Calculating Partial Derivatives and Differentials for u x^y y^x

Introduction

This article provides a detailed explanation of how to calculate the partial derivatives and differentials for the function ( u x^y y^x ). It covers both the process of implicit differentiation and the application of the chain rule. The page also includes examples to help illustrate the concepts.

Understanding the Function

Consider the function ( u x^y y^x ). To understand how to differentiate this function, we need to distinguish between partial derivatives and differentials.

Partial Derivatives

Partial Derivative with Respect to ( x )

To find the partial derivative of ( u ) with respect to ( x ), we treat ( y ) as a constant, and apply the chain rule:

The derivative of ( x^y ) with respect to ( x ) is ( y x^{y-1} x^y ln(x) cdot frac{dy}{dx} ).

The derivative of ( y^x ) with respect to ( x ) is ( y^x ln(y) y^x x ln(y) ).

Combining these, we get:

[frac{partial u}{partial x} y x^{y-1} x^y ln(y) y^x ln(x) y^x x ln(y) cdot frac{dy}{dx}]

Partial Derivative with Respect to ( y )

To find the partial derivative of ( u ) with respect to ( y ), we treat ( x ) as a constant:

The derivative of ( x^y ) with respect to ( y ) is ( x^y ln(x) ).

The derivative of ( y^x ) with respect to ( y ) is ( x y^{x-1} x y^x ln(y) ).

Combining these, we get:

[frac{partial u}{partial y} x^y ln(x) x y^{x-1} x y^x ln(y)]

Differentials

General Form of Differentials

The differential ( du ) is given by:

[frac{du}{dx} frac{partial u}{partial x} dx frac{partial u}{partial y} dy]

Thus, using the partial derivatives found earlier, we can express ( du ) as:

[frac{du}{dx} left(y x^{y-1} x^y ln(y) y^x ln(x) y^x x ln(y) cdot frac{dy}{dx}right) dx left(x^y ln(x) x y^{x-1} x y^x ln(y)right) dy]

Examples and Applications

Example 1: Finding ( frac{du}{dx} ) and ( frac{du}{dy} ) for ( u x^y y^x )

Let's go through the steps to find ( frac{du}{dx} ) and ( frac{du}{dy} ) for the function ( u x^y y^x ) using the chain rule and implicit differentiation.

Example 2: Derivative of ( y x^x )

Consider the function ( y x^x ). To find ( frac{dy}{dx} ), we use the natural logarithm and the chain rule:

(ln y ln x^x x ln x )

(frac{1}{y} frac{dy}{dx} ln x 1)

(frac{dy}{dx} y(1 ln x) x^x(1 ln x))

Using similar steps, we can find ( frac{du}{dx} ) and ( frac{du}{dy} ) for the function ( u x^y y^x ).

Summary

To summarize, the partial derivatives and differentials of the function ( u x^y y^x ) are:

[frac{partial u}{partial x} y x^{y-1} x^y ln(y) y^x ln(x) y^x x ln(y) cdot frac{dy}{dx}] [frac{partial u}{partial y} x^y ln(x) x y^{x-1} x y^x ln(y)] [frac{du}{dx} left(y x^{y-1} x^y ln(y) y^x ln(x) y^x x ln(y) cdot frac{dy}{dx}right) dx left(x^y ln(x) x y^{x-1} x y^x ln(y)right) dy]

Understanding these derivatives and differentials is crucial for solving more complex problems using the chain rule and implicit differentiation techniques.