Question

Derive as funções f (x)= 2^3x-1

Answer 1

$f(x)=2 x^{3} -1 \\ f'(x)=6 x^{2} \\ f''(x)=12x$

Answer 2

$\frac{\partial }{\partial \:x}\left(2^{3x-1}\right)$

Passo 1: Aplicar a regra do exponencial:
$=\frac{\partial \:}{\partial \:x}\left(e^{\left(3x-1\right)\ln \left(2\right)}\right)$

Substituição de w:
$\left(3x-1\right)\ln \left(2\right)=w$

Passo 2: Aplicar a regra da cadeia:
$=\frac{\partial \:}{\partial \:w}\left(e^w\right)\frac{\partial \:}{\partial \:x}\left(\left(3x-1\right)\ln \left(2\right)\right)$

Observação:
$\frac{\partial \:}{\partial \:w}\left(e^w\right)=e^w$

Observação 2:
$\frac{\partial \:}{\partial \:x}\left(\left(3x-1\right)\ln \left(2\right)\right)=3\ln \left(2\right)$

$=\ln \left(2\right)\left(\frac{\partial \:}{\partial \:x}\left(3x\right)-\frac{\partial \:}{\partial \:x}\left(1\right)\right)$

$=\ln \left(2\right)\left(3-0\right)$

$=3\ln \left(2\right)$

Passo 3: Organizar
$=e^w\cdot \:3\ln \left(2\right)$

Substituir o valor de w novamente:
$=e^{\left(3x-1\right)\ln \left(2\right)}\cdot \:3\ln \left(2\right)$

$=\left(e^{\ln \left(2\right)}\right)^{\left(3x-1\right)}\cdot \:3\ln \left(2\right)$

Resposta:
$=2^{3x-1}\cdot \:3\ln \left(2\right)$