This paper aims to develop a novel continuous exact differentiator to determine the first-order derivative of a given signal in case of unknown upper-bounded second-order derivative with a simple tuning process. In addition, an improved convergence time and estimation accuracy can be successfully attained by employing innovative differentiator-generating functions. Furthermore, the stability of the proposed differentiator is theoretically verified through the Lyapunov theory. Finally, several numerical simulations are conducted to demonstrate the advantage of the suggested approach over existing reference methods in the literature.