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What degree of advanced mathematics is needed in modern society?
Basics of differential calculus (e.g. used in finding optimal values).
It is necessary to understand how rates of change work.
Calculus is important in understanding physical phenomena.
Calculus is important for solving optimization problems, which are at the heart of economics and other practical human endeavors.
Derivatives are a protagonist of the subject of (ordinary and partial) differential equations and dynamical systems, which, apart from their obvious applications to physics, can be used to model the evolution of many complex systems.
Specific domains will not need to much in depth knowledge on this topic. For example, a firefighter, a police-officer, or a stewardess don't need to know exactly how to use these things.