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

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

### Step 1: Identify the outer and inner functions.
- The **outer function** is \( g(u) = u^3 \), where \( u = 3x + 3 \).
- The **inner function** is \( h(x) = 3x + 3 \).

### Step 2: Differentiate the outer function.
The derivative of \( g(u) = u^3 \) with respect to \( u \) is:
\[
g'(u) = 3u^2
\]

### Step 3: Differentiate the inner function.
The derivative of \( h(x) = 3x + 3 \) with respect to \( x \) is:
\[
h'(x) = 3
\]

### Step 4: Apply the chain rule.
According to the chain rule, the derivative of \( f(x) = (3x + 3)^3 \) is given by:
\[
f'(x) = g'(h(x)) \cdot h'(x)
\]
Substitute the derivatives from Steps 2 and 3:
\[
f'(x) = 3(3x + 3)^2 \cdot 3
\]

### Step 5: Simplify the expression.
\[
f'(x) = 9(3x + 3)^2
\]

So, the derivative of \( f(x) = (3x + 3)^3 \) is:
\[
f'(x) = 9(3x + 3)^2
\]

Would you like more details on any step?

### Relative questions:
1. How does the chain rule work for functions with more than two layers?
2. What happens when the inner function is a trigonometric function?
3. Can you differentiate \( f(x) = (5x^2 + 4)^4 \) using the chain rule?
4. How does the power rule relate to the chain rule?
5. What is the product rule and how does it differ from the chain rule?

### Tip:
When differentiating composite functions, always start by identifying the inner and outer functions clearly to avoid mistakes.

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

Learn how to differentiate the composite function f(x) = (3x + 3)^3 using the chain rule. This step-by-step guide walks through identifying inner and outer functions and applying differentiation rules. Suitable for students in advanced calculus, Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation