Second Partial Derivatives

Oct 13, 2025 by leolavanderia.com.br 57 views

Introduction

In this article, we will explore the concept of partial derivatives of a multivariable function. A partial derivative is a derivative of a multivariable function with respect to one of its variables, while keeping the other variables constant. This is a fundamental concept in calculus and is used to study the behavior of functions of multiple variables.

The Function

The function we will be working with is given by:

f(x, y) = x^9 - x^5 y - 3x^3 y^2 + 7xy

This is a polynomial function of two variables, x and y.

First Partial Derivatives

To compute the first partial derivatives of f(x, y), we will differentiate the function with respect to x and y separately.

Partial Derivative with Respect to x

To compute the partial derivative of f(x, y) with respect to x, we will treat y as a constant and differentiate the function with respect to x.

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x^9 - x^5 y - 3x^3 y^2 + 7xy)

Using the power rule for differentiation, we get:

\frac{\partial f}{\partial x} = 9x^8 - 5x^4 y - 9x^2 y^2 + 7y

Partial Derivative with Respect to y

To compute the partial derivative of f(x, y) with respect to y, we will treat x as a constant and differentiate the function with respect to y.

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^9 - x^5 y - 3x^3 y^2 + 7xy)

Using the power rule for differentiation, we get:

\frac{\partial f}{\partial y} = -x^5 - 6x^3 y + 7x

Second Partial Derivatives

To compute the second partial derivatives of f(x, y), we will differentiate the first partial derivatives with respect to x and y separately.

Second Partial Derivative with Respect to x

To compute the second partial derivative of f(x, y) with respect to x, we will differentiate the first partial derivative with respect to x with respect to x.

\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} (9x^8 - 5x^4 y - 9x^2 y^2 + 7y)

Using the power rule for differentiation, we get:

\frac{\partial^2 f}{\partial x^2} = 72x^7 - 20x^3 y - 18xy^2

Second Partial Derivative with Respect to y

To compute the second partial derivative of f(x, y) with respect to y, we will differentiate the first partial derivative with respect to y with respect to y.

\frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} (-x^5 - 6x^3 y + 7x)

Using the power rule for differentiation, we get:

\frac{\partial^2 f}{\partial y^2} = -6x^3

Second Partial Derivative with Respect to x and y

To compute the second partial derivative of f(x, y) with respect to x and y, we will differentiate the first partial derivative with respect to x with respect to y.

\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y} (9x^8 - 5x^4 y - 9x^2 y^2 + 7y)

Using the power rule for differentiation, we get:

\frac{\partial^2 f}{\partial y \partial x} = -5x^4 - 18xy^2 + 7

Second Partial Derivative with Respect to y and x

To compute the second partial derivative of f(x, y) with respect to y and x, we will differentiate the first partial derivative with respect to y with respect to x.

\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x} (-x^5 - 6x^3 y + 7x)

Using the power rule for differentiation, we get:

\frac{\partial^2 f}{\partial x \partial y} = -5x^4 - 18xy^2 + 7

Conclusion

Introduction

In our previous article, we explored the concept of partial derivatives of a multivariable function. A partial derivative is a derivative of a multivariable function with respect to one of its variables, while keeping the other variables constant. This is a fundamental concept in calculus and is used to study the behavior of functions of multiple variables.

In this article, we will answer some frequently asked questions about partial derivatives of a multivariable function.

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

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

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

A: To compute the partial derivative of a multivariable function, you will need to differentiate the function with respect to one of its variables, while keeping the other variables constant. This can be done using the power rule for differentiation.

Q: What is the notation for partial derivatives?

A: The notation for partial derivatives is ∂f/∂x, where f is the function and x is the variable with respect to which the derivative is being taken.

Q: Can I compute the partial derivative of a function that is not differentiable?

A: No, you cannot compute the partial derivative of a function that is not differentiable. The partial derivative is only defined for functions that are differentiable.

Q: How do I compute the second partial derivative of a multivariable function?

A: To compute the second partial derivative of a multivariable function, you will need to differentiate the first partial derivative with respect to one of the variables.

Q: What is the relationship between the first and second partial derivatives of a multivariable function?

A: The first partial derivative of a multivariable function is a measure of the rate of change of the function with respect to one of its variables, while the second partial derivative is a measure of the rate of change of the first partial derivative with respect to one of the variables.

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

A: Yes, you can use the partial derivative to find the maximum or minimum of a multivariable function. The partial derivative can be used to determine the critical points of the function, which are the points where the function has a maximum or minimum value.

Q: How do I use the partial derivative to find the critical points of a multivariable function?

A: To use the partial derivative to find the critical points of a multivariable function, you will need to set the partial derivative equal to zero and solve for the variable. This will give you the critical points of the function.

Conclusion

In this article, we have answered some frequently asked questions about partial derivatives of a multivariable function. We have discussed the difference between a partial derivative and a total derivative, how to compute the partial derivative of a multivariable function, and how to use the partial derivative to find the maximum or minimum of a multivariable function.

Additional Resources

For more information on partial derivatives of a multivariable function, please see the following resources:

Practice Problems

To practice computing partial derivatives of a multivariable function, please see the following problems:

Compute the partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to x.
Compute the partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to y.
Compute the second partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to x.
Compute the second partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to y.

Answer Key

To check your answers, please see the following answer key:

The partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to x is 2x + 2y.
The partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to y is 2x + 2y.
The second partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to x is 2.
The second partial derivative of the function f(x, y) = x^2 + 2xy + y^2 with respect to y is 2.