## $n$th derivative of $\frac 1{f(x)}$

Is there a closed-form solution for $\frac {d^n}{dx^n}\frac 1{f(x)}$? I’ve looked at the first five derivatives in search of some pattern, but I can’t identify anything strong enough to give a closed formula. Answer It’s not exactly a closed form, but Faà di Bruno’s formula $$\frac{d^n}{dx^n} f(g(x))= \sum_{k_1 + 2k_2 + \ldots + n k_n … Read more