### Calculus Problem: Finding Derivatives Using the Definition #### Problem Statement: Given \( f(x) = \frac{3}{x - 2} \), find \( f'(-4) \) using the definition of a derivative. #### Solution: Provide your answer below: \[ f'(-4) = \boxed{\phantom{0}} \] --- **Explanation:** To find the derivative of \( f(x) \) at \( x = -4 \) using the definition of a derivative, we use the following formula: \[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \] In this case, \( a = -4 \), so we need to substitute \( f(x) = \frac{3}{x - 2} \) into the formula: \[ f'(-4) = \lim_{h \to 0} \frac{\frac{3}{(-4 + h) - 2} - \frac{3}{-4 - 2}}{h} \] This expression can then be simplified and evaluated to find the desired derivative value at \( x = -4 \). #### Key Points to Remember: - The definition of the derivative is crucial in understanding how rates of change are calculated. - Practicing with the definition helps reinforce concepts of limits and continuity. Feel free to solve this derivative step-by-step to enhance your understanding! --- #### Attribution: All problem statements and solutions are created for educational purposes.
### Calculus Problem: Finding Derivatives Using the Definition #### Problem Statement: Given \( f(x) = \frac{3}{x - 2} \), find \( f'(-4) \) using the definition of a derivative. #### Solution: Provide your answer below: \[ f'(-4) = \boxed{\phantom{0}} \] --- **Explanation:** To find the derivative of \( f(x) \) at \( x = -4 \) using the definition of a derivative, we use the following formula: \[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \] In this case, \( a = -4 \), so we need to substitute \( f(x) = \frac{3}{x - 2} \) into the formula: \[ f'(-4) = \lim_{h \to 0} \frac{\frac{3}{(-4 + h) - 2} - \frac{3}{-4 - 2}}{h} \] This expression can then be simplified and evaluated to find the desired derivative value at \( x = -4 \). #### Key Points to Remember: - The definition of the derivative is crucial in understanding how rates of change are calculated. - Practicing with the definition helps reinforce concepts of limits and continuity. Feel free to solve this derivative step-by-step to enhance your understanding! --- #### Attribution: All problem statements and solutions are created for educational purposes.
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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![### Calculus Problem: Finding Derivatives Using the Definition
#### Problem Statement:
Given \( f(x) = \frac{3}{x - 2} \), find \( f'(-4) \) using the definition of a derivative.
#### Solution:
Provide your answer below:
\[ f'(-4) = \boxed{\phantom{0}} \]
---
**Explanation:**
To find the derivative of \( f(x) \) at \( x = -4 \) using the definition of a derivative, we use the following formula:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
In this case, \( a = -4 \), so we need to substitute \( f(x) = \frac{3}{x - 2} \) into the formula:
\[ f'(-4) = \lim_{h \to 0} \frac{\frac{3}{(-4 + h) - 2} - \frac{3}{-4 - 2}}{h} \]
This expression can then be simplified and evaluated to find the desired derivative value at \( x = -4 \).
#### Key Points to Remember:
- The definition of the derivative is crucial in understanding how rates of change are calculated.
- Practicing with the definition helps reinforce concepts of limits and continuity.
Feel free to solve this derivative step-by-step to enhance your understanding!
---
#### Attribution:
All problem statements and solutions are created for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bc92d6f-8141-453b-b1aa-15dddf73f72d%2F0d24a600-d4ab-40a0-8c14-0227512d3871%2Fje05oq9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem: Finding Derivatives Using the Definition
#### Problem Statement:
Given \( f(x) = \frac{3}{x - 2} \), find \( f'(-4) \) using the definition of a derivative.
#### Solution:
Provide your answer below:
\[ f'(-4) = \boxed{\phantom{0}} \]
---
**Explanation:**
To find the derivative of \( f(x) \) at \( x = -4 \) using the definition of a derivative, we use the following formula:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
In this case, \( a = -4 \), so we need to substitute \( f(x) = \frac{3}{x - 2} \) into the formula:
\[ f'(-4) = \lim_{h \to 0} \frac{\frac{3}{(-4 + h) - 2} - \frac{3}{-4 - 2}}{h} \]
This expression can then be simplified and evaluated to find the desired derivative value at \( x = -4 \).
#### Key Points to Remember:
- The definition of the derivative is crucial in understanding how rates of change are calculated.
- Practicing with the definition helps reinforce concepts of limits and continuity.
Feel free to solve this derivative step-by-step to enhance your understanding!
---
#### Attribution:
All problem statements and solutions are created for educational purposes.
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