Use the four-step definition of the derivative to find f'(x) if f(x) = 5x 4. f(x + h) = f(x + h) – f(x) =| f(x + h) – f(x) h f(x + h) – f(x) Find f'(x) by determining lim h0 h
Use the four-step definition of the derivative to find f'(x) if f(x) = 5x 4. f(x + h) = f(x + h) – f(x) =| f(x + h) – f(x) h f(x + h) – f(x) Find f'(x) by determining lim h0 h
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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![### Understanding the Four-Step Definition of the Derivative
To find the derivative \( f'(x) \) using the four-step definition of the derivative, follow these steps for the function \( f(x) = 5x - 4 \):
1. **Find \( f(x + h) \)**
\[
f(x + h) = \rule{150pt}{0.4pt}
\]
2. **Compute \( f(x + h) - f(x) \)**
\[
f(x + h) - f(x) = \rule{150pt}{0.4pt}
\]
3. **Divide by \( h \)**
\[
\frac{f(x + h) - f(x)}{h} = \rule{150pt}{0.4pt}
\]
4. **Take the limit as \( h \) approaches 0**
\[
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \rule{150pt}{0.4pt}
\]
### Explanation of Each Step
- **Step 1**: Substitute \( x + h \) into the function \( f(x) \).
- **Step 2**: Subtract the original function \( f(x) \) from the result of Step 1.
- **Step 3**: Divide the difference obtained in Step 2 by \( h \).
- **Step 4**: Determine the limit of the expression as \( h \) approaches zero to find the derivative.
By following these steps, you can determine the derivative of any function using its definition. Let’s apply this process to the given function \( f(x) = 5x - 4 \) step-by-step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a9572e1-cd0e-48a9-8ed8-4d34eb3e96b9%2F7180823d-58b4-4fd8-98d7-fda312f24df5%2Foemsb8m_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Four-Step Definition of the Derivative
To find the derivative \( f'(x) \) using the four-step definition of the derivative, follow these steps for the function \( f(x) = 5x - 4 \):
1. **Find \( f(x + h) \)**
\[
f(x + h) = \rule{150pt}{0.4pt}
\]
2. **Compute \( f(x + h) - f(x) \)**
\[
f(x + h) - f(x) = \rule{150pt}{0.4pt}
\]
3. **Divide by \( h \)**
\[
\frac{f(x + h) - f(x)}{h} = \rule{150pt}{0.4pt}
\]
4. **Take the limit as \( h \) approaches 0**
\[
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \rule{150pt}{0.4pt}
\]
### Explanation of Each Step
- **Step 1**: Substitute \( x + h \) into the function \( f(x) \).
- **Step 2**: Subtract the original function \( f(x) \) from the result of Step 1.
- **Step 3**: Divide the difference obtained in Step 2 by \( h \).
- **Step 4**: Determine the limit of the expression as \( h \) approaches zero to find the derivative.
By following these steps, you can determine the derivative of any function using its definition. Let’s apply this process to the given function \( f(x) = 5x - 4 \) step-by-step.
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