The partial derivative of a function \( z = f(x, y) \) with respect to one of its variables, such as \( x \) or \( y \), measures how the function \( z \) changes as that particular variable changes, while keeping the other variable constant.
To denote this, we use the notation:
- The partial derivative of \( z \) with respect to \( x \) is written as \( \frac{\partial z}{\partial x} \) or \( f_x \).
- The partial derivative of \( z \) with respect to \( y \) is written as \( \frac{\partial z}{\partial y} \) or \( f_y \).
Definition:
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Partial Derivative with respect to \( x \): \[ \frac{\partial z}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x} \]
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Partial Derivative with respect to \( y \): \[ \frac{\partial z}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) - f(x, y)}{\Delta y} \]
Interpretation:
- \( \frac{\partial z}{\partial x} \) gives the rate of change of \( z \) in the direction of \( x \) while \( y \) is held constant.
- \( \frac{\partial z}{\partial y} \) gives the rate of change of \( z \) in the direction of \( y \) while \( x \) is held constant.
Partial derivatives are fundamental in multivariable calculus and are extensively used in various fields, including physics, engineering, economics, and optimization.