I have been attempting to understand my math education (as a bachelor in electrical engineering) from a more algebraic perspective recently. I would like to understand more about the link between exponentiation and taking the derivative.

(From Wikipedia) given an algebra $A$ over a ring or field $K(+,\cdot,0_K,1_K)$, consider a $K$-linear map $D : A \rightarrow A$ that satisfies

$$D(a\cdot b) = a\cdot D(b) + D(a)\cdot b$$

Consider also (from Wikipedia) a homomorphism $E$ from the additive group of $K$ to the multiplicative group of $K$ such that:

$$ E(0_K) = 1_K $$ $$ \forall a, b \in K_+ \mid E(a + b) = E(a) \cdot E(b) $$

It seems that the only homomorphism $E$ that satisfies the above for the ring of integers is $ n \mapsto ( -1 )^n $. Let's consider the ring on $\mathbb{Q}$ instead.

Is there any nice way to understand the idea that

$$ \exists E \mid \exists c \in K \mid D ( E(x) ) = c \cdot E(x)$$

and that that homomorphism is the natural exponential function?

It seems to me that the homomorphism $E$ does a good job of capturing the concept of exponentiation. I have a bit of trouble seeing how the map $D$ encodes the idea of a derivative, but that's okay; I think I can work it out with some more reading and thought.

I am mainly curious if we can have this relation between the natural exponential $e$ and the derivative without invoking the idea of a power series. I'll admit, I'm not very opposed to the idea of using power series over the field of $\mathbb{Q}$, but I would like to invoke the reals in my thinking as little as possible.

Specifically I would like to understand functional analysis and signals from as much an algebraic point of view as possible, relying as little as possible on topological/real notions. Most of the spaces I think with have a metric which induces a measure and topology, but for my own sanity and symbol pushing, I would like to understand these things from as algebraic a position as possible.

Also, if there are any resources for understanding signals and electromagnetism from an algebraic perspective, that would be greatly appreciated.