Answer by ziggurism for Why General Leibniz rule and Newton's Binomial are so...
The main feature of the Leibniz law $\left[f(x)g(x)\right]' = f'(x)g(x) + f(x)g'(x)$ is that it turns a product into a sum. Another way to write it to show that more explicitly...
View ArticleAnswer by tryst with freedom for Why General Leibniz rule and Newton's...
Suppose, we have two functions $ f(x) $ and $ g(x)$ , then the Taylor expansion of both around a point $ x= \alpha $ is of the form:$$ f(x) = \sum_{k=0}^{\infty} a_k (x-\alpha)^k$$$$ g(x) =...
View ArticleAnswer by Mathlover for Why General Leibniz rule and Newton's Binomial are so...
Taylor series expansion of $f(x+h)$$$f(x+h)=f(x)+h\frac{d}{dx} \left(f(x) \right)+\frac{h^2 }{2!}\frac{d^2}{dx^2} \left( f(x) \right)+\frac{h^3 }{3!}\frac{d^3}{dx^3} \left( f(x) \right)+....$$Taylor...
View ArticleAnswer by Tad for Why General Leibniz rule and Newton's Binomial are so similar?
Here's one combinatorial way to look at both formulas.For the first one, let $M$ be the operator which eats a polynomial $f(x,y)$ and returns the polynomial $(x+y)f(x,y)$. Note $M$ is linear, since...
View ArticleWhy General Leibniz rule and Newton's Binomial are so similar?
The binomial expansion:$$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$The General Leibniz rule (used as a generalization of the product rule for derivatives):$$(fg)^{(n)} = \sum_{k=0}^{n}...
View Article
