Quotient rule and binomial theorem

[Jhenrique]

If it's possible to relate the product rule with the binomial theorem, so:

$$(x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2$$
$$D^2(fg)=1f^{(2)}g^{(0)}+2f^{(1)}g^{(1)}+1f^{(0)}g^{(2)}$$

So, is it possible to relate the quotient rule with the binomial theorem too?

 

[Jorriss]

Hello,
You may find the following useful:
https://www.physicsforums.com/showpost.php?p=4261179&postcount=7

 

[Mark44]

If it's possible to relate the product rule with the binomial theorem, so:

$$(x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2$$
$$D^2(fg)=1f^{(2)}g^{(0)}+2f^{(1)}g^{(1)}+1f^{(0)}g^{(2)}$$

So, is it possible to relate the quotient rule with the binomial theorem too?

I don't think so.

 

[HallsofIvy]

Write $f/g$ as $fg^{-1}$ and apply the product rule to that.