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:Let R(x, y) = 0.7x³y -5x³ +6y², find the second
partial derivatives:
a²R
dy²
4
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![### Mixed Partial Derivative
In the study of multivariable calculus, partial derivatives involve the rate of change of a function with respect to one variable while keeping others constant. When we take the partial derivative of a function multiple times with respect to different variables, we obtain mixed partial derivatives.
The following equation describes a mixed partial derivative:
\[
\frac{\partial^2 R}{\partial x \partial y} = \frac{\partial}{\partial x } \left(\frac{\partial R}{\partial y}\right)
\]
#### Explanation:
- **\( R \)**: This typically represents a function of multiple variables.
- **\(\frac{\partial^2 R}{\partial x \partial y}\)**: This denotes the second-order partial derivative of \( R \). This specific derivative first considers the change with respect to \( y \) and then the change with respect to \( x \).
- **\(\frac{\partial}{\partial x }\left(\frac{\partial R}{\partial y}\right) \)**: This demonstrates that the second-order mixed partial derivative can be computed by first taking the partial derivative of \( R \) with respect to \( y \) and then taking the partial derivative of the result with respect to \( x \).
#### Key Points:
- **Order of Differentiation**: The order of taking partial derivatives matters. However, under certain conditions, such as when the function \( R \) and its partial derivatives are continuous, the mixed partial derivatives are equal regardless of the order in which they are taken (Clairaut's Theorem).
- **Applications**: Mixed partial derivatives are essential in fields such as physics, engineering, and economics, where functions with multiple variables frequently occur.
Understanding mixed partial derivatives is crucial for analyzing and modeling systems that depend on several changing factors.](https://content.bartleby.com/qna-images/question/0d56d87b-1ee4-40b0-9831-f06655864a4b/583e4858-1ad6-4640-a97c-346bb0d820ae/e0gf6qc_processed.jpeg)
Transcribed Image Text:### Mixed Partial Derivative
In the study of multivariable calculus, partial derivatives involve the rate of change of a function with respect to one variable while keeping others constant. When we take the partial derivative of a function multiple times with respect to different variables, we obtain mixed partial derivatives.
The following equation describes a mixed partial derivative:
\[
\frac{\partial^2 R}{\partial x \partial y} = \frac{\partial}{\partial x } \left(\frac{\partial R}{\partial y}\right)
\]
#### Explanation:
- **\( R \)**: This typically represents a function of multiple variables.
- **\(\frac{\partial^2 R}{\partial x \partial y}\)**: This denotes the second-order partial derivative of \( R \). This specific derivative first considers the change with respect to \( y \) and then the change with respect to \( x \).
- **\(\frac{\partial}{\partial x }\left(\frac{\partial R}{\partial y}\right) \)**: This demonstrates that the second-order mixed partial derivative can be computed by first taking the partial derivative of \( R \) with respect to \( y \) and then taking the partial derivative of the result with respect to \( x \).
#### Key Points:
- **Order of Differentiation**: The order of taking partial derivatives matters. However, under certain conditions, such as when the function \( R \) and its partial derivatives are continuous, the mixed partial derivatives are equal regardless of the order in which they are taken (Clairaut's Theorem).
- **Applications**: Mixed partial derivatives are essential in fields such as physics, engineering, and economics, where functions with multiple variables frequently occur.
Understanding mixed partial derivatives is crucial for analyzing and modeling systems that depend on several changing factors.
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