<s>[INST] Find a formula for $f^{(n)}(x)$ where $f(x) = \dfrac{1}{x^2-a^2}$ 
Let $a$ be a constant. Find a formula for $f^{(n)}(x)$ where $f(x) = \dfrac{1}{x^2-a^2}$. 

After computing a few derivatives, the derivatives seem to have factorials in them sometimes and other times not. For example, $\dfrac{d^5}{dx^5} \left (\dfrac{1}{x^2-a^2} \right) = -\dfrac{240(3a^4x+10a^2x^3+3x^5)}{(x^2-a^2)^6}$ while $\dfrac{d^6}{dx^6} \left (\dfrac{1}{x^2-a^2} \right) = -\dfrac{720(a^6+21a^4x^2+35a^2x^4+7x^6)}{(x^2-a^2)^6}$, so I am not immediately seeing the pattern.
 [/INST] $$
{1 \over {x^{\,2}  - a^{\,2} }} = {1 \over {\left( {x + a} \right)\left( {x - a} \right)}} = {1 \over {2\,a}}\left( {{1 \over {\left( {x - a} \right)}} - {1 \over {\left( {x + a} \right)}}} \right)
$$
... then it goes easily ..
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