A new type of calculus called “multiplicative calculus” is developed and some basic theorems about derivatives, integrals, and infinite products are proved within this calculus. Multiplicative calculus is based on a multiplicative mechanism in the same sense that the usual calculus is based on an additive mechanism. One consequence is that, just as the usual derivative of a linear function is constant, the multiplicative derivative of an exponential function is constant (and just as the usual derivative of a constant function is 0, the multiplicative derivative of a constant function is 1). Similarly, just as two functions that have a constant difference have the same usual derivative, two functions that have a constant ratio have the same multiplicative derivative. Finally, just as many functions have infinite series representations based on the usual derivative, the same functions have infinite product representations based on the multiplicative derivative. These in turn are derived from exponential approximations to functions that are analogous to the linear approximations of the usual calculus. Multiplicative calculus is a useful supplement to the usual calculus in that it is tailored to situations involving exponential functions in the same sense that the usual calculus is tailored to situations involving linear functions.
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Dick Stanley (1999) studied this question.