Understanding the Partial Derivatives of the Function u yx

Introduction to Partial Derivatives:

Partial derivatives are a fundamental concept in multivariable calculus that allow us to understand how a function changes with respect to a single variable, while keeping the other variables constant. This is particularly useful in fields such as economics, physics, and engineering, where we often need to quantify the effects of changes in individual variables on a complex system.

What is the Partial Derivative of u yx?

The function u yx is a power function where the base is a variable and the exponent is another variable. To find the partial derivatives of this function, we use the rules of differentiation and logarithmic properties.

1. Partial Derivative with Respect to x:

To find frac{?u}{?x}, we treat y as a constant. The steps are as follows:

Take the natural logarithm of both sides: ln(u) x ln(y). Apply the differentiation rule: frac{1}{u} . frac{?u}{?x} ln(y) frac{?x}{?x}. Simplify to find frac{?u}{?x} u ln(y).

Substituting back, we have:

frac{?u}{?x} yx ln(y).

2. Partial Derivative with Respect to y:

To find frac{?u}{?y}, we treat x as a constant. The steps are as follows:

Again, take the natural logarithm of both sides: ln(u) x ln(y). Apply the differentiation rule: frac{1}{u} . frac{?u}{?y} x frac{1}{y}. Simplify to find frac{?u}{?y} x yx-1.

Substituting back, we have:

frac{?u}{?y} x yx-1.

3. Summary of Partial Derivatives:

The partial derivatives of the function u yx are:

frac{?u}{?x} yx ln(y) frac{?u}{?y} x yx-1

Simple Approach Using Logarithms

To simplify the process, we can use logarithms directly:

Take the natural logarithm of both sides: ln(u) x ln(y).

For the partial derivative with respect to x, treat y as a constant:

ln(u) x ln(y).

Differentiate both sides with respect to x: frac{1}{u} . frac{?u}{?x} ln(y).

Thus, frac{?u}{?x} yx ln(y).

For the partial derivative with respect to y, treat x as a constant:

ln(u) x ln(y).

Differentiate both sides with respect to y: frac{1}{u} . frac{?u}{?y} x frac{1}{y}.

Thus, frac{?u}{?y} x yx-1.

Understanding these partial derivatives is crucial for analyzing the behavior of functions that are not simple or one-dimensional. By keeping certain variables constant and varying others, we gain insights into the complex relationships between variables in real-world scenarios.

Key Points

frac{?u}{?x} yx ln(y) is the rate of change of u with respect to x when y is constant. frac{?u}{?y} x yx-1 is the rate of change of u with respect to y when x is constant.

Closure

In conclusion, the function u yx has specific partial derivatives that describe how the function changes with respect to each variable. These derivatives are essential for understanding the behavior of systems that involve multiple interdependent variables. By mastering these concepts, you can tackle more complex mathematical and real-world problems with confidence.