What exactly is a derivative? In calculus courses, we learn the classical derivative:
$$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
And the directional derivative:
$$D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}$$
In which the partial derivative is just a slight variation of the previous one. But some days ago, I've read Stopple's Primer of Analytic Number Theory, I've read something about the difference operator:
$$\Delta f(x)=f(x+1)-f(x)$$
Which at least for me, seems quite sems quite familiar with the previous examples. The only difference is that this is not really a limiting process - another one that lacks the limiting process but is also called a derivative is the arithmetic derivative. But there are other examples in which there is a limiting process, in non-newtonian calculus, for example, we have the geometric derivative:
$$f^{*}(x) = \lim_{h \to 0}{ \left({f(x+h)\over{f(x)}}\right)^{1\over{h}} }$$
And the bigeometric derivative:
$$f^{*}(x) = \lim_{h \to 0}{ \left({f((1+h)x)\over{f(x)}}\right)^{1\over{h}} } =  \lim_{k \to 1}{ \left({f(kx)\over{f(x)}}\right)^{1\over{\ln(k)}} }$$
And in this site, the authors argue that:

"There are infinitely many non-Newtonian calculi. Like the classical calculus, each of them possesses, among other things: a derivative, an integral, a natural average, a special class of functions having a constant derivative, and two Fundamental Theorems which reveal that the derivative and integral are 'inversely' related."

I remember from my calculus classes that the classical derivative is a comparison of a certain function with the slope of a straight line. It seems to my limited knowledge that these other derivatives are also comparisons with some other geometrical figures, perhaps? From this book:

During the Renaissance many scholars, including Galileo, discussed the following problem:

Two estimates, $10$ and $1000$, are proposed as the value of a horse. Which estimate, if any, deviates more from the true value of $100$?

The scholars who maintained that the deviations should be measures by the differences concludes that the estimate of $10$ was closer to the actual value. However, Galileo eventual maintained that the deviations should be measured by ratios, and he concluded that the two estimates deviated equally from the true value.

Other examples are also Fréchet's derivatives and Gâteaux derivatives which I'm not exactly sure of what they are.
As you can see further in the book, this yields the previously mentioned geometric derivative. So, assuming there is a class of kinds of derivatives, what binds them all? How can I look at anything and decide if it's a derivative or not? I presume that if something is a derivative, then it must have - just as the authors of the mentioned website said - an integral, a natural average, a special class of functions having a constant derivative, and two Fundamental Theorems which reveal that the derivative and integral are 'inversely' related.
So, given any expresion, is it a derivative if I can come up with all these items? This seems a faint answer to me, I'd like to see if there is a better one.
 A: This may will not answer your question directly, but I'd like to give some hints on the the bigger picture. The derivative in its original sense is a geometrical concept. In conclusion: If you want to learn more about the concept of a derivative, you need to learn about differentiable manifolds and Riemannian metrics.
I mentioned this with a target in mind: Once you have a setup of a Riemannian manifold there is a technical concept that in some sense generalizes the notion of partial derivatives: So-called connections. 
Connections in this setup are a more general concept than the pure derivative and they could be seen like the "class" you asked for encoding the key idea what a derivative makes a derivative.
Also, the terms parallel transport and covariant derivative are closely related.
I do not know how one can relate Fréchet- , Gâteaux- and/or bigeometric derivates to this in detail, but I can imagine there are other resources for this question.
If you want to get a more deep understanding of the term derivative, exploring differential geometry along the terms mentioned can't be a wrong advice.

edit: (August 10, 2016)
It is also worth noting that the Leibniz rule is a central characteristic behavior that every of the mentioned derivatives fulfills (you may want to lookup the term derivation). So if you want, you can see this specific computation rule as common property that "binds" the class of derivatives. In particular this is valid for the connections I've thrown into the pool above.
