## A Product Calculus

The derivative $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ can be thought of as a measure of how much $f(x)$ changes relative to changes in $x$.  But why measure the relationship between the rates of change this way?  Why not a different definition of the derivative that compares changes another way?  For instance, suppose we change subtraction to division and division to taking roots in the definition of $f'(x)$ to get

$\displaystyle f^*(x) = \lim_{\Delta x \to 0} \left(\frac{f(x + \Delta x)}{f(x)}\right)^{1/\Delta x}.$

It turns out that this definition of the derivative leads to something we might call “product calculus.”  This product calculus bears the same relationship to multiplication and exponentiation that the usual calculus does to addition and multiplication, respectively.  It also has its own set of differentiation and integration rules, a fundamental theorem, mean value theorem, l’Hôpital’s rule, Taylor series, and applications, all of which parallel those of the usual calculus.  Several years ago I wrote up a paper on this product calculus, intending it as a survey appropriate for some undergraduate-focused expository journal.  I also had a student do a summer research project picking up where my paper left off.  The ideas weren’t new (they go back at least 100 years to Vito Volterra), but I thought that an exposition on the subject might be worthwhile.  After I wrote the paper, though, I discovered some surveys by others and so gave up trying to publish it.  (See, for example,  Bashirov, Kurpınar, and Özyapıcı [1], Grossman and Katz [2], and Stanley [3], as well as the Wikipedia article on the product integral.)

Here’s a summary of some of my favorite results and applications.

1. The product derivative can be represented in terms of the usual derivative via

$\displaystyle f^*(x) = \exp\left(\frac{d}{dx}\ln |f(x)|\right) = e^{f'(x)/f(x)}.$

This means that the product derivative is just $exp$ of the logarithmic derivative, which I think partly explains its lack of popularity.  (In this sense, is it really anything new?)

2. The derivative of the product is now the product of the derivative: $(fg)^* = f^* g^*$.  (This was the basis of my answer to the math.SE question “When do the freshman’s dream product and quotient rules for differentiation hold?”)

3. The product integral gives the geometric mean of a function over a continuous interval via $\displaystyle (\mathscr{P}_a^b f(x)^{dx})^{1/(b-a)}.$  (This was the basis of  my answer to the math.SE question “What is to geometric mean as integration is to arithmetic mean?“)

4. The product integral gives the future value of an amount over time with a variable growth rate.  The value of $A$ after $T$ time units in which the growth rate is $r(t)$ is $\displaystyle A \mathscr{P}_0^T \left(e^{r(t)}\right)^{dt}.$

For more applications, see my survey paper or the references below.

References

1. Agamirza E. Bashirov, Emine Mısırlı Kurpınar, Ali Özyapıcı, “Multiplicative calculus and its applications,” Journal of Mathematical Analysis and Applications, 337 (1): 36-48, 2008.
2. Michael Grossman and Robert Katz, Non-Newtonian Calculus, Lee Press, 1972.
3. Dick Stanley, “A multiplicative calculus,” Primus 9 (4): 310-326, 1999.