4 ms·

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 var

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.

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.