Differentiate f(x) = (3x + 3)3 using the chain rule
use y as the viariable

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!

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

Learn how to differentiate the function f(x) = (3x + 3)^3 using the chain rule. This step-by-step solution explains key concepts like composite functions and the chain rule, making it ideal for students in Grades 11-12 studying calculus.

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