I have tried to find references that are related to calculus on tropical semiring, but I was not able to find appropriate references.
So, I used to Thompson's approach to define deriviative on tropical semiring as follows:
$\;$ Let $ \forall x \in \mathbb{R} ^{*}, \, x<dx<+\infty $.
$\;$ For $ y = x^{\otimes n}$, where $n \in \mathbb{N} $,
$\;$ $ \frac{dy}{dx} := y \oplus dy = \left( x \oplus dx \right) ^{\otimes n} = x ^{\otimes n} \oplus \left( dx \right) ^{\otimes n} = x ^{\otimes n} $.
However, it is quite confusing that every polynomial and its derivative is the same.
Furthermore, this method is only can be done when the function is polynomial, like for $ x^{\otimes (-1)} $, I am not sure if Thompson's approach is appropriate.
In this case, how to define derivatives on tropical semiring?
And is there any references for this?
Thanks in advance!
