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
![Let \( f(x) = (5 \sin x + 8 \cos x) \tan^{-1} x \)
\[ f'(x) = \]
The task is to differentiate the function \( f(x) = (5 \sin x + 8 \cos x) \tan^{-1} x \). We will apply the product rule, which states that for two functions \( u(x) \) and \( v(x) \), the derivative is given by:
\[ (uv)' = u'v + uv' \]
Here, \( u(x) = 5 \sin x + 8 \cos x \) and \( v(x) = \tan^{-1} x \).
1. Differentiate \( u(x) \):
- \( u'(x) = 5 \cos x - 8 \sin x \)
2. Differentiate \( v(x) \):
- \( v'(x) = \frac{1}{1+x^2} \)
Now substitute these into the product rule formula:
\[ f'(x) = (5 \cos x - 8 \sin x) \tan^{-1} x + (5 \sin x + 8 \cos x) \left( \frac{1}{1+x^2} \right) \]
This is the expression for the derivative \( f'(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f1c5ee8-9f50-4f4c-aa8f-9d252d015e08%2F310a03f4-b788-48e0-8346-3f38fde054f4%2F690dya_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f(x) = (5 \sin x + 8 \cos x) \tan^{-1} x \)
\[ f'(x) = \]
The task is to differentiate the function \( f(x) = (5 \sin x + 8 \cos x) \tan^{-1} x \). We will apply the product rule, which states that for two functions \( u(x) \) and \( v(x) \), the derivative is given by:
\[ (uv)' = u'v + uv' \]
Here, \( u(x) = 5 \sin x + 8 \cos x \) and \( v(x) = \tan^{-1} x \).
1. Differentiate \( u(x) \):
- \( u'(x) = 5 \cos x - 8 \sin x \)
2. Differentiate \( v(x) \):
- \( v'(x) = \frac{1}{1+x^2} \)
Now substitute these into the product rule formula:
\[ f'(x) = (5 \cos x - 8 \sin x) \tan^{-1} x + (5 \sin x + 8 \cos x) \left( \frac{1}{1+x^2} \right) \]
This is the expression for the derivative \( f'(x) \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 7 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