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
![### Calculating the Derivative
**Problem Statement:**
Find \( dy \) for the following equation.
\[ y = 3x \sqrt{12 - x^3} \]
**Objective:**
Express \( \frac{dy}{dx} \) in terms of \( x \).
**Solution Steps:**
To find \( \frac{dy}{dx} \), we will use the product rule and chain rule of differentiation.
1. **Identify the Terms:**
- Let \( u = 3x \).
- Let \( v = \sqrt{12 - x^3} \).
2. **Differentiate Each Term:**
- \( \frac{du}{dx} = 3 \).
- For \( v = (12 - x^3)^{1/2} \):
- Using the chain rule, \( \frac{dv}{dx} = \frac{1}{2}(12 - x^3)^{-1/2} \cdot (-3x^2) \).
- Simplifying gives \( \frac{dv}{dx} = \frac{-3x^2}{2\sqrt{12 - x^3}} \).
3. **Apply the Product Rule:**
- The product rule states: \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \).
- Substituting:
\[
\frac{dy}{dx} = 3x \cdot \frac{-3x^2}{2\sqrt{12 - x^3}} + \sqrt{12 - x^3} \cdot 3
\]
4. **Combine and Simplify:**
- \( \frac{dy}{dx} = \frac{-9x^3}{2\sqrt{12 - x^3}} + 3\sqrt{12 - x^3} \).
**Conclusion:**
Thus, \( dy = \left(\frac{-9x^3}{2\sqrt{12 - x^3}} + 3\sqrt{12 - x^3}\right) dx \).
This expression represents the derivative of \( y \) with respect to \( x \), capturing all changes in \( y \) as \( x \) varies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeadb183-1175-48ee-be1d-be4cbd36e1e9%2Fe20c73ac-9a1d-4e7a-a891-5f478dcb1690%2Firc8qij_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Derivative
**Problem Statement:**
Find \( dy \) for the following equation.
\[ y = 3x \sqrt{12 - x^3} \]
**Objective:**
Express \( \frac{dy}{dx} \) in terms of \( x \).
**Solution Steps:**
To find \( \frac{dy}{dx} \), we will use the product rule and chain rule of differentiation.
1. **Identify the Terms:**
- Let \( u = 3x \).
- Let \( v = \sqrt{12 - x^3} \).
2. **Differentiate Each Term:**
- \( \frac{du}{dx} = 3 \).
- For \( v = (12 - x^3)^{1/2} \):
- Using the chain rule, \( \frac{dv}{dx} = \frac{1}{2}(12 - x^3)^{-1/2} \cdot (-3x^2) \).
- Simplifying gives \( \frac{dv}{dx} = \frac{-3x^2}{2\sqrt{12 - x^3}} \).
3. **Apply the Product Rule:**
- The product rule states: \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \).
- Substituting:
\[
\frac{dy}{dx} = 3x \cdot \frac{-3x^2}{2\sqrt{12 - x^3}} + \sqrt{12 - x^3} \cdot 3
\]
4. **Combine and Simplify:**
- \( \frac{dy}{dx} = \frac{-9x^3}{2\sqrt{12 - x^3}} + 3\sqrt{12 - x^3} \).
**Conclusion:**
Thus, \( dy = \left(\frac{-9x^3}{2\sqrt{12 - x^3}} + 3\sqrt{12 - x^3}\right) dx \).
This expression represents the derivative of \( y \) with respect to \( x \), capturing all changes in \( y \) as \( x \) varies.
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 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.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