The chain and substitution rules of interval-valued intuitionistic fuzzy calculus

The interval-valued intuitionistic fuzzy set proposed by Atanassov is the extension of intuitionistic fuzzy set. It extends the membership degree and non-membership to interval values instead of a single value. So it contains more possible values and maybe more considerate. Among all the researches, the exploration on the calculus of interval-valued intuitionistic fuzzy set is entirely new. Recently, Zhao et al. (Int J Comput Intell Syst 9:36–56, 2016) proposed the concept of interval-valued intuitionistic fuzzy function (IVIFF) and gave a calculation method of derivative and differential of IVIFF. Based on this work, in this paper, firstly, we utilize a new and easier method to express the derivative and differential of IVIFF. Secondly, we propose the chain rules of derivative and the form invariance of differential in the interval-valued intuitionistic fuzzy environment. In addition, some properties of the substation rules for interval-valued intuitionistic fuzzy indefinite integrals and definite integrals are also developed.