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
![**Topic: Differentiation of Trigonometric Functions**
**Problem Statement:**
Calculate the derivative of \( y = \cos(7x^3) \).
(Use symbolic notation and fractions where needed.)
\( y' = \)
**Solution:**
To solve this problem, we'll need to use the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \).
1. Given the function \( y = \cos(7x^3) \), identify the outer and inner functions:
- Outer function: \( f(u) = \cos(u) \)
- Inner function: \( u = 7x^3 \)
2. Find the derivative of the outer function:
\( f'(u) = -\sin(u) \)
3. Find the derivative of the inner function:
\( u = 7x^3 \)
\( u' = 21x^2 \)
4. Apply the chain rule:
\[
y' = f'(u) \cdot u'
\]
\[
y' = -\sin(7x^3) \cdot 21x^2
\]
Simplify the expression:
\[
y' = -21x^2 \sin(7x^3)
\]
Thus, the derivative of \( y = \cos(7x^3) \) is:
\[
y' = -21x^2 \sin(7x^3)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F575d721b-26eb-4db6-af93-50c207cd3fab%2Fdc43bb31-a121-4bfe-a4ac-713a4c2cc61c%2F3cnkbz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Differentiation of Trigonometric Functions**
**Problem Statement:**
Calculate the derivative of \( y = \cos(7x^3) \).
(Use symbolic notation and fractions where needed.)
\( y' = \)
**Solution:**
To solve this problem, we'll need to use the chain rule for differentiation. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \).
1. Given the function \( y = \cos(7x^3) \), identify the outer and inner functions:
- Outer function: \( f(u) = \cos(u) \)
- Inner function: \( u = 7x^3 \)
2. Find the derivative of the outer function:
\( f'(u) = -\sin(u) \)
3. Find the derivative of the inner function:
\( u = 7x^3 \)
\( u' = 21x^2 \)
4. Apply the chain rule:
\[
y' = f'(u) \cdot u'
\]
\[
y' = -\sin(7x^3) \cdot 21x^2
\]
Simplify the expression:
\[
y' = -21x^2 \sin(7x^3)
\]
Thus, the derivative of \( y = \cos(7x^3) \) is:
\[
y' = -21x^2 \sin(7x^3)
\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 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