TITLE: Relationship between factorial and derivatives
QUESTION [5 upvotes]: I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the factorial of $n$: $$\frac{d^n}{dx^n}(x^n)=n!$$

REPLY [3 votes]: Look at the general expression $$y_i = \frac{{\rm d}^i x^n}{{\rm d}x^i}$$
This is the i-th derivative of $x^n$ for $i=1\ldots n$. This can be directly evaluated as
$$ \boxed{ \frac{{\rm d}^i x^n}{{\rm d}x^i} = \frac{n!}{(n-i)!} x^{n-i} } $$
So the i-th derivative of a n-th order polynomial contains terms of $n!$, $(n-1)!$, $(n-2)!$ etc