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Find the derivative of the function g(x) g'(x) = = ex 4 - 3x

Find the derivative of the function g(x) g'(x) = = ex 4 - 3x

Calculus: Early Transcendentals
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 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: \[ g'(x) = \frac{e^x(7 - 3x)}{(4 - 3x)^2} \]
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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