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Partial differentiation is a fundamental technique in calculus when dealing with functions of multiple variables. Unlike ordinary differentiation, which deals with functions of a single variable, partial differentiation works with functions that have two or more variables. To differentiate such functions partially, we hold all but one variable constant and find the derivative with respect to the variable of interest.

For example, consider the function from the exercise, \(f(x, y) = xe^y\). To find the partial derivative of \(f\) with respect to \(x\), \(f_x\), we treat \(y\) as a constant. Similarly, to find the partial derivative with respect to \(y\), \(f_y\), we treat \(x\) as constant. These operations are crucial as they allow us to understand how the function's value changes as one variable changes while the others remain fixed.

Understanding the concept of variables being 'held constant' helps avoid errors in computations, and provides clarity in the interpretation of the derivatives as rates of change in various directions in the space of the function's variables.

Multivariable calculus extends the tools of single-variable calculus to functions of several variables. This field is concerned with partial differentiation as well as multiple integration, gradient vectors, directional derivatives, and more. It provides the mathematical foundation for working with quantities that depend on more than one variable and is essential in many areas of science and engineering.

In our example with \(f(x, y) = xe^y\), we are applying multivariable calculus techniques to examine how the function changes locally with respect to each variable individually. This is just the beginning: multivariable calculus also addresses how these changes interact and provides ways to optimize functions over regions defined by multiple variables.

With multivariable calculus, we can describe how a system changes in a multi-dimensional space, predict outcomes, and solve practical problems that have numerous interacting factors.

The second derivative test is a criterion used in calculus to determine local extrema (minima and maxima) of a function. In the context of multivariable calculus, the second derivative test can help determine whether a critical point is a local minimum, local maximum, or saddle point.

To apply the second derivative test to functions of two variables such as \(f(x, y)\), we compute the second derivatives \(f_{xx}\), \(f_{yy}\), and the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\). These are used to calculate the discriminant \(D = f_{xx}f_{yy} - (f_{xy})^2\). If \(D > 0\) and \(f_{xx} > 0\), there's a local minimum; if \(D > 0\) and \(f_{xx} < 0\), a local maximum; and if \(D < 0\), the point is a saddle point.

The second derivative test gives us a powerful way to understand the curvature and behavior of a surface near critical points without graphing the entire function, which can be especially challenging for complex functions.

Clairaut's theorem, also known as the equality of mixed partial derivatives, is an important result in higher-dimensional calculus. The theorem states that if a multivariable function has continuous second partial derivatives at a point, then the mixed partial derivatives at that point are equal.

To illustrate with our exercise, after finding \(f_{xy} = e^y\) and \(f_{yx} = e^y\), Clairaut's theorem confirms that they should be equal if the function's mixed partials are continuous. This is a crucial verification step because it ensures that the order of differentiation does not affect the outcome of mixed partial derivatives.

Clairaut's theorem helps assure us that the multi-dimensional surfaces represented by these functions behave predictably under differentiation, allowing mathematicians and scientists to approach complex problems with multiple variables with more confidence.