Problem 27 - Calculus and its applications

Question

Find the four second-order partial derivatives. $$f(x, y)=2 x y$$

Step-by-Step Solution

Answer

The second-order partial derivatives are: \( \frac{\partial^2 f}{\partial x^2} = 0 \), \( \frac{\partial^2 f}{\partial y^2} = 0 \), \( \frac{\partial^2 f}{\partial x \partial y} = 2 \), \( \frac{\partial^2 f}{\partial y \partial x} = 2 \).

Step 1: Understand Partial Derivatives

Partial derivatives involve differentiating a multivariable function with respect to one variable while keeping the others constant. Since we need second-order partial derivatives, we'll first find the first-order partial derivatives with respect to each variable.

Step 2: Find First Partial Derivative with Respect to x

Differentiate the function with respect to x treating y as a constant:\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (2xy) = 2y \]

Step 3: Find First Partial Derivative with Respect to y

Differentiate the function with respect to y treating x as a constant:\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (2xy) = 2x \]

Step 4: Find Second Partial Derivative with Respect to x

Differentiate the first-order derivative \( \frac{\partial f}{\partial x} = 2y \) again with respect to x:\[ \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}(2y) = 0 \]

Step 5: Find Second Partial Derivative with Respect to y

Differentiate the first-order derivative \( \frac{\partial f}{\partial y} = 2x \) again with respect to y:\[ \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}(2x) = 0 \]

Step 6: Find Mixed Partial Derivative with Respect to x and Then y

Differentiate \( \frac{\partial f}{\partial x} = 2y \) with respect to y:\[ \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial y}(2y) = 2 \]

Step 7: Find Mixed Partial Derivative with Respect to y and Then x

Differentiate \( \frac{\partial f}{\partial y} = 2x \) with respect to x:\[ \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial x}(2x) = 2 \]

Key Concepts

Multivariable Calculus

Multivariable Calculus is an extension of calculus that involves functions with multiple variables. In our context, we deal with functions like \( f(x, y) = 2xy \), which depend on both \( x \) and \( y \). This field allows us to explore how changes in one variable affect the function, while other variables are held constant.

In single-variable calculus, we consider derivatives as the rate of change of a function with respect to one variable. However, in multivariable calculus, we take partial derivatives. These focus on just one variable at a time, treating other variables as constants. This gives us the flexibility to analyze complex systems in more detail.

Studying multivariable calculus is essential because it lays the groundwork for fields such as economics, engineering, and physics, where systems are not only dependent on a single element but a multitude.

Second-Order Derivatives

Second-order derivatives provide even deeper insights into how functions behave. When we first take a derivative, we are finding the rate of change of a function. Taking a second-order derivative means differentiating the derivative.

In our exercise, finding the second-order derivative involves taking the first derivative of the function \( f(x, y) \) with respect to either \( x \) or \( y \) and then differentiating again with respect to the same variable.

• For \( \frac{\partial f}{\partial x} = 2y \), the second-order derivative with respect to \( x \) is zero, i.e., \( \frac{\partial^2 f}{\partial x^2} = 0 \).
• Similarly, for \( \frac{\partial f}{\partial y} = 2x \), the second-order derivative with respect to \( y \) is also zero, i.e., \( \frac{\partial^2 f}{\partial y^2} = 0 \).

These results show that the rate of change itself does not change if we keep differentiating along the same variable direction in this case.

Mixed Partial Derivatives

Mixed partial derivatives involve differentiating a function first with respect to one variable and then with another. These derivatives provide insights into how two variables interact and influence each other in a function.

The process is straightforward but crucial:

• First, calculate the derivative concerning one variable.
• Then, take another derivative of the result concerning a different variable.

For our function, after finding \( \frac{\partial f}{\partial x} = 2y \), we differentiate again, but now with respect to \( y \):

• This gives \( \frac{\partial^2 f}{\partial x \partial y} = 2 \).
• Similarly, by reversing the order, first \( \frac{\partial f}{\partial y} = 2x \) and then regarding \( x \), leads to \( \frac{\partial^2 f}{\partial y \partial x} = 2 \).

Importantly, mixed partial derivatives often show symmetry; here, \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \) always equal, highlighting consistent interaction between variables.