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Calculus is a branch of mathematics that studies continuous change. It is widely used in various scientific fields, including physics, engineering, and economics. One of the fundamental ideas in calculus is the concept of a derivative, which measures how a function changes as its input changes. This makes calculus a powerful tool for analyzing and predicting both simple and complex systems.

In problems involving functions of multiple variables, such as the function given in the exercise, calculus helps us understand how the output of the function responds to changes in each of its input variables. Calculus provides tools like partial derivatives to analyze these changes. Partial derivatives are crucial for handling functions with more than one variable and can help determine the function's behavior when each variable changes independently.

Differentiation is the process of finding a derivative, which tells us how a function is changing at any given point. In the context of functions with several variables, such as our function \( g(x, y)=\ln \sqrt{x^{2}+y^{2}} \), we can calculate how much the function changes as each variable changes. This process is known as taking partial derivatives.

To differentiate a function of two variables, we perform partial differentiation. This means we take the derivative of the function with respect to one variable while keeping the other variable constant. For instance, when finding the partial derivative with respect to \( x \), you treat \( y \) as a constant, and vice versa.

In our example, after simplifying the logarithmic function to \( \frac{1}{2} \ln (x^{2} + y^{2}) \), you apply differentiation rules to find the partial derivatives: \( g_x = \frac{x}{x^{2}+y^{2}} \) and \( g_y = \frac{y}{x^{2}+y^{2}} \). These expressions show how the function \( g(x, y) \) changes with respect to \( x \) and \( y \), respectively.

Logarithmic functions are the inverse of exponential functions, and they play a pivotal role in many areas of mathematics and science. A key property of logarithmic functions is their ability to transform multiplicative relationships into additive ones, which can simplify complex calculations.

In this exercise, the function \( g(x, y)=\ln \sqrt{x^{2}+y^{2}} \) involves a logarithmic expression. By leveraging the properties of logarithms, such as \( \ln \sqrt{a} = \frac{1}{2} \ln a \), we were able to simplify the expression to \( \frac{1}{2} \ln (x^{2} + y^{2}) \).

Logarithmic differentiation is particularly useful when dealing with products or quotients inside a logarithmic function. It allows for easier manipulation and differentiation of these expressions. In this case, the differentiation of \( \ln (x^{2} + y^{2}) \) involves using the chain rule, further illustrating the flexibility and efficiency provided by understanding logarithmic properties.