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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Question
![**Problem Statement:**
Find the derivative of \( g(x) = \frac{6e^x + 4}{-10x^5 + 5x^3} \).
**Solution:**
To differentiate \( g(x) \), use the quotient rule, which states:
\[
g'(x) = \frac{(v \cdot u') - (u \cdot v')}{v^2}
\]
where \( u = 6e^x + 4 \), \( v = -10x^5 + 5x^3 \), and \( u' \), \( v' \) are the derivatives of \( u \) and \( v \) respectively.
Calculate \( u' \) and \( v' \):
- \( u' = \frac{d}{dx}(6e^x + 4) = 6e^x \)
- \( v' = \frac{d}{dx}(-10x^5 + 5x^3) = -50x^4 + 15x^2 \)
Substitute these into the quotient rule formula:
\[
g'(x) = \frac{((-10x^5 + 5x^3) \cdot 6e^x) - ((6e^x + 4) \cdot (-50x^4 + 15x^2))}{(-10x^5 + 5x^3)^2}
\]
**Final Expression:**
\[
g'(x) = \frac{(-60x^5e^x + 30x^3e^x) - ((-300x^4e^x + 90x^2e^x) + (-200x^4 + 60x^2))}{(-10x^5 + 5x^3)^2}
\]
The expression is organized for further simplification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d09ffa2-b6c2-4e97-915d-74d176d987ab%2Fa0124c7f-8c4a-49c2-9e5a-055fcdf9f3e0%2Fldv34s7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of \( g(x) = \frac{6e^x + 4}{-10x^5 + 5x^3} \).
**Solution:**
To differentiate \( g(x) \), use the quotient rule, which states:
\[
g'(x) = \frac{(v \cdot u') - (u \cdot v')}{v^2}
\]
where \( u = 6e^x + 4 \), \( v = -10x^5 + 5x^3 \), and \( u' \), \( v' \) are the derivatives of \( u \) and \( v \) respectively.
Calculate \( u' \) and \( v' \):
- \( u' = \frac{d}{dx}(6e^x + 4) = 6e^x \)
- \( v' = \frac{d}{dx}(-10x^5 + 5x^3) = -50x^4 + 15x^2 \)
Substitute these into the quotient rule formula:
\[
g'(x) = \frac{((-10x^5 + 5x^3) \cdot 6e^x) - ((6e^x + 4) \cdot (-50x^4 + 15x^2))}{(-10x^5 + 5x^3)^2}
\]
**Final Expression:**
\[
g'(x) = \frac{(-60x^5e^x + 30x^3e^x) - ((-300x^4e^x + 90x^2e^x) + (-200x^4 + 60x^2))}{(-10x^5 + 5x^3)^2}
\]
The expression is organized for further simplification.
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