Which mathematical tool would you use to determine instantaneous rates of change?

The chosen answer, differential calculus, is the appropriate mathematical tool for determining instantaneous rates of change. This branch of calculus focuses specifically on the concept of derivatives, which measure how a function changes at a particular point. The derivative is defined as the limit of the average rate of change of a function over an interval as that interval approaches zero, effectively providing the instantaneous rate of change at any given point.

In the context of various mathematical tools, integral calculus primarily deals with the accumulation of quantities and finding areas under curves, rather than focusing on instantaneous changes. Linear algebra involves the study of vector spaces and linear transformations, and while it has many applications, it doesn't address rates of change directly. While the term "differentiate calculus" might suggest a related concept, it is not a standard term within mathematics; the correct term should be differential calculus, which emphasizes the process of differentiation used to find those instantaneous rates.