Is there a formula for the $n^{\text{th}}$ derivative of the reciprocal of a polynomial $f(x) = \sum_{k=0}^{K} a_k x^k$?
Question 1: In particular, does there exist a formula for $$g_n(x) = \frac{d^n}{dx^n} \left(\frac{1}{\sum_{k=0}^{K} a_k x^k}\right)$$
In my particular case a partial fraction decomposition cannot be found for the denominator. If anyone knows of the answer to the first question, I am also curious about an equation of the following form.
Question 2: Does there also exist a formula (in terms of $n$) for the following derivative? $$\hat{g}_n(x) = \frac{d^n}{dx^n} \left(\frac{x}{\sum_{k=0}^{K} a_k x^k}\right)$$
