Compute the second partial derivatives 2xy (x2+y2,2' Verify the following theorem in this case. 22f дх2 22f Әх ду 22р ?у Әх 22f ду2 || = II а2р агр а2р агр ах2' ах ду ду ах' ду2 , on the region where (x, y) = (0, 0) f(x, y) = 000 - for the following function. 221 Әх ду If f(x, y) is of class c2 (is twice continuously differentiable), then the mixed partial derivatives are equal; that is, 22f ?у ?х
Compute the second partial derivatives 2xy (x2+y2,2' Verify the following theorem in this case. 22f дх2 22f Әх ду 22р ?у Әх 22f ду2 || = II а2р агр а2р агр ах2' ах ду ду ах' ду2 , on the region where (x, y) = (0, 0) f(x, y) = 000 - for the following function. 221 Әх ду If f(x, y) is of class c2 (is twice continuously differentiable), then the mixed partial derivatives are equal; that is, 22f ?у ?х
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...
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![**Topic:** Computing Second Partial Derivatives
**Objective:** Compute the second partial derivatives for the given function and verify a theorem regarding mixed partial derivatives.
**Problem Statement:**
Compute the second partial derivatives \(\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial x \partial y}, \frac{\partial^2 f}{\partial y \partial x}, \frac{\partial^2 f}{\partial y^2}\) for the following function.
Given:
\[
f(x, y) = \frac{2xy}{(x^2 + y^2)^2}
\]
in the region where \((x, y) \neq (0, 0)\).
**Theorem Verification:**
Verify if the following theorem holds true for this function:
If \(f(x, y)\) is of class \(C^2\) (i.e., is twice continuously differentiable), then the mixed partial derivatives are equal:
\[
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}
\]
**Tasks:**
- Compute \(\frac{\partial^2 f}{\partial x^2}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial y^2}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial x \partial y}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial y \partial x}\):
\[
\boxed{\hspace{5em}}
\]
**Explanation for Each Computation:**
1. **\(\frac{\partial^2 f}{\partial x^2}\):** Compute the first partial derivative with respect to \(x\), then differentiate that result again with respect to \(x\).
2. **\(\frac{\partial^2 f}{\partial y^2}\):** Compute the first partial derivative with respect to \(y\), then differentiate that result again with respect to \(y\).
3. **\(\frac{\partial^2 f}{\partial x \partial y}\):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff893ad80-e3fa-4d69-87b2-5d427e54a923%2F98b0469c-fb27-4368-913d-f4f6aa09afd5%2Fyzi8tgq_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic:** Computing Second Partial Derivatives
**Objective:** Compute the second partial derivatives for the given function and verify a theorem regarding mixed partial derivatives.
**Problem Statement:**
Compute the second partial derivatives \(\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial x \partial y}, \frac{\partial^2 f}{\partial y \partial x}, \frac{\partial^2 f}{\partial y^2}\) for the following function.
Given:
\[
f(x, y) = \frac{2xy}{(x^2 + y^2)^2}
\]
in the region where \((x, y) \neq (0, 0)\).
**Theorem Verification:**
Verify if the following theorem holds true for this function:
If \(f(x, y)\) is of class \(C^2\) (i.e., is twice continuously differentiable), then the mixed partial derivatives are equal:
\[
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}
\]
**Tasks:**
- Compute \(\frac{\partial^2 f}{\partial x^2}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial y^2}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial x \partial y}\):
\[
\boxed{\hspace{5em}}
\]
- Compute \(\frac{\partial^2 f}{\partial y \partial x}\):
\[
\boxed{\hspace{5em}}
\]
**Explanation for Each Computation:**
1. **\(\frac{\partial^2 f}{\partial x^2}\):** Compute the first partial derivative with respect to \(x\), then differentiate that result again with respect to \(x\).
2. **\(\frac{\partial^2 f}{\partial y^2}\):** Compute the first partial derivative with respect to \(y\), then differentiate that result again with respect to \(y\).
3. **\(\frac{\partial^2 f}{\partial x \partial y}\):
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