Expression of the Runge function's derivative

Question

I am trying to get the nth derivative of the Runge function i.e. i want : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}.$$

Mathematica gives me the answer : $$\dfrac{d^n}{dx^n} \dfrac{1}{1+25x^2}=\dfrac{n! (-50 x)^n }{(1+25x^2)^{n+1}} {_2F_1}\left(\dfrac{1-n}{2},-\dfrac{n}{2};-n; 1 + \dfrac{1}{25x^2} \right).$$

I don't know how to get Mathematica's result, could you give me some clue ?

Answer 1

Expand the fraction into its binomial series, and simplify the expression. Then, by way of induction, find a general formula for the $n^{th}$ derivative of its general term, and rewrite it using the definition of the Gaussian hypergeometric function $_{_2}F_{_1}$.