## Alternative definitions of the derivative

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COVIZAPIBETEFOKY
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### Alternative definitions of the derivative

What generalized definitions of the derivative are out there? I am talking about real-valued single-variable functions, not partial derivatives, complex analysis, etc. I am talking about derivatives which exist for some functions that are not differentiable by the standard definition (ie, the limit of the quotient (f(x+h)-f(x))/h as h goes to 0). For example, There might be a derivative of the absolute value function at 0.

I have been working on an alternative definition of the derivative myself, which still needs more work before I'm ready to talk to anyone in the math world about, but as I put more work into it, I would like to find out if I am, in fact, reinventing the wheel; has someone else already tried what I'm currently doing? I'm not going to put forward my idea yet (and may not for quite a while), but while I'm working on the idea, I am going to make an effort to check whether or not my idea is really original.

For the record, I have been looking around at the most reliable literature on mathematics (Wikipedia) and seen the Weak Derivative and Subderivative.
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### Re: Alternative definitions of the derivative

The weak derivative and distributions seem to be pretty much good enough for any analyst's work that I have come across. Do you think that your idea is more general?

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### Re: Alternative definitions of the derivative

[f(x+h) - f(x-h)] / 2h is, I believe, consistent with the derivative on all differential points of a function.

It gives a value for |x| @ x=0 of 0.

It ignores point-discontinuities. It fails at jump discontinuities in general, however.
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stephentyrone
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### Re: Alternative definitions of the derivative

Pretty much every "alternative notion of derivative" that I've seen has just been distributions in disguise. That said, without knowing more about what you have in mind, it's hard to know what literature to point you at.
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### Re: Alternative definitions of the derivative

t0rajir0u wrote:Do you think that your idea is more general?

I don't yet know whether my definition is more general or not.

Yakk wrote:[f(x+h) - f(x-h)] / 2h is, I believe, consistent with the derivative on all differential points of a function.

It gives a value for |x| @ x=0 of 0.

It ignores point-discontinuities. It fails at jump discontinuities in general, however.

It would be natural (to me, anyway) that any reasonable definition of a derivative would either fail or spit out an infinite number at jump discontinuities. I do know that this formula (without the limit) is often used as a good approximation for the derivative.

stephentyrone wrote:Pretty much every "alternative notion of derivative" that I've seen has just been distributions in disguise. That said, without knowing more about what you have in mind, it's hard to know what literature to point you at.

I'm confused by what both you and t0rajir0u have referred to as "distributions"? Can you explain what you mean by that? And seriously, just throw out any alternative definitions you've seen before; I don't really care whether they're related to what I'm developing or not. I may still be curious to learn about them.

And, BTW, I'm not really looking for anything that may have any applied usage. I don't care about applications; I'm all for pure mathematics.
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### Re: Alternative definitions of the derivative

Distributions are a generalization of functions that let you do more things with them. For example, the "derivative" of the Heaviside step function is the Dirac delta function. Distributions are defined in terms of their integrals. So, even though the delta function carries an "infinite" value at zero, its integral is still well-defined.
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### Re: Alternative definitions of the derivative

Distributions are defined in terms of their integrals.

More precisely, distributions are linear functionals on the space of smooth functions with compact support, where derivatives are defined via integration-by-parts. This notion of derivative agrees with the usual derivative for differentiable functions if you identify the function f with the functional
$F(\phi) = \int f \phi dx$
but lets you talk meaningfully about the derivatives of a lot of other things as well, such as functions with step discontinuities. (At the cost of some other nice things; two functions that agree almost everywhere are the same when viewed as distributions, so you lose the usual notion of evaluation at a point).

That's the very rough overview; the wikipedia page on them is quite good, and pretty much any upper-level general PDE text will have a chapter or two on the theory of distributions and Sobolev spaces (I'm partial [hah!] to Craig Evans' book). Someone else can probably suggest a more focused text.
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### Re: Alternative definitions of the derivative

COVIZAPIBETEFOKY wrote: I don't care about applications; I'm all for pure mathematics.

That's not what I mean by "application." The definition of a distribution is very useful in analysis for stating nice general theorems that the usual derivative fails to encapsulate, which is why we use distributions. Would your definition of a derivative either

- help put other theory into perspective, or
- actually be useful in resolving a particular question?

Mathematics without any motivation is a shot in the dark. You have no basis for determining whether a definition is interesting until you find out what it can do, so you're either guessing or you're already trying to do something.

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### Re: Alternative definitions of the derivative

t0rajir0u wrote:
COVIZAPIBETEFOKY wrote: I don't care about applications; I'm all for pure mathematics.

That's not what I mean by "application." The definition of a distribution is very useful in analysis for stating nice general theorems that the usual derivative fails to encapsulate, which is why we use distributions.

Perhaps I misled a bit. I am unsure whether there will be any applications, within pure mathematics or outside, but I'm plowing forth anyway.

t0rajir0u wrote:Mathematics without any motivation is a shot in the dark.

There you go. I'm taking a shot in the dark. I can afford to do that, as I'm currently just getting the last of my high school courses out of my way, and I find myself having a lot of spare time nowadays.
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### Re: Alternative definitions of the derivative

stephentyrone wrote:(At the cost of some other nice things; two functions that agree almost everywhere are the same when viewed as distributions, so you lose the usual notion of evaluation at a point).

Is that true? f(x) and f(x) + delta(0) are the same almost everywhere, but clearly are not the same distribution.
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### Re: Alternative definitions of the derivative

Diadem wrote:Is that true? f(x) and f(x) + delta(0) are the same almost everywhere, but clearly are not the same distribution.

Which delta() is that? It's certainly not the Dirac delta!

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### Re: Alternative definitions of the derivative

Diadem, f(x) + delta(0) isn't a function.

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### Re: Alternative definitions of the derivative

I wrote:two functions that agree almost everywhere are the same when viewed as distributions

Diadem wrote:Is that true? f(x) and f(x) + delta(0) are the same almost everywhere, but clearly are not the same distribution.

t0rajir0u wrote:f(x) + delta(0) isn't a function.

What t0rajir0u said. There is some notion of point evaluation for distributions, but it doesn't agree with the usual definition of point evaluation for functions, and changing the function on any measure-zero subset of the domain has no effect on the point evaluation of the distribution to which the function gives rise.
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### Re: Alternative definitions of the derivative

Ah we're talking about functions. I thought we were talking about distributions? Yes, I know he said function, but well, that's a pretty common abuse of terminology. I mean, it's even called the Dirac delta function!

Anyway, in that case you are absolutely right of course.
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### Re: Alternative definitions of the derivative

I should be using math fonts but I forget the syntax. If f : R^n -> R^m is differentiable at a E R^n. There exists a linear transformation L : R^n -> R^m such that lim h-> 0 |(f(a + h) - f(a) - L(h))|/|h| = 0 The function L is refered to as the derivative of f at a.

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### Re: Alternative definitions of the derivative

Mmmmmmmmmk wrote:I should be using math fonts but I forget the syntax. If f : R^n -> R^m is differentiable at a E R^n. There exists a linear transformation L : R^n -> R^m such that lim h-> 0 |(f(a + h) - f(a) - L(h))|/|h| = 0 The function L is refered to as the derivative of f at a.

Original Post wrote:real-valued single-variable functions,

On another note, it took me almost six months before it hit me how this reduced to the standard definition of derivative for m=n=1
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