-level calculus.

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