Leibniz’s formula for the n-th derivative of a product is an extremely important and useful formula. Is an explicit formula for the n-th derivative of a quotient also very important?
This area plays the most important role in fractional integro-differentiation and inverse function as well as Lambert Function which has direct application in viscous flows,chemical engineering and material science.
The Leibnitz Theorem Formula of products is equivalent to the formula for quotients. i.e f/g = (f)(g)-1 .
Dear Issam Kaddoura ,
This involves have an explicit formula for the n-th derivative of (1/g(x)). This is the difficult part. Obtain the n-th derivative of (f/g) having that of (1/g) is easy. I derives such a formula in my article:
Explicit Formula for the n-th Derivative of a Quotient
Dear Roudy El Haddad
The division is a trivial consequence of multiplication.
Observe that in fields or rings, we have the basic operations are + and x addition and multiplication, the division is defined in terms of multiplication.
All results related to the division operation are the trivial consequences of the multiplication operation.
In general, Faa di Bruno's formula is the most elegant formula related to the n-th derivative of the composition of functions. Then we apply
f(x)=1/x, hence f(g(x))= 1/g(x)
and then apply Faa di Bruno's formula that computes the nth derivative of 1/g(x).
See
https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula
https://mathworld.wolfram.com/FaadiBrunosFormula.html
Dear Issam Kaddoura ,
Also check out Bourbaki‘s result in the attached image. It presents a determinant representation of the result.
Cannot easily iterate, because the result is not a simple ratio again, but a combination.
It is not like the CAUCHY rule to evaluate limits.
Learn by memory: Bottom times differentiation of top, minus top times differentiation of bottom, divided by bottom squared.
Absolutely, do you have that? I'm in need of it right now. If you have for example the k-th derivative of 1/f(x) as a formula, I'd love to know that.
The catch is, your formula will only be useful if it's simple.
Your formula is interesting, but impossible to use. It's not useful, unless you can express that with algebra.
@Issam Kadoura Nop, that is cheating, that doesn't work.