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:**
Consider the function \( y = e^{4 - 3x} \). Find \( y' \).
**Solution:**
To find the derivative \( y' \) of the function \( y = e^{4 - 3x} \) with respect to \( x \), apply the chain rule.
The function is in the form \( y = e^{u} \) where \( u = 4 - 3x \).
The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \).
Now, find \( \frac{du}{dx} \), where \( u = 4 - 3x \). The derivative of \( u \) is \( -3 \).
By the chain rule:
\[ y' = \frac{d}{dx}(e^{u}) = e^{u} \cdot \frac{du}{dx} \]
So,
\[ y' = e^{4 - 3x} \cdot (-3) \]
Thus, the derivative is:
\[ y' = -3e^{4 - 3x} \]
**Answer:**
\[ y' = -3e^{4 - 3x} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb775eebb-fd18-4d87-bf30-4fb41f9888f9%2Fced30ab3-6cf4-4bdf-9459-c86d8b010108%2Figyjtfd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider the function \( y = e^{4 - 3x} \). Find \( y' \).
**Solution:**
To find the derivative \( y' \) of the function \( y = e^{4 - 3x} \) with respect to \( x \), apply the chain rule.
The function is in the form \( y = e^{u} \) where \( u = 4 - 3x \).
The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \).
Now, find \( \frac{du}{dx} \), where \( u = 4 - 3x \). The derivative of \( u \) is \( -3 \).
By the chain rule:
\[ y' = \frac{d}{dx}(e^{u}) = e^{u} \cdot \frac{du}{dx} \]
So,
\[ y' = e^{4 - 3x} \cdot (-3) \]
Thus, the derivative is:
\[ y' = -3e^{4 - 3x} \]
**Answer:**
\[ y' = -3e^{4 - 3x} \]
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