This paper establishes exponential convergence to zero, rather than to a residual neighborhood, for a first-order serial differentiator and its higher-order cascade. If the acceleration of the input is locally absolutely continuous with known acceleration and jerk bounds, appropriate gain conditions yield an exponential estimate that is uniform on every prescribed initial energy sublevel. The Lyapunov function is the energy of the error system plus a quadratic correction that makes it strict. For a nominal input observed through a continuous perturbation with a persistent amplitude, a practical exponential estimate with residual holds, without any smallness condition, and exact exponential convergence is retained without perturbation. This amplitude-only transfer applies to the nonsmooth interstage error of a serial differentiator. With an exact first input it closes an exact exponential induction over any fixed finite number of stages under stagewise rate conditions.