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
![### Implicit Differentiation Problem
**Find \(\frac{dy}{dx}\) at the point (0, 2) if \(y^2 + e^{xy} = 4\).**
**Steps to solve:**
1. **Given Equation**: \(y^2 + e^{xy} = 4\)
2. **Differentiate both sides with respect to \(x\)**:
\[ \frac{d}{dx}(y^2) + \frac{d}{dx}(e^{xy}) = \frac{d}{dx}(4) \]
3. **Apply the chain rule and product rule**:
\[ 2y \frac{dy}{dx} + e^{xy} \left( y + x \frac{dy}{dx} \right) = 0 \]
4. **Substitute \(x=0\) and \(y=2\)** into the differentiated equation**:
\[ 2(2) \frac{dy}{dx} + e^{0\cdot2} \left(2 + 0 \cdot \frac{dy}{dx}\right) = 0 \]
Simplifying the equation above, we get:
\[ 4 \frac{dy}{dx} + 1 \cdot 2 = 0 \]
5. **Solve for \(\frac{dy}{dx}\)**:
\[ 4 \frac{dy}{dx} + 2 = 0 \]
\[ 4 \frac{dy}{dx} = -2 \]
\[ \frac{dy}{dx} = -\frac{2}{4} \]
\[ \frac{dy}{dx} = -\frac{1}{2} \]
So, \(\frac{dy}{dx} = -\frac{1}{2}\) at the point (0, 2).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9a5f5a5-4eae-476c-bcab-4a001ac27552%2F23f88d78-038d-4b24-9e41-fb723b7e5ab1%2Fsbzizzi_processed.png&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation Problem
**Find \(\frac{dy}{dx}\) at the point (0, 2) if \(y^2 + e^{xy} = 4\).**
**Steps to solve:**
1. **Given Equation**: \(y^2 + e^{xy} = 4\)
2. **Differentiate both sides with respect to \(x\)**:
\[ \frac{d}{dx}(y^2) + \frac{d}{dx}(e^{xy}) = \frac{d}{dx}(4) \]
3. **Apply the chain rule and product rule**:
\[ 2y \frac{dy}{dx} + e^{xy} \left( y + x \frac{dy}{dx} \right) = 0 \]
4. **Substitute \(x=0\) and \(y=2\)** into the differentiated equation**:
\[ 2(2) \frac{dy}{dx} + e^{0\cdot2} \left(2 + 0 \cdot \frac{dy}{dx}\right) = 0 \]
Simplifying the equation above, we get:
\[ 4 \frac{dy}{dx} + 1 \cdot 2 = 0 \]
5. **Solve for \(\frac{dy}{dx}\)**:
\[ 4 \frac{dy}{dx} + 2 = 0 \]
\[ 4 \frac{dy}{dx} = -2 \]
\[ \frac{dy}{dx} = -\frac{2}{4} \]
\[ \frac{dy}{dx} = -\frac{1}{2} \]
So, \(\frac{dy}{dx} = -\frac{1}{2}\) at the point (0, 2).
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 3 steps with 3 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