2. Quotient Rule. Fill in the Quotient Rule information for T = B = 4- the function, y= Then use the Quotient Rule 7x-5 dy -. Do not simplify your answers. dx formula to write T' = B' =
2. Quotient Rule. Fill in the Quotient Rule information for T = B = 4- the function, y= Then use the Quotient Rule 7x-5 dy -. Do not simplify your answers. dx formula to write T' = B' =
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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![### Topic: Application of the Quotient Rule in Differential Calculus
#### Problem Statement:
Consider the function:
\[ y = \frac{4 - \sqrt[3]{x^2}}{7x^2 - 5} \]
Using the quotient rule, find the derivative \( \frac{dy}{dx} \) without simplifying your answers.
#### Background:
The quotient rule is used in calculus to find the derivative of the ratio of two differentiable functions. If a function \( f(x) \) can be expressed as \( \frac{T(x)}{B(x)} \), then its derivative is given by:
\[ \frac{dy}{dx} = \frac{T'(x)B(x) - T(x)B'(x)}{[B(x)]^2} \]
#### Steps to Follow:
1. Identify the numerator function \( T(x) \) and the denominator function \( B(x) \).
2. Find the derivatives \( T'(x) \) and \( B'(x) \).
3. Apply the quotient rule.
#### Given Function:
\[ y = \frac{4 - \sqrt[3]{x^2}}{7x^2 - 5} \]
1. **Numerator Function (T)**:
\[ T = 4 - \sqrt[3]{x^2} \]
2. **Denominator Function (B)**:
\[ B = 7x^2 - 5 \]
3. **Derivative of the Numerator (T')**:
\[ T' = \frac{d}{dx}(4 - \sqrt[3]{x^2}) \]
4. **Derivative of the Denominator (B')**:
\[ B' = \frac{d}{dx}(7x^2 - 5) \]
#### Box for Substitution:
| **T =** | 4 - \(\sqrt[3]{x^2}\) |
| --------- | ---------------------- |
| **B =** | \(7x^2 - 5\) |
| **T' =** | |
| **B' =** | |
To apply the quotient rule, substitute these expressions into the quotient rule formula and then find the derivatives \( T'(x) \) and \( B'(x) \).
- **Finding T'**:
\[ T' = -\frac{2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8671037f-1fac-4205-998f-acf337f22c66%2Ff0592345-2e61-4618-89b9-e99ad5c6c8b2%2F17meun_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Application of the Quotient Rule in Differential Calculus
#### Problem Statement:
Consider the function:
\[ y = \frac{4 - \sqrt[3]{x^2}}{7x^2 - 5} \]
Using the quotient rule, find the derivative \( \frac{dy}{dx} \) without simplifying your answers.
#### Background:
The quotient rule is used in calculus to find the derivative of the ratio of two differentiable functions. If a function \( f(x) \) can be expressed as \( \frac{T(x)}{B(x)} \), then its derivative is given by:
\[ \frac{dy}{dx} = \frac{T'(x)B(x) - T(x)B'(x)}{[B(x)]^2} \]
#### Steps to Follow:
1. Identify the numerator function \( T(x) \) and the denominator function \( B(x) \).
2. Find the derivatives \( T'(x) \) and \( B'(x) \).
3. Apply the quotient rule.
#### Given Function:
\[ y = \frac{4 - \sqrt[3]{x^2}}{7x^2 - 5} \]
1. **Numerator Function (T)**:
\[ T = 4 - \sqrt[3]{x^2} \]
2. **Denominator Function (B)**:
\[ B = 7x^2 - 5 \]
3. **Derivative of the Numerator (T')**:
\[ T' = \frac{d}{dx}(4 - \sqrt[3]{x^2}) \]
4. **Derivative of the Denominator (B')**:
\[ B' = \frac{d}{dx}(7x^2 - 5) \]
#### Box for Substitution:
| **T =** | 4 - \(\sqrt[3]{x^2}\) |
| --------- | ---------------------- |
| **B =** | \(7x^2 - 5\) |
| **T' =** | |
| **B' =** | |
To apply the quotient rule, substitute these expressions into the quotient rule formula and then find the derivatives \( T'(x) \) and \( B'(x) \).
- **Finding T'**:
\[ T' = -\frac{2
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