>>>>>>>Question<<<<<<<<<

it is easy to find it generally but i was thinking that what if in some question ... say 100th differential is 0 ... there has to be a shortcut ... some help please

i even found out the general term of what differential would be after the third differential (when $\displaystyle 2/3 x^{3}$ goes away)...

its $\displaystyle f_{n}(x) = e^{x} + (-1)^{n-1}e^{-x} -2 sin[n\pi/2 + (-1)^{n}x]$

But i dont know how to find when $\displaystyle f_{n}(x)$ will be non-zero