let-f-x-tanx-find-f-n-x-with-n-integr-natural-

Question Number 201534 by Mathspace last updated on 08/Dec/23

l e t f (x) = t a n x

f i n d f^{(n)} (x) w i t h n i n t e g r

n a t u r a l

Commented by Frix last updated on 09/Dec/23

{There}^{'} s a very complicated formula, you must

search the www

But we can give a nice sequence

t = \tan x

u = {a t}^{b} \Rightarrow u^{'} = (t^{2} + 1) \frac{d f}{d t} = a b (t^{b + 1} + t^{b - 1})

If we start with u_{0} = t (a = b = 1)

u_{0} = t

u_{1} = t^{2} + 1

u_{2} = (t^{2} + 1) \times 2 t = 2 t^{3} + 2 t

u_{3} = (t^{2} + 1) (6 t^{2} + 2) = 6 t^{4} + 8 t^{2} + 2

u_{3} = (t^{2} + 1) (24 t^{3} + 16 t) = 24 t^{5} + 40 t^{3} + 16 t

u_{4} = (t^{2} + 1) (120 t^{4} + 120 t^{2} + 16) =

= 120 t^{6} + 240 t^{4} + 136 t^{2} + 16

\dots

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