Step 1: Understanding the Concept:
This problem involves calculating partial derivatives of a function of two variables and then substituting them into a given expression. Partial differentiation means differentiating with respect to one variable while treating all other variables as constants. Step 2: Key Formula or Approach:
1. Calculate the partial derivative of $z$ with respect to $x$, $\frac{\partial z}{\partial x}$.
2. Calculate the partial derivative of $z$ with respect to $y$, $\frac{\partial z}{\partial y}$.
3. Substitute these derivatives into the given expression and simplify. Step 3: Detailed Explanation:
The given function is $z = x^2 - y^2$.
1. Find the partial derivative with respect to $x$ (treating $y$ as a constant):
\[ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x - 0 = 2x \]
2. Find the partial derivative with respect to $y$ (treating $x$ as a constant):
\[ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}(x^2 - y^2) = 0 - 2y = -2y \]
3. Substitute these results into the expression $\frac{1}{x} \frac{\partial z}{\partial x} + \frac{1}{y} \frac{\partial z}{\partial y}$:
\[ \frac{1}{x}(2x) + \frac{1}{y}(-2y) \]
\[ = 2 - 2 = 0 \]
Step 4: Final Answer:
The value of the expression is 0. Therefore, option (C) is correct.