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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![**Problem Statement:**
Find \(\frac{dy}{dx}\) for the given function.
\[ y = \frac{\csc(x)}{x} \]
**Solution:**
Evaluate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \boxed{\phantom{}} \]
**Explanation:**
This problem involves finding the derivative of a function expressed as a quotient. The function \( y = \frac{\csc(x)}{x} \) involves the cosecant function and the variable \( x \). Solving this requires using the quotient rule, which states that for two functions \( u(x) \) and \( v(x) \), the derivative of their quotient is:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v(u') - u(v')}{v^2} \]
In this case, \( u(x) = \csc(x) \) and \( v(x) = x \). Apply the quotient rule to solve for \(\frac{dy}{dx}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0cd85eb8-5592-4bd0-9b53-cd63c8c5d902%2Fda266311-73ce-4536-8285-a42793980d79%2Fdjk5zo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(\frac{dy}{dx}\) for the given function.
\[ y = \frac{\csc(x)}{x} \]
**Solution:**
Evaluate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \boxed{\phantom{}} \]
**Explanation:**
This problem involves finding the derivative of a function expressed as a quotient. The function \( y = \frac{\csc(x)}{x} \) involves the cosecant function and the variable \( x \). Solving this requires using the quotient rule, which states that for two functions \( u(x) \) and \( v(x) \), the derivative of their quotient is:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v(u') - u(v')}{v^2} \]
In this case, \( u(x) = \csc(x) \) and \( v(x) = x \). Apply the quotient rule to solve for \(\frac{dy}{dx}\).
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