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Understanding the chain rule is fundamental in calculus, especially when dealing with compound functions, where one function is nested within another. Imagine a function 'h' that is composed of two functions 'f' and 'g', such that it can be written as h(x) = f(g(x)). To find the derivative of h with respect to x, the chain rule states that you must first find the derivative of the outer function f at g(x), and then multiply it by the derivative of the inner function g at x.

In our exercise, when finding the partial derivative of \(z = 0.5 \ln (xy)\) with respect to 'x', we treat 'y' as a constant and apply the chain rule to find the derivative of the logarithm function with respect to 'x'. The presence of \(y\) as part of the inner function doesn't affect its derivative with respect to 'x'. It's important to remember that the differentiation process here is akin to peeling layers one by one, treating all variables not under differentiation as constants at each step.

When you encounter logarithmic functions during differentiation, logarithmic differentiation provides a convenient method. It simplifies complex expressions and assists with implicit differentiation of functions. The natural logarithm \(\ln(x)\) has a derivative of \(\frac{1}{x}\) with respect to 'x', which can greatly simplify the derivative of functions that are products, quotients, or powers.

In our exercise, logarithmic differentiation is used to differentiate the function \(z = \ln \sqrt{xy}\). By converting the square root to a power of one-half and then using properties of logarithms to simplify, the derivative becomes more manageable. This illustrates how logarithmic differentiation transforms the initial complex expression into one where traditional derivative rules, such as the chain rule, can be applied more directly and effectively.

Multivariable calculus extends the concepts of single variable calculus to functions with more than one variable. A cornerstone of this field is the concept of partial derivatives, which measure how a function changes as only one of the variables is allowed to vary while the others are held constant.

The application of partial derivatives can be seen in our exercise when finding \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). The function \(z\), which depends on two variables 'x' and 'y', changes differently with respect to each variable. While finding the partial derivative with respect to 'x', 'y' is treated as a constant, and vice versa. This separation into individual derivatives with respect to each variable is what makes multivariable calculus a powerful tool in modeling and understanding systems with multiple influencing factors.

Implementing our function from the exercise using partial derivatives, allows for easy analysis and computation of changes in 'z' with respect to 'x' or 'y' individually. This emphasizes the importance of considering each variable's specific impact within a multivariable function.