Dual numbers are just a means of formalizing the properties of an epsilon (a tiny change in calculus), and are the means of preserving enough information to think of the function and its derivative at the same time. EG: (x + ε)² = x² + 2ε + ε², but ε² = 0, so we get x² + 2ε, (a tiny change squared becomes an insignificantly tiny change)