For $N \in \mathbb{N}$, $\frac{\mathrm{d}^{\mathrm{n}}}{\mathrm{~d} x^{\mathrm{n}}}(\log x)=$

Let $y = \log x$ and consider its $n$th derivative $\frac{d^n}{dx^n}(\log x)$.

The first derivative is $\frac{d}{dx}(\log x) = x^{-1}$.

Differentiating further reveals the pattern:
$\frac{d^2}{dx^2}(\log x) = -x^{-2}$
$\frac{d^3}{dx^3}(\log x) = 2x^{-3}$
$\frac{d^4}{dx^4}(\log x) = -6x^{-4}$

This establishes the general formula:
$\frac{d^n}{dx^n}(\log x) = (-1)^{n-1}\frac{(n-1)!}{x^n}$

Among the options, this corresponds to choice D.