If \( z=f(x, y)=x^{2}+3 x y-y^{2} \), find \( d z \)

Given \[ z = f(x, y) = x^2 + 3xy - y^2, \] we compute the total differential \( dz \) as follows. 1. The total differential is given by \[ dz = \frac{\partial z}{\partial x} \,dx + \frac{\partial z}{\partial y} \,dy. \] 2. Compute the partial derivative with respect to \( x \): \[ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\left( x^2 + 3xy - y^2 \right) = 2x + 3y. \] 3. Compute the partial derivative with respect to \( y \): \[ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}\left( x^2 + 3xy - y^2 \right) = 3x - 2y. \] 4. Substitute the partial derivatives into the total differential formula: \[ dz = (2x + 3y) \,dx + (3x - 2y) \,dy. \] Thus, the total differential is \[ dz = (2x + 3y) \,dx + (3x - 2y) \,dy. \]