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> It satisfies the product rule. It's been many years since I had calculus at uni, and never any abstract math. Why is the product rule picked as the "interest

> It satisfies the product rule. It's been many years since I had calculus at uni, and never any abstract math. Why is the product rule picked as the "interesting" attribute of derivatives, ie to serve as the basis for the generalization? Is there some deeper connection of the product rule in ordinary derivatives that singles out the product rule over the other properties a derivative has? For me, a key aspect of derivatives is that it allows for something like Taylor expansion or integrals to exist. Are there any equivalent things to these product-rule-generalized derivatives?

gizmo686 3y ago
I actually think it is better to not think of derivations as a generalization of derivatives; but to think of derivatives as just one of many examples of a derivation. That the name 'derivative' comes from calculus and became the name for this abstraction is just a quirk of history. You could also consider the notion of a linear function to be a generalization of derivatives. Or even the notion of homomorphisms in general. Having said that, there are many usages of other derivations where the calculus inspiration is clear, even if the geometric meaning that motivated the calculus is lost. For instance, we often talk about polynomials over arbitrary fields. In general, there is no way to graph such polynomials. There is no notion of tangent lines, slope, "continuous", or even "less than". There is, however, still the notion of roots and the multiplicity of roots. These notions turn out to be quite important. When working with any polynomial, you can define the "formal derivative" as a derivation that also satisfies D(x) = 1, D(a) = 0 (where a is an element of the underlying field). This operator behaves as you would naively expect a derivative to behave over polynomials. In Galois theory, it is important to distinguish between polynomials that have repeated roots, and those that do not. If you have a polynomial f(x), you can determine this by taking its formal derivative f'(x). Then, you can easily compute their greatest common divisor [0]. If this is a constant, then you know f has no repeated roots. [0] Using Euclid's algorithm, this is a purely mechanical process that can be done without needing to factor either f or f'. A similar trick has actually been used to attack real word cryptography. If there are secret primes p and q, and a public number pq, many cryptosystems assume that it is infeasible to determine what p and q are. However, if there is a bad random number generator, you might get 2 different keys that share a prime, so have the public numbers pq and ps, then you can easily determine that p is a common factor, from which you can easily recover q and s. This means that you can look for a large collection of public keys and try this attack on possible pair of them.
- magicalhippo 3y ago
  Thanks, that makes more sense. Most of the time these generalizations make sense to me, sometimes it can be really obscure. Flipping the viewpoint makes it more clear in this case.
poizan42 3y ago
The product rule is really the simplest rule for derivatives that apply to operations on numbers that gives something interesting. Other basic rules would be addition: (f(x) + g(x))' = f'(x) + g'(x) This is just linearity (together with constant multiplication) And function composition (the chain rule): (f ∘ g)' = (f' ∘ g)⋅g' We would need to somehow figure out what should correspond to function composition. So if we want something that captures some important algebraic properties of derivatives the product rule would be a good place to look.

gizmo686 3y ago

I actually think it is better to not think of derivations as a generalization of derivatives; but to think of derivatives as just one of many examples of a derivation. That the name 'derivative' comes from calculus and became the name for this abstraction is just a quirk of history. You could also consider the notion of a linear function to be a generalization of derivatives. Or even the notion of homomorphisms in general. Having said that, there are many usages of other derivations where the calculus inspiration is clear, even if the geometric meaning that motivated the calculus is lost. For instance, we often talk about polynomials over arbitrary fields. In general, there is no way to graph such polynomials. There is no notion of tangent lines, slope, "continuous", or even "less than". There is, however, still the notion of roots and the multiplicity of roots. These notions turn out to be quite important. When working with any polynomial, you can define the "formal derivative" as a derivation that also satisfies D(x) = 1, D(a) = 0 (where a is an element of the underlying field). This operator behaves as you would naively expect a derivative to behave over polynomials. In Galois theory, it is important to distinguish between polynomials that have repeated roots, and those that do not. If you have a polynomial f(x), you can determine this by taking its formal derivative f'(x). Then, you can easily compute their greatest common divisor [0]. If this is a constant, then you know f has no repeated roots. [0] Using Euclid's algorithm, this is a purely mechanical process that can be done without needing to factor either f or f'. A similar trick has actually been used to attack real word cryptography. If there are secret primes p and q, and a public number pq, many cryptosystems assume that it is infeasible to determine what p and q are. However, if there is a bad random number generator, you might get 2 different keys that share a prime, so have the public numbers pq and ps, then you can easily determine that p is a common factor, from which you can easily recover q and s. This means that you can look for a large collection of public keys and try this attack on possible pair of them.

magicalhippo 3y ago
Thanks, that makes more sense. Most of the time these generalizations make sense to me, sometimes it can be really obscure. Flipping the viewpoint makes it more clear in this case.

poizan42 3y ago

The product rule is really the simplest rule for derivatives that apply to operations on numbers that gives something interesting. Other basic rules would be addition: (f(x) + g(x))' = f'(x) + g'(x) This is just linearity (together with constant multiplication) And function composition (the chain rule): (f ∘ g)' = (f' ∘ g)⋅g' We would need to somehow figure out what should correspond to function composition. So if we want something that captures some important algebraic properties of derivatives the product rule would be a good place to look.