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
Question
Part A and B
![**Chain Rule Application in Calculus**
Given \( y = f(u) \) and \( u = g(x) \), find \(\frac{dy}{dx} = f'(g(x))g'(x)\) for the following functions.
1. \( y = \tan u \)
2. \( u = 11x - 4 \)
---
\[
\frac{dy}{dx} = f'(g(x))g'(x) = \boxed{\quad}
\]
**Explanation**: This exercise involves applying the chain rule to differentiate composite functions. The chain rule states that if a variable \( y \) depends on a variable \( u \), which itself depends on a variable \( x \), then the derivative of \( y \) with respect to \( x \) is the derivative of \( y \) with respect to \( u \) times the derivative of \( u \) with respect to \( x \).
In this case:
- \( y = \tan(u) \) represents the outer function.
- \( u = 11x - 4 \) is the inner function.
Calculate \( f'(u) \) for \( y = \tan(u) \) and \( g'(x) \) for \( u = 11x - 4 \), and then multiply them to find \(\frac{dy}{dx}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a931645-54a1-474a-bda2-4c874edd548a%2F0e3331b8-141c-4fe7-8b82-fb9c5e069874%2Frtpqmzr_processed.png&w=3840&q=75)
Transcribed Image Text:**Chain Rule Application in Calculus**
Given \( y = f(u) \) and \( u = g(x) \), find \(\frac{dy}{dx} = f'(g(x))g'(x)\) for the following functions.
1. \( y = \tan u \)
2. \( u = 11x - 4 \)
---
\[
\frac{dy}{dx} = f'(g(x))g'(x) = \boxed{\quad}
\]
**Explanation**: This exercise involves applying the chain rule to differentiate composite functions. The chain rule states that if a variable \( y \) depends on a variable \( u \), which itself depends on a variable \( x \), then the derivative of \( y \) with respect to \( x \) is the derivative of \( y \) with respect to \( u \) times the derivative of \( u \) with respect to \( x \).
In this case:
- \( y = \tan(u) \) represents the outer function.
- \( u = 11x - 4 \) is the inner function.
Calculate \( f'(u) \) for \( y = \tan(u) \) and \( g'(x) \) for \( u = 11x - 4 \), and then multiply them to find \(\frac{dy}{dx}\).
![**Problem Statement:**
Given \( y = f(u) \) and \( u = g(x) \), find \(\frac{dy}{dx} = f'(g(x))g'(x)\) for the following functions.
**Functions:**
- \( y = \cos u \)
- \( u = 3x + 2 \)
**Solution:**
\[
\frac{dy}{dx} = f'(g(x))g'(x) = \boxed{\phantom{a}}
\]
**Explanation:**
To solve this problem, we need to apply the chain rule. The function \( y = \cos u \) suggests that \( f(u) = \cos u \), so its derivative \( f'(u) = -\sin u \).
Substituting \( u = g(x) = 3x + 2 \) into \( f'(u) \), we get \( f'(g(x)) = -\sin(3x + 2) \).
Next, we find \( g'(x) \). Given that \( u = 3x + 2 \), differentiate with respect to \( x \) to get \( g'(x) = 3 \).
Therefore, \(\frac{dy}{dx} = (-\sin(3x + 2))(3)\).
This results in \(\frac{dy}{dx} = -3\sin(3x + 2)\).
The boxed area on the right is where you would place the final answer for clarity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a931645-54a1-474a-bda2-4c874edd548a%2F0e3331b8-141c-4fe7-8b82-fb9c5e069874%2Fxp760y4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given \( y = f(u) \) and \( u = g(x) \), find \(\frac{dy}{dx} = f'(g(x))g'(x)\) for the following functions.
**Functions:**
- \( y = \cos u \)
- \( u = 3x + 2 \)
**Solution:**
\[
\frac{dy}{dx} = f'(g(x))g'(x) = \boxed{\phantom{a}}
\]
**Explanation:**
To solve this problem, we need to apply the chain rule. The function \( y = \cos u \) suggests that \( f(u) = \cos u \), so its derivative \( f'(u) = -\sin u \).
Substituting \( u = g(x) = 3x + 2 \) into \( f'(u) \), we get \( f'(g(x)) = -\sin(3x + 2) \).
Next, we find \( g'(x) \). Given that \( u = 3x + 2 \), differentiate with respect to \( x \) to get \( g'(x) = 3 \).
Therefore, \(\frac{dy}{dx} = (-\sin(3x + 2))(3)\).
This results in \(\frac{dy}{dx} = -3\sin(3x + 2)\).
The boxed area on the right is where you would place the final answer for clarity.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Similar questions
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning