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:**
Let \( F(3) = 3 \), \( F'(3) = 5 \), \( H(3) = 5 \), and \( H'(3) = 1 \).
**Questions:**
A. If \( G(z) = F(z) \cdot H(z) \), then \( G'(3) = \)
[Input Box]
B. If \( G(w) = \frac{F(w)}{H(w)} \), then \( G'(3) = \)
[Input Box]
**Explanation of Method:**
To solve for the derivatives in questions A and B, the following rules will be useful:
1. **Product Rule**: If \( G(z) = F(z) \cdot H(z) \), then \( G'(z) = F'(z) \cdot H(z) + F(z) \cdot H'(z) \).
2. **Quotient Rule**: If \( G(w) = \frac{F(w)}{H(w)} \), then \( G'(w) = \frac{F'(w) \cdot H(w) - F(w) \cdot H'(w)}{H(w)^2} \).
By applying these rules to the given functions and their derivatives at \( z = 3 \) (or \( w = 3 \)), students can find \( G'(3) \) for both scenarios.
Feel free to enter your solutions in the provided input boxes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F461e6b03-e26d-48fc-aa6c-204c3b542fc6%2Ff52a1460-60ff-46b8-bfa0-0f4b02732bdd%2Fptyb9t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( F(3) = 3 \), \( F'(3) = 5 \), \( H(3) = 5 \), and \( H'(3) = 1 \).
**Questions:**
A. If \( G(z) = F(z) \cdot H(z) \), then \( G'(3) = \)
[Input Box]
B. If \( G(w) = \frac{F(w)}{H(w)} \), then \( G'(3) = \)
[Input Box]
**Explanation of Method:**
To solve for the derivatives in questions A and B, the following rules will be useful:
1. **Product Rule**: If \( G(z) = F(z) \cdot H(z) \), then \( G'(z) = F'(z) \cdot H(z) + F(z) \cdot H'(z) \).
2. **Quotient Rule**: If \( G(w) = \frac{F(w)}{H(w)} \), then \( G'(w) = \frac{F'(w) \cdot H(w) - F(w) \cdot H'(w)}{H(w)^2} \).
By applying these rules to the given functions and their derivatives at \( z = 3 \) (or \( w = 3 \)), students can find \( G'(3) \) for both scenarios.
Feel free to enter your solutions in the provided input boxes.
![# Calculating Derivatives Using the Quotient Rule
Given the function:
\[ y = \frac{2x}{5x + 3} \]
We need to calculate the derivative using the quotient rule.
The quotient rule formula is:
\[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
### Steps to Follow
1. **Identify \( f(x) \) and \( g(x) \)**:
\[ f(x) = \]
\[ \]
\[ g(x) = \]
2. **Calculate the derivatives \( f'(x) \) and \( g'(x) \)**:
\[ f'(x) = \]
\[ \]
\[ g'(x) = \]
3. **Apply the quotient rule**:
\[ \left[ \frac{2x}{5x + 3} \right]' = \]
\[ \]
After completing these steps, you will find the derivative of the given function. Be sure to simplify your result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F461e6b03-e26d-48fc-aa6c-204c3b542fc6%2Ff52a1460-60ff-46b8-bfa0-0f4b02732bdd%2Fe6rgya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Calculating Derivatives Using the Quotient Rule
Given the function:
\[ y = \frac{2x}{5x + 3} \]
We need to calculate the derivative using the quotient rule.
The quotient rule formula is:
\[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
### Steps to Follow
1. **Identify \( f(x) \) and \( g(x) \)**:
\[ f(x) = \]
\[ \]
\[ g(x) = \]
2. **Calculate the derivatives \( f'(x) \) and \( g'(x) \)**:
\[ f'(x) = \]
\[ \]
\[ g'(x) = \]
3. **Apply the quotient rule**:
\[ \left[ \frac{2x}{5x + 3} \right]' = \]
\[ \]
After completing these steps, you will find the derivative of the given function. Be sure to simplify your result.
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