Third Derivative Quotient Rule/Pascal’s Triangle

So I was looking at the quotient rule for the second derivative and momentarily wondered what the third derivative might look like. Vince commented that it would be really cool if it followed Pascal's triangle. Something like this maybe: $$u'''(x)=\frac{u(x-h)-3 u(x)-3 u'(x)+u(h+x)}{h^2}$$Just a random idea. I'll play around with it after the quiz, see if I can use the same Taylor expansion trick.

Edit: Moved to Uncategorized

So here is the third derivative using a similar method for the second. Instead of adding $u(x+h)$ and $u(x-h)$ we subtract them to pull out the term we want.$$u'''(x)=\frac{3 u(x+h)-3 u(x-h)-6 hu'(x)}{h^3}$$ Ultimately I think there might be a way to relate Pascal's triangle, but it would probably be convoluted.