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In multivariable calculus, partial derivatives help us understand how a function changes with respect to one of its variables when the other variables are kept constant. Imagine you have a multivariable function, like the given function \(g(u, v) = f(x(u, v), y(u, v))\). To find how \(g\) changes with \(u\), we consider \(v\) to be constant. This is called a partial derivative with respect to \(u\). By applying the chain rule, you reach:

• \(\frac{\partial g}{\partial u} = \frac{\partial f}{\partial x}\cdot \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\cdot \frac{\partial y}{\partial u}\)

Breaking down each term, \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) are partial derivatives of \(f\) with respect to \(x\) and \(y\), respectively. The symbols \(\frac{\partial x}{\partial u}\) and \(\frac{\partial y}{\partial u}\) signify how \(x\) and \(y\) change as \(u\) changes.
Partial derivatives are foundational for analyzing the behavior of functions in multidimensional spaces. They help us understand the intricate changes by focusing on one variable at a time.

A second-order derivative involves taking the derivative of a derivative. In this exercise, we're looking for a mixed second-order derivative, \(\frac{\partial^2 g}{\partial u \partial v}\). This derivative tells us how the rate of change of \(g\) with respect to \(u\) changes as \(v\) changes.
To calculate this:

• We start by computing the derivative \(\frac{\partial g}{\partial u}\) and then differentiate this result with respect to \(v\).
• This requires applying the chain rule again for each term obtained from \(\frac{\partial g}{\partial u}\).

Second-order derivatives provide deeper insights into a function's behavior. They reveal curvature and how changes in one variable affect changes in another. For instance, a mixed second-order derivative like this facilitates understanding hidden interdependencies.

Multivariable calculus extends calculus concepts into more than one dimension, allowing for the exploration of functions with multiple variables. In the context of this problem, we focus on a function \(g(u, v)\) that depends on intermediary functions \(x(u, v)\) and \(y(u, v)\), which, in turn, are inputs to another function \(f\).
This setup highlights why tools like the chain rule are crucial. They enable us to connect rates of changes across linked dimensions, uncovering how complex mappings interact. Such techniques are vital when designing models in physics and engineering, where several variables inherently interact.
Mastering multivariable calculus opens a world of possibilities, providing the language for describing the dynamics of real-world processes across various fields. The ability to take partial derivatives and second-order derivatives is just one way to analyze such multifaceted environments.