1. Given the problem: Find the differential $dz$ for the function $$z = x^3 + x^2 y - x y^2 + 4 y^3.$$ 2. The formula for the total differential of a function $z = f(x,y)$ is: $$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy.$$ 3. Calculate the partial derivatives: - $$\frac{\partial z}{\partial x} = 3x^2 + 2xy - y^2$$ - $$\frac{\partial z}{\partial y} = x^2 - 2xy + 12 y^2$$ 4. Substitute these into the total differential formula: $$dz = (3x^2 + 2xy - y^2) dx + (x^2 - 2xy + 12 y^2) dy.$$ 5. This expression gives the infinitesimal change in $z$ in terms of changes in $x$ and $y$. Final answer: $$\boxed{dz = (3x^2 + 2xy - y^2) dx + (x^2 - 2xy + 12 y^2) dy}.$$