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
**Calculate the derivative.**
\[ y = \frac{x^4 - 4}{x^2 - 5} \]
*(Give an exact answer. Use symbolic notation and fractions where needed.)*
\[ \left. \frac{dy}{dx} \right|_{x=8} = \boxed{} \]
---
This question requires you to find the derivative of the function \( y \) with respect to \( x \) and then evaluate it at \( x = 8 \). You should apply the quotient rule for differentiation, which states that if you have a function in the form \( \frac{u}{v} \), the derivative is given by:
\[ \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]
where \( u = x^4 - 4 \) and \( v = x^2 - 5 \). Once you have the derivative, substitute \( x = 8 \) to find the required value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a49795d-3f60-4be9-80d5-b761ab7048e9%2F21c8b3f6-0f1a-4825-bb7e-f32da3ecb478%2Fzvavr8b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Calculate the derivative.**
\[ y = \frac{x^4 - 4}{x^2 - 5} \]
*(Give an exact answer. Use symbolic notation and fractions where needed.)*
\[ \left. \frac{dy}{dx} \right|_{x=8} = \boxed{} \]
---
This question requires you to find the derivative of the function \( y \) with respect to \( x \) and then evaluate it at \( x = 8 \). You should apply the quotient rule for differentiation, which states that if you have a function in the form \( \frac{u}{v} \), the derivative is given by:
\[ \frac{d}{dx} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]
where \( u = x^4 - 4 \) and \( v = x^2 - 5 \). Once you have the derivative, substitute \( x = 8 \) to find the required value.
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