Find The First Partial Derivatives Of The Function.$\[ \begin{array}{c} z = X \sin(xy) \\ \frac{\partial Z}{\partial X} = \square \\ \frac{\partial Z}{\partial Y} = \square \end{array} \\]

Feb 27, 2025 by ADMIN 189 views

**Partial Derivatives of a Multivariable Function**

Introduction

In calculus, partial derivatives are a fundamental concept used to study the behavior of multivariable functions. Given a function of multiple variables, partial derivatives measure the rate of change of the function with respect to one of its variables, while keeping the other variables constant. In this article, we will focus on finding the first partial derivatives of the function $z = x \sin(xy)$ .

The Function

The given function is $z = x \sin(xy)$ . This is a multivariable function, meaning it depends on two variables, $x$ and $y$ . Our goal is to find the partial derivatives of this function with respect to $x$ and $y$ .

Partial Derivative with Respect to x

To find the partial derivative of $z$ with respect to $x$ , we will treat $y$ as a constant. This means that we will differentiate the function with respect to $x$ , while keeping $y$ fixed. Using the product rule and the chain rule, we can write:

\frac{\partial z}{\partial x} = \sin(xy) + x \cos(xy) \cdot y

Simplifying the expression, we get:

\frac{\partial z}{\partial x} = \sin(xy) + xy \cos(xy)

Partial Derivative with Respect to y

To find the partial derivative of $z$ with respect to $y$ , we will treat $x$ as a constant. This means that we will differentiate the function with respect to $y$ , while keeping $x$ fixed. Using the product rule and the chain rule, we can write:

\frac{\partial z}{\partial y} = x^2 \cos(xy)

Discussion

In this article, we have found the first partial derivatives of the function $z = x \sin(xy)$ with respect to $x$ and $y$ . The partial derivative with respect to $x$ is given by $\frac{\partial z}{\partial x} = \sin(xy) + xy \cos(xy)$ , while the partial derivative with respect to $y$ is given by $\frac{\partial z}{\partial y} = x^2 \cos(xy)$ .

Conclusion

In conclusion, partial derivatives are a powerful tool used to study the behavior of multivariable functions. By finding the partial derivatives of a function, we can gain insight into its behavior and make predictions about its behavior in different regions of the function's domain. In this article, we have seen how to find the first partial derivatives of a multivariable function using the product rule and the chain rule.

Applications

Partial derivatives have numerous applications in various fields, including physics, engineering, economics, and computer science. Some examples of applications include:

Optimization: Partial derivatives are used to find the maximum or minimum of a function subject to certain constraints.
Physics: Partial derivatives are used to describe the behavior of physical systems, such as the motion of objects in space.
Economics: Partial derivatives are used to study the behavior of economic systems, such as the behavior of supply and demand curves.
Computer Science: Partial derivatives are used in machine learning and optimization algorithms to train models and make predictions.

Future Work

In future work, we plan to explore more advanced topics in partial derivatives, such as higher-order partial derivatives and partial derivatives of vector-valued functions. We also plan to apply partial derivatives to real-world problems in physics, engineering, economics, and computer science.

References

Calculus: Michael Spivak, "Calculus", 4th edition, Cambridge University Press, 2008.
Partial Derivatives: James Stewart, "Calculus: Early Transcendentals", 8th edition, Brooks Cole, 2015.

Glossary

Partial Derivative: A measure of the rate of change of a function with respect to one of its variables, while keeping the other variables constant.
Product Rule: A rule used to differentiate the product of two functions.
Chain Rule: A rule used to differentiate the composition of two functions.

FAQs

Q: What is the partial derivative of a function? A: The partial derivative of a function is a measure of the rate of change of the function with respect to one of its variables, while keeping the other variables constant.
Q: How do I find the partial derivative of a function? A: To find the partial derivative of a function, you can use the product rule and the chain rule to differentiate the function with respect to one of its variables, while keeping the other variables constant.
Partial Derivatives: A Q&A Guide =====================================

Introduction

Partial derivatives are a fundamental concept in calculus, used to study the behavior of multivariable functions. In our previous article, we explored the partial derivatives of the function $z = x \sin(xy)$ . In this article, we will answer some frequently asked questions about partial derivatives.

Q&A

Q: What is the difference between a partial derivative and a total derivative?

A: A partial derivative measures the rate of change of a function with respect to one of its variables, while keeping the other variables constant. A total derivative, on the other hand, measures the rate of change of a function with respect to all of its variables.

Q: How do I find the partial derivative of a function?

A: To find the partial derivative of a function, you can use the product rule and the chain rule to differentiate the function with respect to one of its variables, while keeping the other variables constant.

Q: What is the notation for partial derivatives?

A: The notation for partial derivatives is $\frac{\partial z}{\partial x}$ , where $z$ is the function and $x$ is the variable with respect to which we are differentiating.

Q: Can I use the power rule to find partial derivatives?

A: Yes, you can use the power rule to find partial derivatives. For example, if $z = x^2$ , then $\frac{\partial z}{\partial x} = 2x$ .

Q: How do I find the partial derivative of a function with multiple variables?

A: To find the partial derivative of a function with multiple variables, you can use the product rule and the chain rule to differentiate the function with respect to one of its variables, while keeping the other variables constant.

Q: What is the relationship between partial derivatives and the gradient?

A: The gradient of a function is a vector of partial derivatives. In other words, if $z = f(x,y)$ , then the gradient of $z$ is $\nabla z = \left(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}\right)$ .

Q: Can I use partial derivatives to find the maximum or minimum of a function?

A: Yes, you can use partial derivatives to find the maximum or minimum of a function. This is known as optimization.

Q: How do I use partial derivatives to find the maximum or minimum of a function?

A: To use partial derivatives to find the maximum or minimum of a function, you can set the partial derivatives equal to zero and solve for the variables.

Q: What is the relationship between partial derivatives and the second derivative test?

A: The second derivative test is a method used to determine whether a function has a maximum or minimum at a given point. It involves finding the second partial derivatives of the function and using them to determine the nature of the function at that point.

Q: Can I use partial derivatives to solve systems of equations?

A: Yes, you can use partial derivatives to solve systems of equations. This is known as the method of Lagrange multipliers.

Q: How do I use partial derivatives to solve systems of equations?

A: To use partial derivatives to solve systems of equations, you can set up a system of equations using the partial derivatives and then solve for the variables.

Conclusion

In conclusion, partial derivatives are a powerful tool used to study the behavior of multivariable functions. By understanding the concepts and techniques of partial derivatives, you can gain insight into the behavior of functions and make predictions about their behavior in different regions of the function's domain.

Glossary

Partial Derivative: A measure of the rate of change of a function with respect to one of its variables, while keeping the other variables constant.
Product Rule: A rule used to differentiate the product of two functions.
Chain Rule: A rule used to differentiate the composition of two functions.
Gradient: A vector of partial derivatives.
Optimization: The process of finding the maximum or minimum of a function.
Second Derivative Test: A method used to determine whether a function has a maximum or minimum at a given point.

References

Calculus: Michael Spivak, "Calculus", 4th edition, Cambridge University Press, 2008.
Partial Derivatives: James Stewart, "Calculus: Early Transcendentals", 8th edition, Brooks Cole, 2015.

FAQs

Q: What is the partial derivative of a function? A: The partial derivative of a function is a measure of the rate of change of the function with respect to one of its variables, while keeping the other variables constant.
Q: How do I find the partial derivative of a function? A: To find the partial derivative of a function, you can use the product rule and the chain rule to differentiate the function with respect to one of its variables, while keeping the other variables constant.