f-x-sin-5-x-calculate-f-5-pi-2-

Question Number 146899 by mathmax by abdo last updated on 16/Jul/21

f (x) = \sin^{5} x calculate f^{(5)} (\frac{π}{2})

Answered by Olaf_Thorendsen last updated on 17/Jul/21

f (x) = \sin^{5} x

f (x) = {(\frac{e^{i x} - e^{i x}}{2 i})}^{5}

f (x) = \frac{1}{16 \times 2 i} (e^{5 i x} - 5 e^{3 i x} + 10 e^{i x} - 10 e^{- i x} + 5 e^{- 3 i x} - e^{- 5 i x})

f (x) = \frac{\sin 5 x - 5 \sin 3 x + 10 \sin x}{16}

f^{(5)} (x) = \frac{5^{5} \cos 5 x - 5 . 3^{5} \cos 3 x + 10 \cos x}{16}

f^{(5)} (\frac{π}{2}) = 0