If the idea behind a normal derivative might be describe as the amount one needs to ad to move forward. If so, then the idea behind a geometric derivative is the amount one needs to multiply by.
The definition of a geometric derivative is
One thing one might wonder is whether we can use them to approximate functions, like we can use derivatives to approximate functions with Taylor series. A quick Google search didn’t reveal anything, but it isn’t too hard to figure out.
Where as Taylor series are a sum of (which have their nth derivative as the constant a) our approximation should be a product with terms that have analogous properties regarding geometric derivatives (ie. the nth geometric derivative is a constant).
Since , . By induction, .
So, our approximation at 0 would be where . The approximation about a point y will be .
As a test, let’s try approximating about $1$:
Update: In my discussion with George G. bellow, I realised that we can understand the geometric Taylor Series as the exponentiated Taylor series of the logarithm of the original function…