Partial Derivatives and Simplification of Complex Functions

Partial derivatives are a core concept in multivariate calculus, allowing us to understand the rate of change of a function with respect to a specific variable while holding other variables constant. This article provides a detailed exploration of the partial derivatives of a specific function, z xy / (x^2y^2^2). We will walk through the step-by-step process of computing the partial derivatives with respect to x and y, discussing the intricacies of working with exponents and simplifying complex expressions.

Introduction to Partial Derivatives

A partial derivative is a derivative taken with respect to one variable out of a function of multiple variables while treating all other variables as constants. In the context of the function z xy / (x^2y^2^2), we will evaluate the rate of change of z with respect to x and y. This process is crucial for understanding and optimizing functions, especially in fields such as physics, engineering, and economics.

Function Analysis: z xy / (x^2y^2^2)

The given function is z xy / (x^2y^2^2). At first glance, this expression might appear complex due to the high exponents. However, by simplifying the function, we can make the partial derivatives more manageable.

Partial Derivative with Respect to x

To find the partial derivative of z with respect to x, we need to treat y as a constant. Let's begin by simplifying the original function first:

[ z frac{xy}{x^2y^2} cdot frac{1}{y^2} frac{y}{x^2y^2} cdot frac{x}{y^2} frac{y}{xy^2} frac{1}{xy} ]

Now, we derive z with respect to x using the rule for the derivative of a rational function:

[ z_x frac{partial}{partial x} left( frac{1}{xy} right) ]

Applying the quotient rule, which states that if u/v is a function, then:

[ frac{partial}{partial x} left( frac{u}{v} right) frac{v frac{partial u}{partial x} - u frac{partial v}{partial x}}{v^2} ]

Let u 1 and v xy, then u_x 0 and v_x y.

[ z_x frac{(xy) cdot 0 - 1 cdot y}{(xy)^2} -frac{y}{x^2y^2} -frac{1}{x^2y} ]

Partial Derivative with Respect to y

Next, let's find the partial derivative of z with respect to y, treating x as a constant:

[ z_y frac{partial}{partial y} left( frac{1}{xy} right) ]

Again, applying the quotient rule:

[ z_y frac{(xy) cdot 0 - 1 cdot x}{(xy)^2} -frac{x}{x^2y^2} -frac{1}{xy^2} ]

Simplifications and Applications

The process of finding the partial derivatives of a complex function involves several steps, such as simplification, application of derivative rules, and careful manipulation of exponents. By simplifying the original function to z 1 / xy, we made the computation of the partial derivatives more straightforward and understood the behavior of the function with respect to each variable.

Conclusion

Understanding and computing partial derivatives, especially for complex functions, is essential in many fields of science and engineering. The ability to simplify expressions and apply appropriate rules is critical for these computations. In this article, we demonstrated the process of finding the partial derivatives of the function z xy / (x^2y^2^2), showing how simplification and application of derivative rules can lead to clearer and more manageable expressions.

References and Further Reading

For a deeper understanding of partial derivatives and related topics, consider exploring the following resources:

Apostol, T. M. (1967). Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wiley. Hass, J., Heil, C., Watson, G. (2016). University Calculus: Early Transcendentals (4th Edition). Pearson. Roger, T. (2013). Calculus: Early Transcendentals (2nd Edition). W. H. Freeman.