A mixed partial derivative is the second derivative of a function with respect to two different variables, taken in succession. This derivative provides insight into how the function changes when varying one variable while holding another variable constant, and then observing how that change affects the first variable. Mixed partial derivatives are crucial for understanding the behavior of functions of multiple variables and are often denoted as \( f_{xy} \) or \( \frac{\partial^2 f}{\partial y \partial x} \).
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