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
This should clear up. Its asking to prove y'.
![The equation given is:
\[
y = \frac{x}{\sqrt{a^2 - 1}} - \frac{2}{\sqrt{a^2 - 1}} \arctan \left( \frac{\sin(x)}{a + \sqrt{a^2 - 1} \cdot (1 + \cos(x))} \right)
\]
1. **Expression Explanation**:
- The equation is presented in a format involving trigonometric and inverse trigonometric functions.
- The first term is a fraction, with the numerator \( x \) and the denominator \( \sqrt{a^2 - 1} \).
- The second term involves the inverse tangent function \(\arctan\).
- Inside the \(\arctan\), the fraction has \(\sin(x)\) as the numerator.
- The denominator inside the \(\arctan\) is structured as \( a + \sqrt{a^2 - 1} \cdot (1 + \cos(x)) \).
2. **Variables**:
- \( y \) is expressed in terms of \( x \) and \( a \).
- Both \( x \) and \( a \) influence the trigonometric components of the expression.
This equation can be used in educational contexts to explain complex expressions involving trigonometric and inverse trigonometric functions as part of a precalculus or calculus curriculum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76e325e3-3783-401d-9226-dc54b50637b3%2F86a3aa5d-ef59-46c2-a433-a37199d98ede%2Fhzrtqo_processed.png&w=3840&q=75)
Transcribed Image Text:The equation given is:
\[
y = \frac{x}{\sqrt{a^2 - 1}} - \frac{2}{\sqrt{a^2 - 1}} \arctan \left( \frac{\sin(x)}{a + \sqrt{a^2 - 1} \cdot (1 + \cos(x))} \right)
\]
1. **Expression Explanation**:
- The equation is presented in a format involving trigonometric and inverse trigonometric functions.
- The first term is a fraction, with the numerator \( x \) and the denominator \( \sqrt{a^2 - 1} \).
- The second term involves the inverse tangent function \(\arctan\).
- Inside the \(\arctan\), the fraction has \(\sin(x)\) as the numerator.
- The denominator inside the \(\arctan\) is structured as \( a + \sqrt{a^2 - 1} \cdot (1 + \cos(x)) \).
2. **Variables**:
- \( y \) is expressed in terms of \( x \) and \( a \).
- Both \( x \) and \( a \) influence the trigonometric components of the expression.
This equation can be used in educational contexts to explain complex expressions involving trigonometric and inverse trigonometric functions as part of a precalculus or calculus curriculum.
Transcribed Image Text:Transcription:
"Show that \( y' = \frac{1}{a + \cos(x)} \)."
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