11 ms·

Shouldn't derivatives by respect to change of another variable. For example d/dx of f(x).

pfortuny 12y ago
No, it can be an operator on any kind of (usually) ring or more general algebraic structure. What you refer to is the "usual" derivative of functions of one variable, just one of many derivatives one can define.
- sp332 12y ago
  Where would it be used? It seems like declaring that the derivative of every prime number = 1 is entirely arbitrary.
  - wyager 12y ago
    Yes, it is completely arbitrary. However, this arbitrary definition follows certain rules and properties and can therefore be used for certain types of mathematical reasoning. This is why, when we talk about rings and fields and such we say "multiplication-like" or "addition-like" operators. The operators defined for the algebraic structure may not be exactly like "standard" operators, but they still follow rules and you can still do cool things with them.
  - pfortuny 12y ago
    Not entirely: it is the only way to do it coherently, but it is not so easy to explain.
    - wyager 12y ago
      >it is the only way to do it coherently There are many consistent ways to define the derivative of a number. The way we are all familiar with is to define a number as a zeroth-order polynomial.
- quotemstr 12y ago
  You can even take the derivative of a grammar to get a parser! http://matt.might.net/papers/might2011derivatives.pdf http://matt.might.net/papers/might2011derivatives.pdf
  - kazinator 12y ago
    Yes, this seems to build on the idea of regex derivatives. If regex derivatives can be used to transform a regular expression into a recognizer for strings, why not transform a more general grammar into a recognizer of strings.
  - kazinator 12y ago
    I implemented Brzozowski's regex derivatives to build a regex implementation back-end. That back-end is used whenever exotic constructs (negation, intersection) appear in the abstract syntax of the regex; in their absence, the implementation falls back on the NFA-graph-based back end.
nilkn 12y ago
The standard way of generalizing the derivative is to require it to be linear and to satisfy a version of the product rule. See, for instance, derivations: http://en.wikipedia.org/wiki/Derivation_(differential_algebra) http://en.wikipedia.org/wiki/Derivation_(differential_algebr... In the case of this article, the proposed definition is not linear, so it is indeed a bizarre candidate for a derivative.
tel 12y ago
That's the intuition used to develop the concept, but it becomes increasingly difficult to apply that intuition to in more exotic locales. Thus, it's important to eventually seek out more abstract ways of characterizing the derivative (and integral). In more advanced mathematics, you usually state that the derivative is any operation which follows two rules 1. Linearity, d(ax + by) = a d(x) + b d(y) 2. The Product Rule, d(xy) = x d(y) + d(x) y and then try to squeeze things until that operation is defined uniquely.[0] Likewise, it's often valuable to define integration as nothing more than the relationship such that I(region, derivative(quantity)) = I(boundary(region), quantity) which is known as the Generalized Stokes Rule. It basically is the "Fundamental Theorem of Calculus" on steroids and it gives a characterization of integration in terms of nothing more than it's algebraic/topological relationship with derivation... which is itself abstracted as mentioned above. --- Why do all this? Because you can squeeze most of Calculus so that it depends only upon this "abstract interface" and then apply things you learned from calculus all over the place. --- Finally, note that this is more like a "proposed" derivative than "the" derivative on natural numbers. The author notes that linearity fails, for instance. Thus, some intuition might "port over" but we shouldn't expect too much of it to do so. Which echoes back to your original question—there's not really a notion of instantaneous change for us to be talking about... so how much sense does it make to talk about a derivative here? Apparently, more than no sense at all, but less than you might want. [0] Note that all we need to state this property of the derivative is a notion of multiplication and addition. This structure is, at its most abstract usually called a ring (but can be made even weaker if needed). An example "exotic" ring might be concurrent processes. If P and Q are two processes then P*Q is P "followed by" Q and P + Q is P and Q "together". Can we write a derivative here? Who knows? (As another comment in this thread suggests, this kind of formulation can be used to consider the "derivative of a grammar" to be a parser! It's also well-known that the derivative of an algebraic data type is its "zipper"!)
- AnimalMuppet 12y ago
  > That's the intuition used to develop the concept, but it becomes increasingly difficult to apply that intuition to in more exotic locales. Then maybe it's better to use a different term for things that are different. Maybe it's better to keep the term "derivative" for the rate of change of one thing with respect to another thing, and to let the generalizations that aren't that be called something else.
Chinjut 12y ago
If you like, think of this as first turning a number into the unique monic polynomial with negated prime roots whose value at 0 is that number, then taking the derivative of that polynomial at 0. ...Not that there's any particular reason you should like this.
cperciva 12y ago
In a sense this is d/d{primes}. D(8) is how fast 8 changes with respect to 2, while D(60) is how fast 60 changes if 2, 3, and 5 simultaneously increase (at the same rate).
- e12e 12y ago
  So... we can use it to determine the age of the universe, if we know at which point in time 6 * 9 = 42 held, by extrapolating back to at what point 6 * 9 = 0? ;-) [ed: looks like my multiplication signs got eaten by hn]
joeframbach 12y ago
Think of Derivative as a function which inputs a function and outputs a function. When you think "derivative of 5 is 0", you imply "derivative of f(x)=5 is f'(x)=0" This article seems to define a function called Derivative which inputs a number and outputs a number. In this case, "derivative of 5 is 1" actually translates to "derivative of 5 is 1".

No, it can be an operator on any kind of (usually) ring or more general algebraic structure. What you refer to is the "usual" derivative of functions of one variable, just one of many derivatives one can define.

sp332 12y ago
Where would it be used? It seems like declaring that the derivative of every prime number = 1 is entirely arbitrary.
- wyager 12y ago
  Yes, it is completely arbitrary. However, this arbitrary definition follows certain rules and properties and can therefore be used for certain types of mathematical reasoning. This is why, when we talk about rings and fields and such we say "multiplication-like" or "addition-like" operators. The operators defined for the algebraic structure may not be exactly like "standard" operators, but they still follow rules and you can still do cool things with them.
- pfortuny 12y ago
  Not entirely: it is the only way to do it coherently, but it is not so easy to explain.
  - wyager 12y ago
    >it is the only way to do it coherently There are many consistent ways to define the derivative of a number. The way we are all familiar with is to define a number as a zeroth-order polynomial.
quotemstr 12y ago
You can even take the derivative of a grammar to get a parser! http://matt.might.net/papers/might2011derivatives.pdf http://matt.might.net/papers/might2011derivatives.pdf
- kazinator 12y ago
  Yes, this seems to build on the idea of regex derivatives. If regex derivatives can be used to transform a regular expression into a recognizer for strings, why not transform a more general grammar into a recognizer of strings.
- kazinator 12y ago
  I implemented Brzozowski's regex derivatives to build a regex implementation back-end. That back-end is used whenever exotic constructs (negation, intersection) appear in the abstract syntax of the regex; in their absence, the implementation falls back on the NFA-graph-based back end.

nilkn 12y ago

The standard way of generalizing the derivative is to require it to be linear and to satisfy a version of the product rule. See, for instance, derivations: http://en.wikipedia.org/wiki/Derivation_(differential_algebra) http://en.wikipedia.org/wiki/Derivation_(differential_algebr... In the case of this article, the proposed definition is not linear, so it is indeed a bizarre candidate for a derivative.

tel 12y ago

That's the intuition used to develop the concept, but it becomes increasingly difficult to apply that intuition to in more exotic locales. Thus, it's important to eventually seek out more abstract ways of characterizing the derivative (and integral). In more advanced mathematics, you usually state that the derivative is any operation which follows two rules 1. Linearity, d(ax + by) = a d(x) + b d(y) 2. The Product Rule, d(xy) = x d(y) + d(x) y and then try to squeeze things until that operation is defined uniquely.[0] Likewise, it's often valuable to define integration as nothing more than the relationship such that I(region, derivative(quantity)) = I(boundary(region), quantity) which is known as the Generalized Stokes Rule. It basically is the "Fundamental Theorem of Calculus" on steroids and it gives a characterization of integration in terms of nothing more than it's algebraic/topological relationship with derivation... which is itself abstracted as mentioned above. --- Why do all this? Because you can squeeze most of Calculus so that it depends only upon this "abstract interface" and then apply things you learned from calculus all over the place. --- Finally, note that this is more like a "proposed" derivative than "the" derivative on natural numbers. The author notes that linearity fails, for instance. Thus, some intuition might "port over" but we shouldn't expect too much of it to do so. Which echoes back to your original question—there's not really a notion of instantaneous change for us to be talking about... so how much sense does it make to talk about a derivative here? Apparently, more than no sense at all, but less than you might want. [0] Note that all we need to state this property of the derivative is a notion of multiplication and addition. This structure is, at its most abstract usually called a ring (but can be made even weaker if needed). An example "exotic" ring might be concurrent processes. If P and Q are two processes then P*Q is P "followed by" Q and P + Q is P and Q "together". Can we write a derivative here? Who knows? (As another comment in this thread suggests, this kind of formulation can be used to consider the "derivative of a grammar" to be a parser! It's also well-known that the derivative of an algebraic data type is its "zipper"!)

AnimalMuppet 12y ago
> That's the intuition used to develop the concept, but it becomes increasingly difficult to apply that intuition to in more exotic locales. Then maybe it's better to use a different term for things that are different. Maybe it's better to keep the term "derivative" for the rate of change of one thing with respect to another thing, and to let the generalizations that aren't that be called something else.

Chinjut 12y ago

If you like, think of this as first turning a number into the unique monic polynomial with negated prime roots whose value at 0 is that number, then taking the derivative of that polynomial at 0. ...Not that there's any particular reason you should like this.

cperciva 12y ago

In a sense this is d/d{primes}. D(8) is how fast 8 changes with respect to 2, while D(60) is how fast 60 changes if 2, 3, and 5 simultaneously increase (at the same rate).

e12e 12y ago
So... we can use it to determine the age of the universe, if we know at which point in time 6 * 9 = 42 held, by extrapolating back to at what point 6 * 9 = 0? ;-) [ed: looks like my multiplication signs got eaten by hn]

joeframbach 12y ago

Think of Derivative as a function which inputs a function and outputs a function. When you think "derivative of 5 is 0", you imply "derivative of f(x)=5 is f'(x)=0" This article seems to define a function called Derivative which inputs a number and outputs a number. In this case, "derivative of 5 is 1" actually translates to "derivative of 5 is 1".