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
![### Topic: Calculus - Derivatives
**Objective:** Find the derivative of \( y \) with respect to \( s \).
**Problem Statement:**
Given the function:
\[ y = \sec^{-1}(7s^5 + 4) \]
Find \( \frac{dy}{ds} \).
---
**Solution:**
To find the derivative of \( y = \sec^{-1}(7s^5 + 4) \) with respect to \( s \), we apply the chain rule and the derivative formula for the inverse secant function.
The derivative is given by:
\[
\frac{dy}{ds} = \frac{35s^4}{\sqrt{(7s^5 + 4)^2 - 1} \cdot (7s^5 + 4)}
\]
**Explanation of the Derivative:**
1. **Numerator**:
\[ 35s^4 \]
This is obtained by differentiating the inner function \( 7s^5 + 4 \), which results from applying the chain rule.
2. **Denominator**:
- The term \(\sqrt{(7s^5 + 4)^2 - 1}\) represents the standard part of the derivative of the inverse secant function.
- \((7s^5 + 4)\) is the original argument of the inverse secant function, which appears as a factor in the denominator.
**Note**: The derivative for inverse secant functions involves both the inner function and its derivative, adjusted for the arcsecant identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d4ac937-eddb-4217-be88-b7c3e9181e82%2F3a82a89a-f2ff-4a80-b721-d0c4fa54767b%2Fl74rlwq_processed.png&w=3840&q=75)
Transcribed Image Text:### Topic: Calculus - Derivatives
**Objective:** Find the derivative of \( y \) with respect to \( s \).
**Problem Statement:**
Given the function:
\[ y = \sec^{-1}(7s^5 + 4) \]
Find \( \frac{dy}{ds} \).
---
**Solution:**
To find the derivative of \( y = \sec^{-1}(7s^5 + 4) \) with respect to \( s \), we apply the chain rule and the derivative formula for the inverse secant function.
The derivative is given by:
\[
\frac{dy}{ds} = \frac{35s^4}{\sqrt{(7s^5 + 4)^2 - 1} \cdot (7s^5 + 4)}
\]
**Explanation of the Derivative:**
1. **Numerator**:
\[ 35s^4 \]
This is obtained by differentiating the inner function \( 7s^5 + 4 \), which results from applying the chain rule.
2. **Denominator**:
- The term \(\sqrt{(7s^5 + 4)^2 - 1}\) represents the standard part of the derivative of the inverse secant function.
- \((7s^5 + 4)\) is the original argument of the inverse secant function, which appears as a factor in the denominator.
**Note**: The derivative for inverse secant functions involves both the inner function and its derivative, adjusted for the arcsecant identity.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
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