Problem 25 - Applied Calculus

Question

For Problems calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$f=5+x^{2} y^{2}$$

Step-by-Step Solution

Answer

The mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are equal; both are \( 4xy \).

Step 1: First-Order Partial Derivative with respect to x

The function is given as \( f(x, y) = 5 + x^2 y^2 \). First, find the partial derivative with respect to \( x \), denoted as \( f_x \).\[ f_x = \frac{\partial}{\partial x}(5 + x^2 y^2) = 2xy^2 \].

Step 2: First-Order Partial Derivative with respect to y

Next, find the partial derivative of \( f(x, y) \) with respect to \( y \), denoted as \( f_y \).\[ f_y = \frac{\partial}{\partial y}(5 + x^2 y^2) = 2x^2y \].

Step 3: Second-Order Partial Derivative with respect to x twice

Now, find the second-order partial derivative of \( f \) with respect to \( x \) twice, denoted as \( f_{xx} \).\[ f_{xx} = \frac{\partial}{\partial x}(f_x) = \frac{\partial}{\partial x}(2xy^2) = 2y^2 \].

Step 4: Second-Order Partial Derivative with respect to y twice

Find the second-order partial derivative of \( f \) with respect to \( y \) twice, denoted as \( f_{yy} \).\[ f_{yy} = \frac{\partial}{\partial y}(f_y) = \frac{\partial}{\partial y}(2x^2y) = 2x^2 \].

Step 5: Mixed Partial Derivative with respect to x and y

Find the mixed partial derivative with respect to \( x \) and then \( y \), denoted as \( f_{xy} \).\[ f_{xy} = \frac{\partial}{\partial y}(f_x) = \frac{\partial}{\partial y}(2xy^2) = 4xy \].

Step 6: Mixed Partial Derivative with respect to y and x

Finally, find the mixed partial derivative with respect to \( y \) and then \( x \), denoted as \( f_{yx} \).\[ f_{yx} = \frac{\partial}{\partial x}(f_y) = \frac{\partial}{\partial x}(2x^2y) = 4xy \].

Step 7: Confirm Equality of Mixed Partials

The mixed partial derivatives are equal if \( f_{xy} = f_{yx} \). Since we found \( f_{xy} = 4xy \) and \( f_{yx} = 4xy \), the mixed partial derivatives are indeed equal, confirming the condition that mixed partials \( f_{xy} = f_{yx} \).

Key Concepts

Partial Derivatives

In calculus, when we deal with functions of multiple variables, such as two variables in the function \( f(x, y) = 5 + x^2 y^2 \), we use partial derivatives to understand how the function changes with respect to just one of these variables while keeping the other constant.
For example, to find the partial derivative of \( f \) with respect to \( x \), denoted as \( f_x \), we treat \( y \) as a constant and differentiate only in terms of \( x \). In this case:

• The first-order partial derivative \( f_x \) is \( 2xy^2 \).
• Similarly, the first-order partial derivative with respect to \( y \), denoted as \( f_y \), results in \( 2x^2y \).

This partial differentiation allows us to analyze the function along one axis in the coordinate plane, providing insights into how changes in \( x \) or \( y \) alone affect the function.

Mixed Partial Derivatives

When we move beyond first-order partial derivatives, we encounter second-order partial derivatives. These involve taking the derivative of a partial derivative.
Sometimes, we need to differentiate with respect to different variables, creating mixed partial derivatives. For the function \( f(x, y) = 5 + x^2 y^2 \), we calculate the mixed partial derivatives as follows:

• First, calculate the derivative of \( f_x \) with respect to \( y \), resulting in \( f_{xy} = 4xy \).
• Conversely, differentiating \( f_y \) with respect to \( x \) gives \( f_{yx} = 4xy \).

In many cases, like this one, the mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are equal, illustrating a fundamental aspect of calculus called Clairaut's theorem, which states that the order of differentiation does not affect the outcome, provided the function is well-behaved and continuous.

Applied Calculus

Partial derivatives are a crucial tool in applied calculus, used in fields such as engineering, physics, and economics to optimize functions and model real-world phenomena.
By using partial derivatives, we can explore how changing one variable while keeping others constant impacts a system or process.

• For instance, in thermodynamics, partial derivatives help understand how changing pressure affects volume and temperature.
• In economics, they can be used to determine how changing the cost of one commodity affects total production costs.

The knowledge of mixed partial derivatives further enhances our ability to predict interactions between variables, offering a more comprehensive view of dynamic systems. This mathematical approach aids in making informed decisions and solving complex problems across various disciplines.