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.)

 

[blue_raver22]

I can provide some insight into the relationship between the Leibniz rule and Newton's binomial theorem. Both of these concepts are fundamental in calculus and are used to solve complex mathematical problems. The Leibniz rule, also known as the generalized product rule, is used to find the derivative of a product of two functions. On the other hand, Newton's binomial theorem is used to expand a binomial expression raised to a power.

The similarity between the two lies in the use of summation notation and the coefficients {n \choose r} in both formulas. These coefficients represent the number of ways to choose r objects from a set of n objects, which is also known as the binomial coefficient. In Newton's binomial theorem, these coefficients are used to expand the binomial expression into a sum of terms. In the Leibniz rule, these coefficients are used to determine the contributions of each term in the product to the final derivative.

In terms of the basic idea, both of these concepts rely on the concept of combinations and permutations. In Newton's binomial theorem, we are looking at all possible ways of arranging n objects into groups of r. In the Leibniz rule, we are considering all possible ways of combining two functions to find the derivative of their product. This connection between combinations and derivatives can be further explored through the concept of combinatorial calculus.

Overall, the Leibniz rule and Newton's binomial theorem are powerful tools in mathematics and their similarity lies in their use of combinations and permutations to solve complex problems. By understanding this connection, we can gain a deeper understanding of these concepts and their applications in various fields of science and mathematics.