\( \frac{\partial z}{\partial x} \), 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 rearrange it to express \( z \) in terms of \( x \) and \( y \): \[ z = xy - 4. \] Now, we can differentiate both sides with respect to \( x \): \[ \frac{\partial z}{\partial x} = y. \] So, \( \frac{\partial z}{\partial x} = y \). This highlights how differentiating one variable can give insight into how another related variable behaves, creating a dynamic relationship in calculus! To double-check our work, you could use implicit differentiation on the original equation. By treating \( z \) as a function of both \( x \) and \( y \), you differentiate \( x y - z = 4 \) with respect to \( x \), arriving at a similar conclusion. It's a great way to validate your findings and reinforce your understanding of partial derivatives!