Talk:List of derivatives and integrals in alternative calculi

Based on what is now in the article's table, I think the "multiplicative" derivative and integrals are . ..
. . . are things I recognize from a high school project of mine in 1963. (These have without a doubt been rediscovered both before and after that.) One approach is to take the usual definitions of derivative and integral and replace addition with multiplication, and multiplication with exponentiation.

For the multiplicative derivative, instead of the limit as h → 0 of (f(x + h) - f(x)) / h one may take instead the limit of (f(x+h) / f(x))1/h. The multiplicative integral is the inverse operator of the multiplicative derivative.

One may show that under suitable conditions, these multiplicative derivative and integral operators are the same as conjugating the ordinary operators with the exponential function. The multiplicative integral, at least, has an application in the field of survival statistics.

An amusing corollary is that the definite multiplicative integral may be used to calculate the geometric mean of a function over an interval. For example, the geometric mean [of the function f(x) = x over the interval [a,b] where 0 < a < b] is equal to (1/e)(bb/aa)1/(b-a), and so in particular the limiting case of the geometric mean of the numbers in the unit interval (0,1] is 1/e.Daqu (talk) 07:35, 11 March 2010 (UTC)


 * Er, um, I just discovered there is an entire article Multiplicative calculus that seems to explain some of these things fairly well.Daqu (talk) 08:04, 11 March 2010 (UTC)

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