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In the realm of calculus, multivariable calculus is an extension of single-variable calculus to higher dimensions. It involves functions that have more than one variable, like the example function, which depends on both variables, 'x' and 'y'. The goal in this branch of mathematics is to analyze changes in a function when there are multiple inputs. This is particularly useful in fields like physics, engineering, economics, and any other area that deals with systems influenced by multiple factors at once.

When dealing with such functions, the concept of partial derivatives comes into play. Unlike the standard derivative of a single-variable function, a partial derivative tells you how the function changes as one of the variables changes, while all other variables are held constant. Understanding how to compute and interpret partial derivatives is a foundational skill in multivariable calculus, as you learn not just how a function moves, but how it moves in multiple dimensions simultaneously. This multidimensional movement can be critical for optimizing functions, understanding physical laws, or even predicting economic trends when multiple variables are involved.

A commonly used technique in calculus is the power rule, which simplifies taking the derivative of a function that is a power of a variable. When you have a term like \(x^n\), where 'n' is any real number, the power rule states that the derivative of that term with respect to 'x' is \(nx^{n-1}\).

Applied to multivariable functions, the power rule must be paired with the understanding that the variable with respect to which you are differentiating must be treated differently than the others. So, in the context of partial derivatives for multivariable functions, if you have a function like \( f(x, y) = x^6 \), the partial derivative with respect to 'x' is found by applying the power rule directly, getting \( 6x^5 \), while the partial derivative with respect to 'y' is zero because the term does not depend on 'y'.

It’s essential to recognize that each term in the function must be differentiated separately, and the power rule simplifies this process greatly when terms of the function are in the form of a variable raised to a power.

Another fundamental concept in calculus is the sum rule. This rule is particularly straightforward: when finding the derivative of a function that is the sum of several terms, you simply take the derivative of each term individually and then sum those derivatives. The sum rule applies just as well to partial derivatives as it does to regular derivatives.

In our example with the function \(f(x, y) = y^8 + 2x^6 + 2xy\), we used the sum rule to break the problem into manageable parts. We differentiated \(y^8\), \(2x^6\), and \(2xy\) with respect to 'x' and 'y' separately. This made the process easier and more methodical, as opposed to trying to differentiate the entire function at once. By understanding and applying the sum rule, you can simplify complex derivative problems into a series of easier computations, which is a valuable technique not only in mathematics but also in breaking down complex problems in real-world applications.