Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
1. Consider the function \( f(x) = -8x^7 + 6x^3 - 25x - 21 \).
Find the derivative, \( f'(x) \).
### Solution Approach
To find the derivative \( f'(x) \) of the given polynomial function, apply the power rule. The power rule states that the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \).
### Step-by-Step Calculation
1. **Differentiate each term**:
- For \(-8x^7\), the derivative is \(-56x^6\).
- For \(6x^3\), the derivative is \(18x^2\).
- For \(-25x\), the derivative is \(-25\).
- The derivative of a constant, \(-21\), is \(0\).
2. **Combine the derivatives**:
\[
f'(x) = -56x^6 + 18x^2 - 25
\]
### Conclusion
The derivative of the function \( f(x) = -8x^7 + 6x^3 - 25x - 21 \) is given by:
\[
f'(x) = -56x^6 + 18x^2 - 25
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f39937e-08d5-4dd9-9b09-d4f7e1a5c538%2F828dc578-438f-4eac-aa7b-de433a673f0b%2Fqqmb1hd_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
1. Consider the function \( f(x) = -8x^7 + 6x^3 - 25x - 21 \).
Find the derivative, \( f'(x) \).
### Solution Approach
To find the derivative \( f'(x) \) of the given polynomial function, apply the power rule. The power rule states that the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \).
### Step-by-Step Calculation
1. **Differentiate each term**:
- For \(-8x^7\), the derivative is \(-56x^6\).
- For \(6x^3\), the derivative is \(18x^2\).
- For \(-25x\), the derivative is \(-25\).
- The derivative of a constant, \(-21\), is \(0\).
2. **Combine the derivatives**:
\[
f'(x) = -56x^6 + 18x^2 - 25
\]
### Conclusion
The derivative of the function \( f(x) = -8x^7 + 6x^3 - 25x - 21 \) is given by:
\[
f'(x) = -56x^6 + 18x^2 - 25
\]
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