We propose an adaptive scheme which is a counterpart of existing high gain control techniques based on control Lyapunov functions. Given a control Lyapunov function, the main idea is that of tuning the feedback gain according to a suitably-chosen Lyapunov time-derivative. The control gain is not monotonically non-decreasing as in existing techniques, but it is increased or decreased depending on the imposed derivative, thus avoiding the well-known issue of actuator over-exploitation. We are able to show robust convergence of the proposed adaptive control scheme as well as other interesting properties. For instance, it is possible to guarantee an a-priori given upper bound for the transient mode of behavior during adaptation. Furthermore, if the control Lyapunov function is designed based on an optimal control problem, then the control action is nominally optimal, precisely it yields the optimal trajectory for any initial condition, if the actual plant matches the nominal system.