Leibniz rule and Newton's bionomial theorem

[ShayanJ]

Remember the Newton's binomial theorem which says:
$(x+y)^n = \sum_{r=0}^n {n \choose r} x^{n-r} y^r$

where ${n \choose r}=\frac{n!}{r! (n-r)!}$
Now let's take a look at the Leibniz rule:
$\frac{d^n}{dz^n}(xy)=\sum_{r=0}^n {n \choose r} \frac{d^{n-r} x}{dz^{n-r}} \frac{d^r y}{dz^r}$

I think the similarity is unignorable!

I've had some studies about abstract algebra and so am familiar with generalization of operations on elements of a set.so I think the similarity mentioned above,means that the operations of exponentiation and derivation are somehow similar to each other as e.g. the special kind of similarity between multiplication of matrices and doing symmetry operations in sequence.
But I don't know what's the basic idea.
I'll appreciate any idea.
Thanks

 

[Tenshou]

What do you mean by the basic idea? as in the basic idea of using binomial formula and the poduct rule?, I have actually asked a very similar question, believe it or not; and the answer I got was : The binomial formula and the product rule are the same(to an extent https://www.physicsforums.com/showthread.php?t=670008) Hope that helped.

 

[ShayanJ]

Thanks for the link,that was an interesting discussion.
And what I mean by basic idea is this:
Lets take the example of matrix multiplication and symmetry operation multiplication(i.e. doing them in sequence),the basic idea is that there may be a non-commutative associative operation on the elements of a set which we give the name multiplication and we can have different instances of it on different sets.I mean sth like this.
I should tell that although the discussion in your thread was interesting,it didn't achieve a result.There was only another proof of Newton's binomial theorem assuming Leibniz rule which just hinted more on their connection.
But I'm seeking a general operation(like the general definition of multiplication)for which the Newton's and Leibniz rules are special cases.

 

[Stephen Tashi]

But I'm seeking a general operation(like the general definition of multiplication)for which the Newton's and Leibniz rules are special cases.

I suggest you search on the topic of "operator algebras" or "algebras of operators". People have written books that develop the analogy between the operations of arithmetic and the operations of operators. ( I haven't studied operator algebras, but I have the impression that in spite of the nice analogies not much practical use is made of such systems. Perhaps people who write computer software like Mathematica do make use of such ideas.)