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In mathematics, we often encounter functions that depend on more than one variable. These are called functions of several variables. A simple real-life example is the temperature at different points of the Earth's surface, which depends on variables such as latitude and longitude.

Functions of several variables can be expressed as \(f(x, y, z)\), where \(x, y, z\) are independent variables. For the given exercise, \(f(x, y, z) = xy + yz + xz\). This expression shows how the output depends on the combination of its input variable values.

Working with such functions is crucial in fields like physics, economics, and engineering because it allows us to model complex phenomena involving multiple inputs.

Multivariable calculus extends the principles of single-variable calculus to functions that depend on multiple variables. It helps us analyze how variables interact with one another. For instance, in our function \(f(x, y, z) = xy + yz + xz\), each part of the equation relates each variable to another.

The study of multivariable calculus involves several concepts, including partial derivatives, gradients, and multiple integrations. We focus on partial derivatives here, which show how changing one variable affects the function, while keeping others constant. This analysis is key in optimizing functions to find maxima or minima, critical in economics and engineering problems.

Understanding these relationships helps in predicting and optimizing complex systems, such as climate models or financial forecasts.

Differentiation is a process that finds the rate at which a function changes with respect to one of its variables. In multivariable functions, when we differentiate with respect to a particular variable, it's known as taking a partial derivative.

For example, in the problem's function \(f(x, y, z) = xy + yz + xz\), finding \(f_x\) means differentiating with respect to \(x\) while treating \(y\) and \(z\) as constants. Therefore, \(f_x = y + z\). Similarly, \(f_y = x + z\) and \(f_z = y + x\).

This process is invaluable because it lets us isolate and understand the influence of individual factors on a system. This insight is particularly useful in fields that depend on extensive data analysis, like machine learning and statistics.