Calculate fz (0, 0, 1, 1), where f(x, y, z, w) = exz+y z² + w
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please show all work!
![### Problem Statement
Calculate \( f_z(0, 0, 1, 1) \), where \( f(x, y, z, w) = \frac{e^{xz+y}}{z^2 + w} \).
---
**Explanation:**
The problem requires calculating the partial derivative of the function \( f \) with respect to the variable \( z \) at the point \( (0, 0, 1, 1) \). The function \( f(x, y, z, w) \) is given as:
\[ f(x, y, z, w) = \frac{e^{xz + y}}{z^2 + w} \]
To solve this, you will need to follow these steps:
1. **Differentiate \( f \) with respect to \( z \).**
2. **Substitute \( x = 0 \), \( y = 0 \), \( z = 1 \), and \( w = 1 \) into the derivative.**
#### Differentiation Step-by-Step:
1. **Differentiate the numerator and the denominator separately and then apply the quotient rule:**
\[ \frac{\partial}{\partial z}\left(\frac{e^{xz + y}}{z^2 + w}\right) = \frac{ (z^2+w) \frac{\partial e^{xz+y}}{\partial z} - e^{xz+y} \frac{\partial (z^2+w)}{\partial z} }{(z^2+w)^2} \]
2. **Compute \(\frac{\partial e^{xz+y}}{\partial z}\):**
\[ \frac{\partial e^{xz + y}}{\partial z} = xe^{xz + y} \]
3. **Compute \(\frac{\partial (z^2 + w)}{\partial z}\):**
\[ \frac{\partial (z^2 + w)}{\partial z} = 2z \]
4. **Substitute these into the quotient rule expression:**
\[ \frac{\partial}{\partial z}\left(\frac{e^{xz + y}}{z^2 + w}\right) = \frac{(z^2 + w) \cdot (xe^{xz + y}) - e^{xz + y} \cdot 2z}{(z^2 + w)^2} \]
5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F717f4d87-5a32-4cdf-855f-90bb556622e3%2F3397412d-5f1f-448e-ba93-cae6a5e819aa%2Fon5v0m_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Calculate \( f_z(0, 0, 1, 1) \), where \( f(x, y, z, w) = \frac{e^{xz+y}}{z^2 + w} \).
---
**Explanation:**
The problem requires calculating the partial derivative of the function \( f \) with respect to the variable \( z \) at the point \( (0, 0, 1, 1) \). The function \( f(x, y, z, w) \) is given as:
\[ f(x, y, z, w) = \frac{e^{xz + y}}{z^2 + w} \]
To solve this, you will need to follow these steps:
1. **Differentiate \( f \) with respect to \( z \).**
2. **Substitute \( x = 0 \), \( y = 0 \), \( z = 1 \), and \( w = 1 \) into the derivative.**
#### Differentiation Step-by-Step:
1. **Differentiate the numerator and the denominator separately and then apply the quotient rule:**
\[ \frac{\partial}{\partial z}\left(\frac{e^{xz + y}}{z^2 + w}\right) = \frac{ (z^2+w) \frac{\partial e^{xz+y}}{\partial z} - e^{xz+y} \frac{\partial (z^2+w)}{\partial z} }{(z^2+w)^2} \]
2. **Compute \(\frac{\partial e^{xz+y}}{\partial z}\):**
\[ \frac{\partial e^{xz + y}}{\partial z} = xe^{xz + y} \]
3. **Compute \(\frac{\partial (z^2 + w)}{\partial z}\):**
\[ \frac{\partial (z^2 + w)}{\partial z} = 2z \]
4. **Substitute these into the quotient rule expression:**
\[ \frac{\partial}{\partial z}\left(\frac{e^{xz + y}}{z^2 + w}\right) = \frac{(z^2 + w) \cdot (xe^{xz + y}) - e^{xz + y} \cdot 2z}{(z^2 + w)^2} \]
5
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 with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
