Use the Chain Rule to evaluate the partial derivative (Use symbolic notation and fractions where needed.) dh da (gr) = at the point (q, r) = (3, 3), where h(u, v) = ueº, u = q¹, v = qr³.
Use the Chain Rule to evaluate the partial derivative (Use symbolic notation and fractions where needed.) dh da (gr) = at the point (q, r) = (3, 3), where h(u, v) = ueº, u = q¹, v = qr³.
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
100%
Need help please.
![### Chain Rule Application for Partial Derivatives
**Problem Statement:**
Use the Chain Rule to evaluate the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (3, 3)\), where \(h(u, v) = ue^v\), \(u = q^4\), \(v = qr^3\).
**Equation and Variables:**
\[
\frac{\partial h}{\partial q} \bigg|_{(q, r)}
\]
**Given:**
- \((q, r) = (3, 3)\)
- \(h(u, v) = ue^v\)
- \(u = q^4\)
- \(v = qr^3\)
(Use symbolic notation and fractions where needed.)
**Solution:**
To determine the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (3, 3)\), apply the chain rule, taking into account the intermediate variables \(u\) and \(v\).
1. **Express \(h\) in terms of \(q\) and \(r\):**
\[h(u, v) = ue^v\]
2. **Calculate the partial derivatives \( \frac{\partial h}{\partial u} \) and \( \frac{\partial h}{\partial v} \):**
\[ \frac{\partial h}{\partial u} = e^v \]
\[ \frac{\partial h}{\partial v} = ue^v \]
3. **Calculate the partial derivatives \(\frac{\partial u}{\partial q}\) and \(\frac{\partial v}{\partial q}\):**
\[ \frac{\partial u}{\partial q} = \frac{\partial}{\partial q}(q^4) = 4q^3 \]
\[ \frac{\partial v}{\partial q} = \frac{\partial}{\partial q}(qr^3) = r^3 \]
4. **Apply the Chain Rule:**
\[ \frac{\partial h}{\partial q} = \frac{\partial h}{\partial u} \cdot \frac{\partial u}{\partial q} + \frac{\partial h}{\partial v} \cdot \frac{\partial v}{\partial q} \]
5. **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa25589fa-5419-4615-a98c-c838d88c4e7f%2Fc03f40eb-5b5c-4249-8ca2-c64d28503082%2Fbq66ky8_processed.png&w=3840&q=75)
Transcribed Image Text:### Chain Rule Application for Partial Derivatives
**Problem Statement:**
Use the Chain Rule to evaluate the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (3, 3)\), where \(h(u, v) = ue^v\), \(u = q^4\), \(v = qr^3\).
**Equation and Variables:**
\[
\frac{\partial h}{\partial q} \bigg|_{(q, r)}
\]
**Given:**
- \((q, r) = (3, 3)\)
- \(h(u, v) = ue^v\)
- \(u = q^4\)
- \(v = qr^3\)
(Use symbolic notation and fractions where needed.)
**Solution:**
To determine the partial derivative \(\frac{\partial h}{\partial q}\) at the point \((q, r) = (3, 3)\), apply the chain rule, taking into account the intermediate variables \(u\) and \(v\).
1. **Express \(h\) in terms of \(q\) and \(r\):**
\[h(u, v) = ue^v\]
2. **Calculate the partial derivatives \( \frac{\partial h}{\partial u} \) and \( \frac{\partial h}{\partial v} \):**
\[ \frac{\partial h}{\partial u} = e^v \]
\[ \frac{\partial h}{\partial v} = ue^v \]
3. **Calculate the partial derivatives \(\frac{\partial u}{\partial q}\) and \(\frac{\partial v}{\partial q}\):**
\[ \frac{\partial u}{\partial q} = \frac{\partial}{\partial q}(q^4) = 4q^3 \]
\[ \frac{\partial v}{\partial q} = \frac{\partial}{\partial q}(qr^3) = r^3 \]
4. **Apply the Chain Rule:**
\[ \frac{\partial h}{\partial q} = \frac{\partial h}{\partial u} \cdot \frac{\partial u}{\partial q} + \frac{\partial h}{\partial v} \cdot \frac{\partial v}{\partial q} \]
5. **
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