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
10
![**Problem 10**
Given the following information:
- \( g(-1) = 2 \)
- \( g'(-1) = 3 \)
Find \( h'(-1) \) for the function:
\[ h(x) = [g(x)]^5 \]
**Explanation:**
To solve for \( h'(-1) \), we use the chain rule. Since \( h(x) = [g(x)]^5 \), the derivative \( h'(x) \) can be found as follows:
1. Differentiate the outside function: \( [g(x)]^5 \) becomes \( 5[g(x)]^4 \).
2. Multiply by the derivative of the inside function \( g(x) \), which is \( g'(x) \).
So, \( h'(x) = 5[g(x)]^4 \cdot g'(x) \).
Substitute \( x = -1 \):
\[ h'(-1) = 5[g(-1)]^4 \cdot g'(-1) \]
Plug in the given values:
\[ h'(-1) = 5[2]^4 \cdot 3 \]
\[ h'(-1) = 5 \cdot 16 \cdot 3 \]
\[ h'(-1) = 240 \]
Therefore, \( h'(-1) = 240 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbc57344-aee0-4c7a-83a5-ce0e8c0e16d3%2Ffb1f5216-8a95-42e2-a0c4-5eeaa4b158c4%2Fzzcwkpi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 10**
Given the following information:
- \( g(-1) = 2 \)
- \( g'(-1) = 3 \)
Find \( h'(-1) \) for the function:
\[ h(x) = [g(x)]^5 \]
**Explanation:**
To solve for \( h'(-1) \), we use the chain rule. Since \( h(x) = [g(x)]^5 \), the derivative \( h'(x) \) can be found as follows:
1. Differentiate the outside function: \( [g(x)]^5 \) becomes \( 5[g(x)]^4 \).
2. Multiply by the derivative of the inside function \( g(x) \), which is \( g'(x) \).
So, \( h'(x) = 5[g(x)]^4 \cdot g'(x) \).
Substitute \( x = -1 \):
\[ h'(-1) = 5[g(-1)]^4 \cdot g'(-1) \]
Plug in the given values:
\[ h'(-1) = 5[2]^4 \cdot 3 \]
\[ h'(-1) = 5 \cdot 16 \cdot 3 \]
\[ h'(-1) = 240 \]
Therefore, \( h'(-1) = 240 \).
Expert Solution
Step 1 Given :
We need to find value of :
given that
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
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