https://file.aiquickdraw.com/mathbot/file/1729898780131tpk2vdqr.jpg

To differentiate the function \( f(x) = x^3 (x + 6) \), we can proceed by using the **product rule**. Recall that the product rule states:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]

In this case:
- \( f(x) = x^3 \) and \( g(x) = x + 6 \).
- We need to find \( f'(x) \) by differentiating each part and then applying the product rule.

### Step 1: Differentiate \( f(x) = x^3 \)
\[
f'(x) = 3x^2
\]

### Step 2: Differentiate \( g(x) = x + 6 \)
\[
g'(x) = 1
\]

### Step 3: Apply the Product Rule
Now substitute into the product rule formula:
\[
f'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\]
\[
f'(x) = (3x^2)(x + 6) + (x^3)(1)
\]

### Step 4: Simplify
Expand and simplify the expression:
\[
f'(x) = 3x^3 + 18x^2 + x^3
\]
Combine like terms:
\[
f'(x) = 4x^3 + 18x^2
\]

### Final Answer
\[
f'(x) = 4x^3 + 18x^2
\]

Would you like a more detailed explanation, or do you have any questions?

---

Here are some related questions to further your understanding:
1. How do you differentiate functions involving products of multiple terms?
2. What is the power rule for differentiation, and how does it apply here?
3. How do you apply the product rule to functions with more than two factors?
4. What other differentiation rules can be combined with the product rule?
5. How does differentiation work for functions with higher powers, like \( x^n \)?

**Tip:** Always simplify your expression fully after applying differentiation rules to make further steps or analysis easier.

Differentiate the function f(x) = x^3(x + 6).

Find the derivative of f(x) = x^3(x + 6) by applying the product rule in calculus. This step-by-step solution explains how to differentiate each part and combine terms. Suitable for advanced high school or college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation