20th derivative of a rational function

Question

I could not find the 20th derivative of the function below :

$$f(x) = \frac{2x}{x^2 - 4}$$

I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.

Answer 1

we have $f(x)=\frac{1}{x-2}+\frac{1}{x+2}$ then we get $$f'(x)=- \left( x-2 \right) ^{-2}- \left( x+2 \right) ^{-2}$$ $$f''(x)=2\, \left( x-2 \right) ^{-3}+2\, \left( x+2 \right) ^{-3}$$ $$f'''(x)=-6\, \left( x-2 \right) ^{-4}-6\, \left( x+2 \right) ^{-4}$$ can you proceed? for your control the answer is $$f(x)^{(20)}=2432902008176640000\, \left( x-2 \right) ^{-21}+2432902008176640000\, \left( x+2 \right) ^{-21} $$

Answer 2

Hint:-

Decompose the given fraction into Partial Fraction and then use the following identity,$$y=\dfrac{1}{x} \implies y^{(n)}=(-1)^nn!\left(\dfrac{1}{x^{n+1}}\right)$$

Hint to a proof of the identity:-

Apply Induction.