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When we deal with functions that depend on more than one variable, we often need to understand how the function changes with respect to one variable while keeping the others constant. This is where the concept of partial derivatives comes into play. Partial derivatives are crucial for optimizing functions in economics, physics, and engineering where variables often interact with each other.

For example, in the exercise given, the function for problem (a) is a multivariable function of three variables, \(x, y, z\). To find how this function changes as we adjust only the variable \(x\) while holding \(y\) and \(z\) fixed, we calculate the partial derivative with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\).

By calculating \(\frac{\partial f}{\partial x} = 2xyz\), we have found how the output of the function changes in response to small changes in \(x\) along a plane where \(y\) and \(z\) are constant. Similarly, the partial derivatives with respect to \(y\) and \(z\) show the change along planes where the other two variables are held constant.

Multivariable calculus extends the concepts of single-variable calculus to functions with more than one input. Two key aspects of multivariable calculus are the understanding of curves and surfaces in three dimensions, and how functions change over these shapes.

In the world of multivariable functions, not only do we have curves, but we also have surfaces and even higher-dimensional analogs. These added dimensions make the calculus significantly richer (and a bit more complicated). For instance, while a single-variable function has only one derivative, a function of three variables, such as \(f(x, y, z) = x^{2}yz\), has three partial derivatives - one for each variable.

The calculation of the total differential, as practiced in the provided exercise, is a fundamental operation in multivariable calculus. It sums the effects of changes in all variables, giving us a linear approximation of the function's change around a point.

The differential of a function represents how a function's output \(f\) changes as its inputs change slightly, which is a cornerstone concept in calculus. In single-variable calculus, the differential \(df\) gives us the slope of the tangent to the function's curve at a given point, suggesting how \(f\) changes over an infinitesimal interval around that point.

However, when we're looking at a function of several variables, such as in our examples \(f(x, y, z)\) and \(f(x, y)\), the differentials must consider changes along each input variable. The total differential combines the partial derivatives with respect to each variable to provide a best linear approximation of the function around a given point. It tells us how the function value would change in response to small changes in the input variables, which is especially useful when evaluating the sensitivity of systems in fields like engineering and economics.