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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![### Calculating Derivatives using the Limit Definition
To find the derivative of a function \( f(x) \) at any point \( x \), we use the limit definition of the derivative. The formula is given by:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h} \]
Let’s apply this method to a specific function:
\[ s(x) = 6x + 7 \]
### Steps to Find the Derivative using the Limit Definition:
1. **Identify the function \( f(x) \)**: In this case, \( f(x) = 6x + 7 \).
2. **Set up the difference quotient**: Substitute \( f(x) \) into the limit definition formula.
3. **Simplify the expression inside the limit**.
4. **Evaluate the limit as \( h \) approaches 0**.
By following these steps, you will be able to find the derivative of the function \( s(x) \) at any point \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2da21713-669c-4fb3-a696-2121fe318bc8%2F99a87db6-a6df-4e4b-aea6-28fe0b6f74d6%2F95hrc5b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating Derivatives using the Limit Definition
To find the derivative of a function \( f(x) \) at any point \( x \), we use the limit definition of the derivative. The formula is given by:
\[ f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h} \]
Let’s apply this method to a specific function:
\[ s(x) = 6x + 7 \]
### Steps to Find the Derivative using the Limit Definition:
1. **Identify the function \( f(x) \)**: In this case, \( f(x) = 6x + 7 \).
2. **Set up the difference quotient**: Substitute \( f(x) \) into the limit definition formula.
3. **Simplify the expression inside the limit**.
4. **Evaluate the limit as \( h \) approaches 0**.
By following these steps, you will be able to find the derivative of the function \( s(x) \) at any point \( x \).
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