Find the total differential. \[ z=x \cos (y)-y \cos (x) \]

To find the total differential \( dz \) of the function \( z = x \cos(y) - y \cos(x) \), we first need to compute the partial derivatives with respect to \( x \) and \( y \). 1. **Calculate \( \frac{\partial z}{\partial x} \)**: \[ \frac{\partial z}{\partial x} = \cos(y) + y \sin(x) \] 2. **Calculate \( \frac{\partial z}{\partial y} \)**: \[ \frac{\partial z}{\partial y} = -x \sin(y) - \cos(x) \] Now, we can express the total differential \( dz \) using these partial derivatives: \[ dz = \frac{\partial z}{\partial x} \, dx + \frac{\partial z}{\partial y} \, dy \] Substituting the partial derivatives we calculated, we have: \[ dz = \left( \cos(y) + y \sin(x) \right) dx + \left( -x \sin(y) - \cos(x) \right) dy \] Therefore, the total differential of \( z \) is: \[ dz = \left( \cos(y) + y \sin(x) \right) dx + \left( -x \sin(y) - \cos(x) \right) dy \]