let f(x)=tanx find f^((n)) (x) with n integr natural


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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

${There}^{'} s a very complicated formula, you must$ $search the www$ $But we can give a nice sequence$ $t = \tan x$ $u = {a t}^{b} \Rightarrow u^{'} = (t^{2} + 1) \frac{d f}{d t} = a b (t^{b + 1} + t^{b - 1})$ $If we start with u_{0} = t (a = b = 1)$ $u_{0} = t$ $u_{1} = t^{2} + 1$ $u_{2} = (t^{2} + 1) \times 2 t = 2 t^{3} + 2 t$ $u_{3} = (t^{2} + 1) (6 t^{2} + 2) = 6 t^{4} + 8 t^{2} + 2$ $u_{3} = (t^{2} + 1) (24 t^{3} + 16 t) = 24 t^{5} + 40 t^{3} + 16 t$ $u_{4} = (t^{2} + 1) (120 t^{4} + 120 t^{2} + 16) =$ $= 120 t^{6} + 240 t^{4} + 136 t^{2} + 16$ $. . .$
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