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Abstract
Yager's $q$ -rung orthopair fuzzy set ( $q$ -ROFS) is a powerful tool to handle uncertainty and vagueness in real life. It expands the spatial scope of membership and nonmembership, and therefore has a wider range of constraints and stronger modeling capabilities. However, to date, there is no investigation for $q$ -rung orthopair fuzzy derivatives and differentials, which are very important for further developing $q$ -rung orthopair fuzzy calculus ( $q$ -ROFC). The basic elements of a $q$ -ROFS are $q$ -rung orthopair fuzzy numbers ( $q$ -ROFNs), based on which we propose the $q$ -rung orthopair fuzzy functions ( $q$ -ROFFs) and discuss their continuities in detail. Subsequently, we study the derivative of the $q$ -ROFF, which reveals an accurate description on rate of change for continuous $q$ -ROFF. Next, the differential operation of $q$ -ROFF is established; thereby providing an effective approximation on nonlinear problem in the $q$ -ROFF environment. Finally, we present numerical examples as explicit applications of $q$ -ROFC.
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