<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] Just observe that
$$f(x)=\frac{1}{2a}\left(\frac{1}{x-a}-\frac{1}{x+a}\right)$$
Then
$$f^{(n)}(x)=\frac{(-1)^nn!}{2a}\left[\frac{1}{(x-a)^{n+1}}-\frac{1}{(x+a)^{n+1}}\right]$$
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