91Ó°ÊÓ

In calculus, the mixed partial derivatives provide insight into how a function changes with respect to two or more variables. When we talk about mixed partial derivatives, we're referring to the derivatives of a function taken first with respect to one variable and then with respect to another.
In the context of our exercise, consider the function \( f(x, y, z) = x^3 y z^2 \). We are interested in evaluating the mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \), also denoted as \( \frac{\partial^2 f}{\partial y \partial x} \).
By first differentiating with respect to \(x\), holding \(y\) and \(z\) constant, we obtain \( 3x^2 y z^2 \). Differentiating this result with respect to \(y\), while holding \(x\) and \(z\) constant, we arrive at \( 3x^2 z^2 \).

• This derivative indicates how the function \(f\) behaves as both \(x\) and \(y\) change while \(z\) remains fixed.
• As a polynomial function, \(f(x, y, z)\) ensures that the order of differentiation does not matter due to the Clairaut's Theorem, which means \( \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x} \).

This equality is valid since polynomial functions are typically continuous and well-behaved.

A polynomial function is a mathematical expression consisting of variables, coefficients, and exponents arranged in a specific manner. In our exercise, the function \( f(x, y, z) = x^3 y z^2 \) showcases these attributes with \(x, y,\) and \(z\) as variables, combined using multiplication and exponentiation.
These types of functions are incredibly important because they have properties that simplify calculus operations:

• They are differentiable everywhere in the domain of real numbers, meaning derivatives can be found without encountering undefined expressions.
• Polynomials are continuous, making it easier to apply calculus theorems like Clairaut's on mixed partial derivatives.
• The power rule is often used to differentiate polynomial terms, allowing for straightforward calculations.

As seen in this exercise, working with a polynomial function allows us to be confident that the mixed partial derivatives will produce consistent and reliable outcomes, thanks to their stable and predictable nature.

The process of finding derivatives, including mixed partial derivatives, involves a systematic approach where we treat certain variables as constants while differentiating with respect to others. This step-by-step progression is essential for tackling functions with multiple variables. Let's break down the process:

• **Step 1: Partial Derivative with Respect to a Single Variable** - Begin by selecting one variable to differentiate with respect to, treating all other variables as constants. For instance, we find \( \frac{\partial f}{\partial x} \) of \( f(x, y, z) = x^3 y z^2 \) by applying the power rule to obtain \( 3x^2 y z^2 \).
• **Step 2: Mixed Partial Derivative** - To find the next derivative with respect to another variable, differentiate the result of the first step, treating the initial variable as a constant. Here, doing this with respect to \(y\) gives \(3x^2 z^2\).
• **Step 3: Evaluate at Given Points or Limits** - Substitute specific values into the derived expressions to evaluate the derivative at particular points, like at \( P(0, 1, z) \) and \( (1, 1, 1) \).

This structured method not only ensures accuracy but also clarifies the behavior of a function as individual variables change, a fundamental aspect in multivariable calculus.