This chapter discusses the construction of rational cubic Ball (cubic/quadratic) which consists of three parameters with C1 continuity. These three parameters can be used to refine the interpolating curve as well as to obtain the best approximation in error estimation. This scheme was tested by interpolate data from the true function. To improve the accuracy, a new method to estimate the first derivative is proposed as it is significant in curve interpolation. Based on error measurement such as Maximum Absolute Error (emax ), Coefficient of Determination (R2), Root Mean Square Error (RMSE ) and Central Processing Unit (CPU) time in second, the proposed scheme is better than existing schemes. Notably, we achieved emax= 0.01331105, R2= 0.91398405, RMSE= 0.00448758 and CPU time = 0.00009958 by using the new derivative estimation. Thus, the proposed scheme with construction of a new derivative estimation has improved the existing rational interpolant. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.