\( \frac{\partial z}{\partial x} \), where \( x y-z=4 \) \( \frac{\partial z}{\partial x}=\square \)

To find \( \frac{\partial z}{\partial x} \) from the equation \( x y - z = 4 \), we can firstly rearrange it to express \( z \) in terms of \( x \) and \( y \): \[ z = xy - 4 \] Now, we differentiate \( z \) with respect to \( x \): \[ \frac{\partial z}{\partial x} = y \] So, \( \frac{\partial z}{\partial x} = y \). Keep in mind that this result indicates how \( z \) changes with \( x \) when \( y \) is held constant. For further exploration, remember that this procedure uses the concept of partial differentiation, which is foundational in multivariable calculus and is widely used in fields such as physics, engineering, and economics, modeling how systems respond to changes in independent variables. Also, ensure you're comfortable with implicit differentiation, especially when dealing with functions where variables are intertwined. Many students mistakenly forget to treat other variables as constants when performing partial derivatives, which can lead to incorrect results!