100th Derivative

Let, $y = f(x) = 4x^{3} - 8\cos x$

Let $y_n$ be the $n_{th}$ derivative.

It is clear that. $100_{th}$ derivative of. $4x^{3}$ will be zero.

$-8\cos x$ follows a pattern in its derivatives. $8\sin x , 8\cos x , -8\sin x , -8\cos x , ........$ After every 4 derivatives it repaeats.

$\dfrac {100}{4} = 25$ .Hence, after 25 cycles its repeats to the same, i.e, $y_{100} = -8\cos x = -8$ .Since, $x = 0.$

$ANSWER :\boxed{-8}$