So I've recently came across this question, which is about defining the differential operator $D_x$ in terms of axioms rather than the traditional "taking the limit"-way. This made me think about a new operator, which I will denote with $E_x$ (for no definite reason), that has the following axiomatic definition ($f,g : \mathbb{R} \mapsto \mathbb{R}$ for the sake of simplicity):
$$E_x\{f \cdot g\}=E_x\{f\} \cdot E_x\{g\}$$
$$E_x\{f \times g\} = E_x\{f\} \times g \space \cdot \space E_x\{g\} \times f$$
$$E_x\{(f \circ \vec{g})(x)\} = \prod_{\forall g \in \vec{g}} \Big( E_g\{f\} \circ \vec{g} \Big)(x) \times E_x\{g\}$$
where
$$a \times b = \exp{(\ln{a} \cdot \ln{b})} = a^{\ln{b}}=b^{\ln{a}}$$
which is a commutative and associative binary operator and moreover it is distributive with multiplication (and higher in order):
$$ a \times (b \cdot c) = \exp{\Big(\ln{a} \cdot \ln{(b \cdot c)}\Big)} = \exp{\Big(\ln{a} \cdot \ln{b}\Big)} \cdot \exp{\Big(\ln{a} \cdot \ln{c}\Big)} = (a \times b) \cdot (a \times c)$$
So the first axiom is the "additive" linearity, the second is the Leibnitz-rule while the latter one is the chain rule. I have several questions regarding this new $E_x$ operator:
- Is this a viable axiomatic system or it breaks down somewhere? Does it already exist or it is a new concept?
- For what functions can it be applied to?
- Does it also have a geometrical representation like the slope of the function in the derivative's case?
- What happenes whet it is applied to the "basic" functions ($x^{n}$, $\sin$, $\cos$, $\exp$, etc...)?