The logarithmic differential is famous $$\frac{\partial \log(f)}{\partial x} = \frac{1}{f(x)}\cdot\frac{\partial f}{\partial x}$$
If we instead consider $k$th derivative of $n$th power of logarithm of a function:
$$\frac{\partial^k (\log(f(x))^n)}{{\partial x}^k}$$
It quickly becomes more advanced expressions
It seems we do get terms with factors like $\frac{k!}{(k-m)!},m\leq n$, binomial coefficients(?),
$$\prod_{\forall l}{f^{(l)}(x)}^{k_l} , k=\sum_{\forall l} k_l$$ $$\frac{1}{f(x)^k}$$ and other nasty things.
Does there exist some nice trick or formula to try and simplify this expression we will get?