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Multivariable calculus extends the concepts of single variable calculus to higher dimensions. It involves functions of two or more variables, like the example function f(x, y) = 2xy. In this field of mathematics, we explore the landscape of surfaces and curves that exist in higher-dimensional spaces.

When dealing with functions of multiple variables, the graphical interpretations become more complex than in single-variable calculus. Yet, the underlying intuition is similar: while derivatives in single-variable calculus represent the slope of the tangent line to the curve, in multivariable calculus, partial derivatives describe the slope of the tangent plane to the surface, along the axis of interest.

Understanding how to operate within this expanded universe is essential for fields such as physics, engineering, economics, and more, where multivariable functions naturally arise.

Partial differentiation is a fundamental technique in multivariable calculus used to investigate how a multivariable function changes as one variable varies while the others are held constant. It gives us the rate of change of the function with respect to each variable independently.

For instance, in the sample problem where f(x, y) = 2xy, to find the partial derivative with respect to x, we consider y to be a constant. This gives us f_x(x, y) = 2y. Similarly, to obtain the partial derivative with respect to y, we treat x as constant, resulting in f_y(x, y) = 2x.

These partial derivatives can convey information about the function's behavior along the respective axes, which is invaluable in optimizing processes, predicting outcomes and understanding complex systems.

The power rule is an essential differentiation technique from single-variable calculus that also plays a vital role in partial differentiation for multivariable calculus.

When applying the power rule to a function in the form f(x) = xⁿ, where n is any real number, the derivative is given by f'(x) = nx^n-1. But in the context of partial derivatives, we use the power rule in a slightly different manner.

As shown in the step-by-step solution, for the function f(x, y) = 2xy, we apply the power rule on one variable at a time, treating the others as constants. This simple yet powerful tool enables us to compute the rate of change with respect to each variable quickly and can be invaluable when working on problems that require the handling of multiple varying quantities simultaneously.