Understanding Partial Derivatives: A Guide to Multivariable Calculus

Partial derivatives are a crucial concept in multivariable calculus, which allows us to understand how the rate of change of a function varies with respect to one of its variables while keeping others constant. This guide will walk you through the process of finding the partial derivative of a function with respect to one variable, using the function $f_{xy}(x, y) frac{x - y}{x^2 y}$ as an example.

Step-by-Step Calculation of Partial Derivative

Given the function:

$f_{xy}(x, y) frac{x - y}{x^2 y}$

Our goal is to find the partial derivative of this function with respect to $y$. Here’s the step-by-step approach:

Step 1: Understanding the Definition

The partial derivative of $f_{xy}(x, y)$ with respect to $y$ can be defined using the limit:

$frac{partial f}{partial y} lim_{k to 0} frac{f_{xy} (x, y k) - f_{xy} (x, y)}{k}$

Step 2: Applying the Function and Transformation

To use the limit definition, we need to substitute $k$ into the function:

$f_{xy}(x, y k) frac{x - (y k)}{x^2 (y k)} frac{x - y - k}{x^2 (y k)} frac{x - y - k}{x^2 y x^2 k}$

Step 3: Simplifying the Expression

Now, subtract the original function from the transformed one:

$f_{xy}(x, y k) - f_{xy}(x, y) frac{x - y - k}{x^2 y x^2 k} - frac{x - y}{x^2 y}$

Factor out the common denominator:

$ frac{(x - y - k) - (x - y)}{x^2 y (1 frac{k}{y})} frac{-k}{x^2 y (1 frac{k}{y})}$

Now, take the limit as $k to 0$:

$frac{partial f}{partial y} lim_{k to 0} frac{-k}{x^2 y (1 frac{k}{y})} frac{-3x}{x^2 y^2}$

Step 4: Final Simplification

The final result is:

$frac{partial f}{partial y} frac{-3x}{x^2 y^2}$

Why Do We Use Partial Derivatives?

Partial derivatives are essential in understanding the behavior of functions with multiple variables. They provide insights into how small changes in one variable affect the function while holding others constant. This concept is widely applicable in various fields, including physics, engineering, and economics.

For example, in physics, partial derivatives are used to describe the relationship between different physical quantities in multidimensional space. In economics, they are used to analyze the impact of changes in variables like cost, price, and demand on the overall function.

Understanding the Limit Definition of Partial Derivatives

The limit definition of partial derivatives helps us to understand these concepts more precisely. By keeping one variable constant and letting the other vary, we can determine the rate of change of the function with respect to that variable. This process is similar to the definition of the derivative in single-variable calculus but applied to multivariable functions.

The process of finding the partial derivative can be summarized as:

Replace $y$ with $y k$ in the function. Subtract the original function from the transformed one. Divide by $k$ and take the limit as $k to 0$.

Conclusion

In summary, understanding partial derivatives is a fundamental skill in multivariable calculus. By following the defined steps, we can accurately determine how a function changes with respect to one of its variables. This knowledge plays a critical role in various applications across different disciplines, making it a valuable tool for students and professionals alike.