Problem 25 - Applied Calculus

Question

Calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$ f=e^{x y} $$

Step-by-Step Solution

Answer

All second-order partial derivatives confirm mixed partials: \(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\).

Step 1: Compute the First Partial Derivatives

Let's find the first partial derivatives of the function. For the partial derivative of the function with respect to \(x\), keep \(y\) constant:\[\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(e^{xy}) = y e^{xy}\]For the partial derivative with respect to \(y\), keep \(x\) constant:\[\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(e^{xy}) = x e^{xy}\]

Step 2: Compute Second-Order Partial Derivatives with Respect to x

Now, take the partial derivative of \(\frac{\partial f}{\partial x} = y e^{xy}\) with respect to \(x\) again:\[\frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}(y e^{xy}) = y^2 e^{xy}\]

Step 3: Compute Second-Order Partial Derivatives with Respect to y

Next, take the partial derivative of \(\frac{\partial f}{\partial y} = x e^{xy}\) with respect to \(y\) again:\[\frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}(x e^{xy}) = x^2 e^{xy}\]

Step 4: Compute Mixed Partial Derivative \(\frac{\partial^2 f}{\partial x \partial y}\)

Now compute the mixed partial derivative taking the first derivative with respect to \(x\) and the second with respect to \(y\):\[\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}(y e^{xy}) = e^{xy} + xy e^{xy}\]

Step 5: Compute Mixed Partial Derivative \(\frac{\partial^2 f}{\partial y \partial x}\)

And compute the mixed partial derivative taking the first derivative with respect to \(y\) and the second with respect to \(x\):\[\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}(x e^{xy}) = e^{xy} + xy e^{xy}\]

Step 6: Confirm Mixed Partials Are Equal

Finally, compare the two mixed partial derivatives:\[\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} = e^{xy} + xy e^{xy}\]Both mixed partial derivatives are equal, confirming the symmetry of second-order mixed partials.

Key Concepts

First Partial Derivatives

Understanding first partial derivatives is crucial for tackling problems with several variables. The first partial derivative of a function measures how the function changes as one of the variables changes, while keeping the other variables constant. Let's break this down with an example:
Consider the function given by \(f = e^{xy}\). When we find the partial derivative with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\), we treat \(y\) as if it were a constant. This yields:

• \( \frac{\partial f}{\partial x} = y e^{xy} \)

Similarly, for the partial derivative with respect to \(y\), represented as \(\frac{\partial f}{\partial y}\), we consider \(x\) as a constant, resulting in:

• \( \frac{\partial f}{\partial y} = x e^{xy} \)

These derivatives show how the function \(f\) changes along the axis of each variable independently. This foundational concept helps in developing second-order derivatives.

Mixed Partial Derivatives

Mixed partial derivatives are derivatives of functions involving more than one independent variable, which are differentiated with respect to different variables successively. They are termed "mixed" because they combine two different variables in their differentiation process. For the function \(f = e^{xy}\), mixed partial derivatives involve taking the derivative first with respect to one variable and then taking the second derivative with respect to another variable.
Let's see this in action:

• First, differentiate with respect to \(x\) and then \(( y \) again: \(\frac{\partial^2 f}{\partial y \partial x} = e^{xy} + xy e^{xy}\).
• Similarly, if you swap the order and differentiate first with respect to \(y\) and then \(x\), you get: \(\frac{\partial^2 f}{\partial x \partial y} = e^{xy} + xy e^{xy}\).

Both these mixed partial derivatives confirm the same result, leading us to explore the concept of symmetry in partial derivatives.

Symmetry of Partial Derivatives

The symmetry of partial derivatives, also known as Clairaut's theorem, states that for most functions that are continuous and have continuous second derivatives, the mixed partial derivatives are equal regardless of the order in which the differentiation is performed. In simpler terms, for the function \(f(x, y)\), it holds that:

• \(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \).

This property was aptly demonstrated in the solution where we found:

• Both mixed derivatives for \(f = e^{xy}\) were equal to \(e^{xy} + xy e^{xy}\).

The equality of these mixed partials is essential in mathematical analysis and the theoretical foundations of multivariable calculus. It allows us to interchange the order of differentiation without affecting the final outcome.