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
![**Differentiate the function:**
\[ y = \frac{\sin x}{2 - \tan x} \]
To tackle this problem, we need to find the derivative of the function \( y \) with respect to \( x \). This requires applying the quotient rule of differentiation, which is generally expressed as:
\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \]
Here, \( u = \sin x \) and \( v = 2 - \tan x \).
1. **Differentiate \( u = \sin x \)**:
\[ u' = \cos x \]
2. **Differentiate \( v = 2 - \tan x \)**:
\[ v' = -\sec^2 x \]
3. **Apply the quotient rule**:
\[ y' = \frac{(\cos x)(2 - \tan x) - (\sin x)(-\sec^2 x)}{(2 - \tan x)^2} \]
**Simplifying further, the derivative is:**
\[ y' = \frac{\cos x (2 - \tan x) + \sin x \sec^2 x}{(2 - \tan x)^2} \]
This is the differentiated function of the given expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff80585f9-eac0-4aa8-81f9-ee704a35044a%2Fec21d915-2bac-4c11-a772-6cd3098711ae%2F3lk78pa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differentiate the function:**
\[ y = \frac{\sin x}{2 - \tan x} \]
To tackle this problem, we need to find the derivative of the function \( y \) with respect to \( x \). This requires applying the quotient rule of differentiation, which is generally expressed as:
\[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \]
Here, \( u = \sin x \) and \( v = 2 - \tan x \).
1. **Differentiate \( u = \sin x \)**:
\[ u' = \cos x \]
2. **Differentiate \( v = 2 - \tan x \)**:
\[ v' = -\sec^2 x \]
3. **Apply the quotient rule**:
\[ y' = \frac{(\cos x)(2 - \tan x) - (\sin x)(-\sec^2 x)}{(2 - \tan x)^2} \]
**Simplifying further, the derivative is:**
\[ y' = \frac{\cos x (2 - \tan x) + \sin x \sec^2 x}{(2 - \tan x)^2} \]
This is the differentiated function of the given expression.
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