The n-th derivative of 1/(x² − 1)

Step 1: the building block. For $g(x)=\frac1{x-a}=(x-a)^{-1}$, successive derivatives are $-(x-a)^{-2}$, then $2(x-a)^{-3}$, then $-6(x-a)^{-4}$, and by induction
$$g^{(n)}(x)=\frac{(-1)^nn!}{(x-a)^{n+1}}.$$

Step 2: partial fractions.
$$\frac1{x^2-1}=\frac12\left(\frac1{x-1}-\frac1{x+1}\right).$$

Step 3: conclusion.
$$f^{(n)}(x)=\frac{(-1)^nn!}2\left(\frac1{(x-1)^{n+1}}-\frac1{(x+1)^{n+1}}\right).$$

Check. For $n=1$: $-\frac12\left(\frac1{(x-1)^2}-\frac1{(x+1)^2}\right)=-\frac12\cdot\frac{4x}{(x^2-1)^2}=-\frac{2x}{(x^2-1)^2}$, which is indeed $f'$.