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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DERIVATIVES
![**Problem 7: Derivative Calculation**
Calculate the derivative of the function \( f(x) = 10x^4 - 32x + e^2 \).
**Solution:**
To find the derivative \( f'(x) \), we'll apply the rules of differentiation:
1. **Power Rule**: If \( f(x) = ax^n \), then \( f'(x) = nax^{n-1} \).
2. **Constant Rule**: The derivative of a constant is zero.
Applying these rules to each term in the function:
- For \( 10x^4 \):
- The derivative is \( 4 \times 10x^{4-1} = 40x^3 \).
- For \( -32x \):
- The derivative is \(-32 \).
- For \( e^2 \):
- Since \( e^2 \) is a constant, its derivative is \( 0 \).
**Final Derivative**:
\[ f'(x) = 40x^3 - 32 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dca08d-d7c1-4511-81d7-e3b17171b463%2Fdee26f5d-6de0-4256-893b-40c2b4fb1e13%2Fvg9rxdv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7: Derivative Calculation**
Calculate the derivative of the function \( f(x) = 10x^4 - 32x + e^2 \).
**Solution:**
To find the derivative \( f'(x) \), we'll apply the rules of differentiation:
1. **Power Rule**: If \( f(x) = ax^n \), then \( f'(x) = nax^{n-1} \).
2. **Constant Rule**: The derivative of a constant is zero.
Applying these rules to each term in the function:
- For \( 10x^4 \):
- The derivative is \( 4 \times 10x^{4-1} = 40x^3 \).
- For \( -32x \):
- The derivative is \(-32 \).
- For \( e^2 \):
- Since \( e^2 \) is a constant, its derivative is \( 0 \).
**Final Derivative**:
\[ f'(x) = 40x^3 - 32 \]
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