Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2. If f (x, y) = In(x²y³) + xy², find дудх"

2. If f (x, y) = In(x²y³) + xy², find дудх"

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please give a full explanation to the answer.

 

### Problem Statement: **2.** If \( f(x, y) = \ln(x^2 y^3) + xy^2 \), find \( \frac{\partial^2 f}{\partial y \partial x} \). ### Explanation: This problem involves finding the second mixed partial derivative of a function \( f(x, y) \). The function provided is a combination of a natural logarithm and a polynomial expression. #### Step-by-step Approach: 1. **Differentiate the function partially with respect to \( x \).** - For \( f(x, y) = \ln(x^2 y^3) + xy^2 \): - Differentiate \( \ln(x^2 y^3) \) with respect to \( x \). - Differentiate \( xy^2 \) with respect to \( x \). 2. **After finding \( \frac{\partial f}{\partial x} \), differentiate the result with respect to \( y \) to find \( \frac{\partial^2 f}{\partial y \partial x} \).** ### Important Concepts: - **Partial Derivatives:** Calculating the derivative with respect to one variable while keeping others constant. - **Chain Rule:** Useful in differentiating compositions of functions. - **Mixed Partial Derivatives:** Mixed second partial derivatives involve taking the derivative with respect to different variables in succession. This type of problem tests understanding of differential calculus and is often encountered in courses related to multivariable calculus and mathematical analysis.
Transcribed Image Text:### Problem Statement: **2.** If \( f(x, y) = \ln(x^2 y^3) + xy^2 \), find \( \frac{\partial^2 f}{\partial y \partial x} \). ### Explanation: This problem involves finding the second mixed partial derivative of a function \( f(x, y) \). The function provided is a combination of a natural logarithm and a polynomial expression. #### Step-by-step Approach: 1. **Differentiate the function partially with respect to \( x \).** - For \( f(x, y) = \ln(x^2 y^3) + xy^2 \): - Differentiate \( \ln(x^2 y^3) \) with respect to \( x \). - Differentiate \( xy^2 \) with respect to \( x \). 2. **After finding \( \frac{\partial f}{\partial x} \), differentiate the result with respect to \( y \) to find \( \frac{\partial^2 f}{\partial y \partial x} \).** ### Important Concepts: - **Partial Derivatives:** Calculating the derivative with respect to one variable while keeping others constant. - **Chain Rule:** Useful in differentiating compositions of functions. - **Mixed Partial Derivatives:** Mixed second partial derivatives involve taking the derivative with respect to different variables in succession. This type of problem tests understanding of differential calculus and is often encountered in courses related to multivariable calculus and mathematical analysis.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Basics of Inferential Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS