This note describes a Lyapunov-Razumikhin function with an indefinite derivative that can be used to determine the input-to-state stability (ISS) of a nonlinear time-delay system. First, an extended comparison principle for estimating the solution of a differential inequality with an upper bound that is not (semi)negative definite is presented. Then, that principle and a Lyapunov-Razumikhin function with an indefinite derivative are employed to derive an ISS criterion. Finally, two examples show the effectiveness of our method and its superiority over others.