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...
Question
![**Problem Statement:**
Find the derivative of the function \( g(x) = \frac{e^x}{4 - 3x} \).
**Solution:**
We need to calculate \( g'(x) \), which requires the use of the quotient rule. The quotient rule states that if you have a function \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) \) is given by:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
In this case:
- \( u(x) = e^x \) and \( u'(x) = e^x \)
- \( v(x) = 4 - 3x \) and \( v'(x) = -3 \)
Plug these into the quotient rule:
\[
g'(x) = \frac{e^x(4 - 3x) - e^x(-3)}{(4 - 3x)^2}
\]
This simplifies to:
\[
g'(x) = \frac{e^x(4 - 3x + 3)}{(4 - 3x)^2}
\]
Finally, the answer is:
\[
g'(x) = \frac{e^x(7 - 3x)}{(4 - 3x)^2}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70295faf-d188-48da-8b46-ee8193f4f21a%2Fd5731c18-8797-41b8-91bb-eacaeb79eab0%2F0d2e4tb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the function \( g(x) = \frac{e^x}{4 - 3x} \).
**Solution:**
We need to calculate \( g'(x) \), which requires the use of the quotient rule. The quotient rule states that if you have a function \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) \) is given by:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
In this case:
- \( u(x) = e^x \) and \( u'(x) = e^x \)
- \( v(x) = 4 - 3x \) and \( v'(x) = -3 \)
Plug these into the quotient rule:
\[
g'(x) = \frac{e^x(4 - 3x) - e^x(-3)}{(4 - 3x)^2}
\]
This simplifies to:
\[
g'(x) = \frac{e^x(4 - 3x + 3)}{(4 - 3x)^2}
\]
Finally, the answer is:
\[
g'(x) = \frac{e^x(7 - 3x)}{(4 - 3x)^2}
\]
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