Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Problem Statement
Find \( f'(x) \).
Given:
\[ f(x) = 9x^3 (x^2 - 9) \]
### Solution
To find the derivative \( f'(x) \), apply the product rule and chain rule.
#### Steps:
1. **Identify the functions**:
- Let \( u(x) = 9x^3 \)
- Let \( v(x) = (x^2 - 9) \)
2. **Find the derivatives**:
- \( u'(x) = \frac{d}{dx}(9x^3) = 27x^2 \)
- \( v'(x) = \frac{d}{dx}(x^2 - 9) = 2x \)
3. **Apply the product rule**:
- The product rule states that if \( f(x) = u(x) \cdot v(x) \), then:
\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
4. **Calculate \( f'(x) \)**:
\[
f'(x) = 27x^2 \cdot (x^2 - 9) + 9x^3 \cdot 2x
\]
\[
f'(x) = 27x^2 \cdot x^2 - 27x^2 \cdot 9 + 18x^4
\]
\[
f'(x) = 27x^4 - 243x^2 + 18x^4
\]
\[
f'(x) = 45x^4 - 243x^2
\]
Therefore:
\[ f'(x) = 45x^4 - 243x^2 \]
The derivative is found and the blue box signifies where this answer should be entered.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e03da4d-971c-4230-a070-21ed988e0d0e%2F53628a02-319f-48e1-989e-e4c0c89d8efe%2Fi2c59g8_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find \( f'(x) \).
Given:
\[ f(x) = 9x^3 (x^2 - 9) \]
### Solution
To find the derivative \( f'(x) \), apply the product rule and chain rule.
#### Steps:
1. **Identify the functions**:
- Let \( u(x) = 9x^3 \)
- Let \( v(x) = (x^2 - 9) \)
2. **Find the derivatives**:
- \( u'(x) = \frac{d}{dx}(9x^3) = 27x^2 \)
- \( v'(x) = \frac{d}{dx}(x^2 - 9) = 2x \)
3. **Apply the product rule**:
- The product rule states that if \( f(x) = u(x) \cdot v(x) \), then:
\[ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \]
4. **Calculate \( f'(x) \)**:
\[
f'(x) = 27x^2 \cdot (x^2 - 9) + 9x^3 \cdot 2x
\]
\[
f'(x) = 27x^2 \cdot x^2 - 27x^2 \cdot 9 + 18x^4
\]
\[
f'(x) = 27x^4 - 243x^2 + 18x^4
\]
\[
f'(x) = 45x^4 - 243x^2
\]
Therefore:
\[ f'(x) = 45x^4 - 243x^2 \]
The derivative is found and the blue box signifies where this answer should be entered.
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