The goal of machine unlearning (MU) is to remove previously learned data from a model. However, existing MU techniques struggle to balance computational cost with performance. Inspired by the effects of label smoothing on model confidence and by differential privacy, this chapter introduces UGradSL, a simple, gradient-based MU method that applies an inverse form of label smoothing. UGradSL is plug-and-play and theoretically grounded, with analyses showing that appropriately applying label smoothing enhances MU effectiveness. Extensive experiments across datasets of varying sizes and modalities confirm its robustness and efficiency. Notably, UGradSL exhibits a strong connection to improvements in local differential privacy. It achieves consistent MU performance gains with only minimal additional computation. For example, UGradSL improves unlearning accuracy by 66% over the gradient ascent baseline without reducing efficiency. A self-adaptive version is also presented to enable straightforward parameter selection.