Understanding Partial Derivatives: A Detailed Guide

When tackling functions of multiple variables, partial derivatives play a crucial role in understanding how a function changes relative to individual variables. This article will delve into the concepts and techniques required to find partial derivatives, specifically for the function z arctan(xy) / (1 x^2).

What Are Partial Derivatives?

Partial derivatives are derivatives taken of a function of several variables, where the other variables are held constant. Essentially, they measure the rate of change of a function in one direction while keeping all other directions constant. This makes partial derivatives invaluable in fields like physics, engineering, and economics, where multi-dimensional functions are common.

How to Find Partial Derivatives

Let's break down the process of finding partial derivatives step by step. The function in question is:

z arctan(xy) / (1 x^2)

To find the partial derivatives with respect to x and y, we treat the other variable as a constant.

1. Differentiating with Respect to y

When differentiating with respect to y, we treat all other terms (including x) as constants. The function simplifies to a form where we can directly apply the derivative rules for arctan with respect to y.

The function can be written as:

z arctan(xy)

The derivative of arctan(u) with respect to u is 1 / (1 u^2). Applying the chain rule:

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

2. Differentiating with Respect to x

When differentiating with respect to x, we treat y as a constant. The function in question is:

z arctan(xy) / (1 x^2)

Here, we are dealing with a quotient rule. The quotient rule states that if we have f/g, then the derivative is:

[frac{d}{dx}(frac{f}{g}) frac{g cdot f' - f cdot g'}{g^2}]

In our case, let:

[f arctan(xy), quad g 1 x^2]

The derivative of arctan(u) with respect to u is 1 / (1 u^2). Applying the chain rule to arctan(xy):

[frac{partial (arctan(xy))}{partial x} frac{1}{1 (xy)^2} cdot frac{partial (xy)}{partial x} frac{1}{1 x^2y^2} cdot y frac{y}{1 x^2y^2}]

And the derivative of g with respect to x is:

[frac{partial (1 x^2)}{partial x} 2x]

Now, applying the quotient rule:

[frac{partial z}{partial x} frac{(1 x^2) cdot frac{y}{1 x^2y^2} - arctan(xy) cdot 2x}{(1 x^2)^2}]

Simplifying this expression:

[frac{partial z}{partial x} frac{y(1 x^2) - 2x arctan(xy)}{(1 x^2)^2(1 x^2y^2)}]

Conclusion

Mastering the technique of finding partial derivatives can greatly enhance your understanding of multi-variable functions and their applications in various fields. The process involves treating all other variables as constants and applying standard differentiation rules.

By following the examples and techniques discussed above, you can confidently find partial derivatives for various functions, providing a powerful tool for further mathematical analysis.