Question

calcular as derivadas sucessivas de f(x)=cos (4x)

Answer 1

$cos(4x)$
$u=4x$
$u'=4$
$f(x)=cos(u)$
$f'(x)=-4sin(u)$
$-4sin(4x)$

f(x)=cos(4x)
f'(x)=-4sin(4x)
f''(x)=-16cos(4x)
f'''(x)=64sin(4x)
f''''(x)=256cos(4x)

Answer 2

Função:

$f_{(x)} = \cos(4x)$


Calculando derivadas sucessivas para n=4, utilizando a regra da cadeia.


$Para\ n=1:\\\\ f'_{(x)}=- \sin(4x) \times 4\\\\ \boxed{f'_{(x)}=-4\sin(4x)}\\\\\\\\ Para\ n=2:\\\\ f''_{(x)}=- 4\cos(4x) \times 4\\\\ \boxed{f''_{(x)}=-16\cos(4x)}\\\\\\\\ Para\ n=3:\\\\ f'''_{(x)}=16\sin(4x) \times 4\\\\ \boxed{f'''_{(x)}=64\sin(4x)}\\\\\\\\ Para\ n=4:\\\\ f''''_{(x)}=64\cos(4x) \times 4\\\\ \boxed{f''''_{(x)}=256\cos(4x)}$


Espero ter ajudado.
Bons estudos!