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If we have a field $F$ and a metric $d$ on $F$, given a function $f:F \rightarrow F$, we could define the derivative of the function at $x \in F$, as being the limit (if it exists) $$\lim_{h \rightarrow 0} (f(x+h)-f(x))h^{-1},$$ where a limit is defined using the usual $\epsilon-\delta$ definition, except with the Euclidean metric replaced by $d$.

From here some obvious things can be concluded such as if the field is finite, everything in the field could be said to be the derivative (taking $\delta$ small enough, the implication in the definition of a limit would be vacuously true). Similarly, the same holds if $d$ is the discrete metric.

My question is, is such a general notion of derivative useful? Has it already been defined and applied in some area that I'm unaware of? Etc.

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Obviously $\mathbb{F} = \mathbb{R}$ your definition gives us the usual real derivative at a point. For $\mathbb{F} = \mathbb{C}$ your definition gives us the usual complex derivative at a point.

If you were to try the skew-field of quaternions then there is trouble. For example,the quaternionic function $f(z)=z^2$ is not differentiable. Look at

A. Sudbery (1979) "Quaternionic Analysis", Mathematical Proceedings of the Cambridge Philosophical Society 85:199–225.

To gain some appreciation of the trouble.

Personally, I like to think of derivatives as Frechet derivatives when I can. We use the norm on the space to define differentiability. This is important as we extend differentiation for functions from $\mathbb{R}^m$ to $\mathbb{R}^n$

Recall that $f: \mathbb{R} \rightarrow \mathbb{R}$ has a derivative $f'(a)$ at $x=a$ if $$ f'(a) = \lim_{ h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ Alternatively, we can express the condition above as $$ \lim_{ h \rightarrow 0} \frac{f(a+h)-f(a)-f'(a)h}{h} =0. $$ This gives an implicit definition for $f'(a)$. To generalize this to higher dimensions we have to replace $h$ with its length $||h||$ since there is no way to divide by a vector in general. Recall that $v \in \mathbb{R}^n$ has $||v|| = \sqrt{v \cdot v}$. Consider $F: U \subseteq \mathbb{R}^m \rightarrow \mathbb{R}^n$ if $dF_{a}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is a linear transformation such that $$ \lim_{ h \rightarrow 0} \frac{F(a+h)-F(a)-dF_a(h)}{||h||} =0 $$ then we say that $F$ is differentiable at $a$ with differential $dF_a$. The matrix of the linear transformation $dF_{a}: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is called the Jacobian matrix $F'(a) \in \mathbb{R}^{n\times m}$ or simply the derivative of $F$ at $a$.

I've discussed the Frechet derivative for finite-dimensional euclidean space. I think you can see that this is easily generalized for finite-dimensional normed spaces. Moreover, this idea of differentiation is also useful in infinite dimensions.

Perhaps you are only interested in abstracting the one-dimensional case? Even in that context the view which is not based on a difference quotient gives you the option of considering objects where not all nonzero elements are units. For example, you could study differentiable functions on $\mathbb{Z}_6$.

There are other ways to abstract the derivative. But, I'll stop here since perhaps you only care specifically about the difference quotient's abstraction. My apologies if these comments are distracting to your post.

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Distracting? Not at all, this is just the kind of answer I was looking for; I only wrote down the one-dimensional case in my question but in general I was just curious on how the derivative can be or has been abstracted, thanks a lot. – Monty Gill Aug 30 '12 at 6:00

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