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
Need help
![**Objective:** Find the derivative \( f'(x) \).
**Function:**
\[ f(x) = \frac{2 \tan x}{1 + \cos x} \]
**Instructions:**
1. **Identify the function** given and observe that it is a quotient, which means the quotient rule is likely a helpful method to find the derivative.
2. **Apply the Quotient Rule**: If you have a function of the form \(\frac{u(x)}{v(x)}\), the derivative is given by:
\[
f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
where \( u(x) = 2 \tan x \) and \( v(x) = 1 + \cos x \).
3. **Differentiate \(u(x)\) and \(v(x)\)**:
- \( u'(x) = 2 \sec^2 x \) (derivative of \( \tan x \) is \( \sec^2 x \))
- \( v'(x) = -\sin x \) (derivative of \( \cos x \) is \( -\sin x \))
4. **Substitute back into the Quotient Rule formula** to find \( f'(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf04e4de-240d-4b28-b57c-3fde9eaa273e%2F9206c6c3-3f87-4bdd-b3cf-0554a01d9acd%2F9n9i966_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Objective:** Find the derivative \( f'(x) \).
**Function:**
\[ f(x) = \frac{2 \tan x}{1 + \cos x} \]
**Instructions:**
1. **Identify the function** given and observe that it is a quotient, which means the quotient rule is likely a helpful method to find the derivative.
2. **Apply the Quotient Rule**: If you have a function of the form \(\frac{u(x)}{v(x)}\), the derivative is given by:
\[
f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
where \( u(x) = 2 \tan x \) and \( v(x) = 1 + \cos x \).
3. **Differentiate \(u(x)\) and \(v(x)\)**:
- \( u'(x) = 2 \sec^2 x \) (derivative of \( \tan x \) is \( \sec^2 x \))
- \( v'(x) = -\sin x \) (derivative of \( \cos x \) is \( -\sin x \))
4. **Substitute back into the Quotient Rule formula** to find \( f'(x) \).
Expert Solution
Step 1
To differentiate the given function.
Step by step
Solved in 2 steps with 1 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