Is there an alternative way to simplify derivative/integral rules for products or does there exist a continuous product analog to an integral? These answers are in the realm of multiplicative calculus, specifically geometric calculus, a non-Newtonian treatment of the matter where products play nice instead of sums. There are many applications in fields where multiplicative thinking and analysis of growth/decay is prevalent like economics, finance, epidemiology, and probability/statistics.
Timecodes:
0:00 - The multiplicative integral
11:50 - The multiplicative derivative
22:11 - Applications, arithmetic systems, and generalizations
Sources and other tidbits:
Multiplicative calculus and applications: https://www.sciencedirect.com/science/article/pii/S0022247X07003824
Product integral: https://en.wikipedia.org/wiki/Product_integral
Overview of geometric and bi-geometric calculus: https://www.statisticshowto.com/non-newtonian-calculus
List of derivatives/integrals in other calculi (including discrete): https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi
Tangent exponential curve Desmos: https://www.desmos.com/calculator/79dp828jnb
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