6

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.

Community
  • 1
Red Banana
  • 23,956
  • 20
  • 91
  • 192
  • 1
    I don't think asking "given an expression, is it a derivative?" is really the right question. It is more "in what frameworks can I measure rates of change?" – Ian Aug 09 '16 at 19:57
  • 1
    There is no copyright on the name "derivative". You can come up with all sorts of things that might have some tenuous connection to the classical derivative and call them "derivatives". For example, you might look at the arithmetic derivative. – Robert Israel Aug 09 '16 at 20:03
  • 1
    "...is it a derivative if I can come up with all these items?" Yes, why wouldn't it be? I don't know if that approach will work in practice, i.e., looking at something, wondering if it's a derivative, and then actively seeking out the corresponding integral, average, etc. IIRC from that book (I read it a few years ago), the authors introduce a concept of "difference" which is used to build "difference quotients" and ultimately the derivative. I think having that "difference" first is perhaps a better way to go in practice. –  Aug 09 '16 at 20:05
  • @RobertIsrael Yes. I mentioned that one in my question. I'm just wondering if there is some mathematical reason for people call things derivatives. – Red Banana Aug 09 '16 at 20:08
  • 1
    @Voyska, for every mathematical construct, look for physical or engineering roots. Think in the context of the question, "What purpose does that construct serve?" The usual derivative (one-variable) is instantaneous velocity. For multivariate derivatives (gradients), one source of physical intuition is elasticity theory: a gradient describes how a block of material, upon becoming stressed, deforms locally. (See, e.g., Feynman's Lectures on Physics, now free on the Internet.) See Arnol'd's article "On teaching mathematics". – avs Aug 09 '16 at 20:24
  • 1
    You may be interested by derivations on K-algebras. See https://en.wikipedia.org/wiki/Derivation_(differential_algebra) – paf Aug 09 '16 at 20:35
  • Closed related question: Is there a garden of derivatives? – Dave L. Renfro Aug 09 '16 at 20:49
  • @DaveL.Renfro I made that question. – Red Banana Aug 09 '16 at 20:49
  • Oh, I didn't notice! – Dave L. Renfro Aug 09 '16 at 20:51
  • 1
    Other types of differentiation: Radon-Nikodym derivatives in measure theory; differentiation of set functions; differentiation in Euclidean spaces via contracting sets of rectangles and other types of regions (see Andrew M. Bruckner's 1971 survey paper Differentiation of Integrals) – Dave L. Renfro Aug 10 '16 at 21:34

1 Answers1

2

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.

Robin Stammer
  • 765