The product rule states, if $\displaystyle f(x)$ and $\displaystyle g(x)$ are two differentiable functions, then:

$\displaystyle \displaystyle [f(x)\cdot g(x)]' = f'(x)\cdot g(x)+g'(x)\cdotf(x).$

If $\displaystyle f(x)$ and $\displaystyle g(x)$ are $\displaystyle n$-times differentiable, then Liebniz's rule generalises this to:

$\displaystyle \displaystyle (f(x)\cdot g(x))^{(n)}=\sum_{k=0}^n {n \choose k} f^{(k)} {(x)g^{(n-k)}(x)$

Now, the quotient rule states that, if $\displaystyle f(x)$ and $\displaystyle g(x)$ are two differentiable functions, then:

$\displaystyle \displaystyle \left[\frac{f(x)}{g(x)}\right]' = \frac{f'(x)g(x)-g'(x)f(x)}{[g(x)]^2}$

Perhaps I've misused the word 'inverse' for what I want, but what does this generalise to?