Write the function in the form y- f(u) and u- g(x). Then find dyldx as a function of x. y (2x + 7) 4 O y = 4u +7; u = x4. aY = 8x3 %3D dx dy Oy=u, u = 2x + 7; 4(2x+7)3 %3D dx O y=ut, u= 2x + 7; = 8(2x + 7)3 %3D O y = u4; u = 2x + 7; = 2(2x + 7)4 dy dx %3D
Write the function in the form y- f(u) and u- g(x). Then find dyldx as a function of x. y (2x + 7) 4 O y = 4u +7; u = x4. aY = 8x3 %3D dx dy Oy=u, u = 2x + 7; 4(2x+7)3 %3D dx O y=ut, u= 2x + 7; = 8(2x + 7)3 %3D O y = u4; u = 2x + 7; = 2(2x + 7)4 dy dx %3D
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...
Related questions
Topic Video
Question
please explain so i know how to solve
![## Problem Statement:
Rewrite the function in the form \( y = f(u) \) and \( u = g(x) \). Then find \(\frac{dy}{dx}\) as a function of \(x\).
Given function: \( y = (2x + 7)^4 \)
### Options:
1. \( y = 4u + 7; \, u = x^4; \, \frac{dy}{dx} = 8x^3 \)
2. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 4(2x + 7)^3 \)
3. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 8(2x + 7)^3 \)
4. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 2(2x + 7)^4 \)
### Solution:
To solve this problem, we'll use the chain rule of differentiation. The chain rule states that if a variable \(y\) depends on \(u\), which in turn depends on \(x\), then the derivative of \(y\) with respect to \(x\) is:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]
1. Identify the inner function \(u\) and the outer function \(y\).
Let's set:
\[ u = 2x + 7 \]
Thus, the function can be rewritten as:
\[ y = u^4 \]
2. Differentiate \(y\) with respect to \(u\):
\[ \frac{dy}{du} = 4u^3 \]
3. Differentiate \(u\) with respect to \(x\):
\[ \frac{du}{dx} = 2 \]
4. Apply the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4u^3 \cdot 2 = 8u^3
\]
Since \(u = 2x + 7\), substitute \(u\) back into the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1de489e-8c51-4ba0-8d15-cf142b4b6c4d%2F2be68c3d-7ff3-4cf1-b235-415e92218169%2Fruse26.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement:
Rewrite the function in the form \( y = f(u) \) and \( u = g(x) \). Then find \(\frac{dy}{dx}\) as a function of \(x\).
Given function: \( y = (2x + 7)^4 \)
### Options:
1. \( y = 4u + 7; \, u = x^4; \, \frac{dy}{dx} = 8x^3 \)
2. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 4(2x + 7)^3 \)
3. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 8(2x + 7)^3 \)
4. \( y = u^4; \, u = 2x + 7; \, \frac{dy}{dx} = 2(2x + 7)^4 \)
### Solution:
To solve this problem, we'll use the chain rule of differentiation. The chain rule states that if a variable \(y\) depends on \(u\), which in turn depends on \(x\), then the derivative of \(y\) with respect to \(x\) is:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]
1. Identify the inner function \(u\) and the outer function \(y\).
Let's set:
\[ u = 2x + 7 \]
Thus, the function can be rewritten as:
\[ y = u^4 \]
2. Differentiate \(y\) with respect to \(u\):
\[ \frac{dy}{du} = 4u^3 \]
3. Differentiate \(u\) with respect to \(x\):
\[ \frac{du}{dx} = 2 \]
4. Apply the chain rule:
\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4u^3 \cdot 2 = 8u^3
\]
Since \(u = 2x + 7\), substitute \(u\) back into the
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781285741550/9781285741550_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
![Thomas' Calculus (14th Edition)](https://www.bartleby.com/isbn_cover_images/9780134438986/9780134438986_smallCoverImage.gif)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
![Calculus: Early Transcendentals (3rd Edition)](https://www.bartleby.com/isbn_cover_images/9780134763644/9780134763644_smallCoverImage.gif)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
![Calculus: Early Transcendentals](https://www.bartleby.com/isbn_cover_images/9781319050740/9781319050740_smallCoverImage.gif)
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
![Precalculus](https://www.bartleby.com/isbn_cover_images/9780135189405/9780135189405_smallCoverImage.gif)
![Calculus: Early Transcendental Functions](https://www.bartleby.com/isbn_cover_images/9781337552516/9781337552516_smallCoverImage.gif)
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning