Is it possible to define the operation D of differentiation of real functions in an abstract way, as for example by the fundamental properties of the derivative:
D(f+g) = D(f) + D(g)
D(fg) = fD(g) + gD(f)
if f(x) = x, D(f) = 1 (to avoid the trivial D(f) = 0).
So if the above conditions hold, and assuming we are in the set of real differentiable functions, does it follow that D(f) is the known derivative of f?
Possibly more assumptions are required, so i'm asking for guidance.
It would be amazing if one could define differentiation in such an algebraic way.
Also possibly do the same for other operations, like integration and exponentiation.
Answer
Yes it is a standard argument in differential geometry that these rules uniquely specify the derivative. Indeed, fix a differentiable function $f$ and a point $p$.
By Taylor's theorem, we have $f(x)=f(p)+(x-p)f'(p)+(x-p)h(x)$ where $h(p)=0$. Then by applying the axioms in your question, it is simple to see that $(Df)(p)=f'(p)$ as desired.
Also, note that this is not a definition of the derivative, since simply writing down a list of properties does not guarantee that there actually exists an operator with those properties.
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