o find f(x, y) <=201 Check Answer/Save What is fz (z,y) at the point (0,-) If ƒ(-, y) = Check Answer/Save 1+: of(x,y) 81 What is fx (2, 1) (0.) : you need to find derivative by holding ✓constant Step-By-Step Example constant Both and constant Neither nor y constant ✔ of(z,y) От 1 (1+z)² Y (1+2)² sin V (1+z)³ sin 1 (1+z)³ (0,m) : O Step-By-Step Example sin 1+= sin Live Help Live Help X
o find f(x, y) <=201 Check Answer/Save What is fz (z,y) at the point (0,-) If ƒ(-, y) = Check Answer/Save 1+: of(x,y) 81 What is fx (2, 1) (0.) : you need to find derivative by holding ✓constant Step-By-Step Example constant Both and constant Neither nor y constant ✔ of(z,y) От 1 (1+z)² Y (1+2)² sin V (1+z)³ sin 1 (1+z)³ (0,m) : O Step-By-Step Example sin 1+= sin Live Help Live Help X
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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Transcribed Image Text:The image presents a mathematical problem focused on partial derivatives, specifically finding the partial derivative \( \frac{\partial f(x, y)}{\partial z} \).
### Transcription:
1. **Expression:** The problem asks, "What is \( \frac{\partial f(x, y)}{\partial z} = \frac{\partial}{\partial z} \left( \frac{y}{1 + xz} \right) \)?"
2. **Options for the Partial Derivative:**
- \( \frac{y}{(1 + xz)^2} \sin \left( \frac{1}{1 + z} \right) \)
- \( \frac{x}{(1 + yz)^2} \sin \left( \frac{1}{1 + z} \right) \)
- \( \frac{y}{(1 + yz)^2} \sin \left( \frac{1}{1 + z} \right) \)
- \( \frac{1}{(1 + yz)^2} \sin \left( \frac{1}{1 + z} \right) \)
3. **Constant Identification:** To find \( \frac{\partial f}{\partial z} \), you need to keep certain variables constant:
- **Options:**
- \( x \) constant
- \( y \) constant
- Both \( x \) and \( y \) constant
- Neither \( x \) nor \( y \) constant
- The correct choice is \( y \) constant.
4. **Buttons/Options Available:**
- **Check Answer/Save**
- **Step-by-Step Example**
- **Live Help**
This problem involves understanding how to correctly apply the concept of partial differentiation, focusing on holding the appropriate variables constant while differentiating with respect to \( z \). This is a critical skill in calculus, particularly useful for multivariable functions.
![### Understanding Partial Derivatives
In this educational guide, we explore the concept of partial derivatives, using a classic example involving functions of two variables. This is a fundamental concept in multivariable calculus, essential for fields like engineering, physics, and mathematics.
#### Problem Statement
The problem asks to find the partial derivative of a function \( f(x, y) \) with respect to \( y \):
\[ \frac{\partial f(x, y)}{\partial y} \]
The solution involves calculating:
\[ \frac{\partial f(x, y)}{\partial y} = \frac{\partial}{\partial y} \left( \text{expression involving } x \text{ and } y \right) \]
#### Process and Solution
1. **Initial Setup:**
- Input the derivative function or expression in the designated box provided.
- The notation specifically asks for the derivation with respect to \( y \).
2. **Equation Solving Steps:**
- Identify terms dependent on \( y \).
- Hold \( x \) constant during differentiation.
- Apply differentiation rules (e.g., power rule, constant rule) to solve.
3. **Answer Submission:**
- Enter the fully simplified derivative in the answer box.
- Click the 'Check Answer/Save' button to validate your solution or save it for review.
4. **Diagram Explanation:**
- A diagram presents multiple terms, each enclosed in parentheses, indicating the process of differentiation.
- The plus and minus signs denote the algebraic operations within the function that need differentiation.
- Arrows illustrate steps or connections between derived terms.
5. **Tool Assistance:**
- Use the “Step-by-Step Example” and “Live Help” for guidance.
- A checklist is provided for operations to confirm if the function is constant or involves multiple variables.
#### Error Handling and Corrections
- If an incorrect entry is made, it is indicated by a red 'X' mark. Double-check the mathematical operations and try again.
This exercise emphasizes both the mechanical computation of derivatives and the conceptual understanding of holding variables constant. Mastery here will build a solid foundation for tackling more advanced applications in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2F15ccbf6a-e44e-43e8-ace6-e0e1db32d508%2Foc8g2br_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Partial Derivatives
In this educational guide, we explore the concept of partial derivatives, using a classic example involving functions of two variables. This is a fundamental concept in multivariable calculus, essential for fields like engineering, physics, and mathematics.
#### Problem Statement
The problem asks to find the partial derivative of a function \( f(x, y) \) with respect to \( y \):
\[ \frac{\partial f(x, y)}{\partial y} \]
The solution involves calculating:
\[ \frac{\partial f(x, y)}{\partial y} = \frac{\partial}{\partial y} \left( \text{expression involving } x \text{ and } y \right) \]
#### Process and Solution
1. **Initial Setup:**
- Input the derivative function or expression in the designated box provided.
- The notation specifically asks for the derivation with respect to \( y \).
2. **Equation Solving Steps:**
- Identify terms dependent on \( y \).
- Hold \( x \) constant during differentiation.
- Apply differentiation rules (e.g., power rule, constant rule) to solve.
3. **Answer Submission:**
- Enter the fully simplified derivative in the answer box.
- Click the 'Check Answer/Save' button to validate your solution or save it for review.
4. **Diagram Explanation:**
- A diagram presents multiple terms, each enclosed in parentheses, indicating the process of differentiation.
- The plus and minus signs denote the algebraic operations within the function that need differentiation.
- Arrows illustrate steps or connections between derived terms.
5. **Tool Assistance:**
- Use the “Step-by-Step Example” and “Live Help” for guidance.
- A checklist is provided for operations to confirm if the function is constant or involves multiple variables.
#### Error Handling and Corrections
- If an incorrect entry is made, it is indicated by a red 'X' mark. Double-check the mathematical operations and try again.
This exercise emphasizes both the mechanical computation of derivatives and the conceptual understanding of holding variables constant. Mastery here will build a solid foundation for tackling more advanced applications in calculus.
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