The partial derivative of a function \( z = f(x, y) \) with respect to one of its variables (either \( x \) or \( y \)) measures how the function \( z \) changes as the variable changes, while holding the other variable constant.
-
Partial Derivative with Respect to \( x \): The partial derivative of \( z \) with respect to \( x \) is denoted as \( \frac{\partial z}{\partial x} \) or \( f_x(x, y) \). It is defined as: \[ \frac{\partial z}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x} \] This expression gives the rate of change of \( z \) relative to \( x \) while keeping \( y \) fixed.
-
Partial Derivative with Respect to \( y \): Similarly, the partial derivative of \( z \) with respect to \( y \) is denoted as \( \frac{\partial z}{\partial y} \) or \( f_y(x, y) \). It is defined as: \[ \frac{\partial z}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y} \] This expression gives the rate of change of \( z \) relative to \( y \), while keeping \( x \) constant.
In summary, the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) provide insight into how the function \( z \) varies with respect to each variable independently.