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 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.

Master the chain rule with this step-by-step guide to differentiate f(x) = (3x + 3)^3. Learn how to break down composite functions and apply the power rule. Ideal for high school and college calculus students.

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