Question

A derivada da função
( )f'(0) = 3/2
( )f'(0) = 7
( )f'(0) = 1/3
( )f'(0) = 1/4
( )f'(0) = 0

Answer 1

Explicação passo-a-passo:

Temos

$f(x)=\sqrt[3]{x^{2}+e^{3x}+7}=(x^{2}+e^{3x}+7)^{\frac{1}{3}}$

fazendo u = x^2 + e^3x + 7 => u' = 2x + 3e^3x

E

f(x) = u⅓ => f'(x) = ⅓u-⅔.u' = ⅓.1/u⅔.u' = $\frac{1}{3}.\frac{1}{\sqrt[3]{u^{2}}}.u'=\frac{1}{3}.\frac{1}{\sqrt[3]{(x^{2}+e^{3x}+7)^{2}}}.(2x + 3e^{3x})$

$f'(0)=\frac{1}{3}.\frac{1}{\sqrt[3]{(0^{2}+e^{3.0}+7)^{2}}}.(2.0+3e^{3.0})=\frac{1}{3}.\frac{1}{\sqrt[3]{(0+e^{0}+7)^{2}}}.(0+3e^{0})=\frac{1}{3}.\frac{1}{\sqrt[3]{(1+7})^{2}}}.(0+3.1)=\frac{1}{3}.\frac{1}{\sqrt[3]{8^{2}}}.3=\frac{1}{\sqrt[3]{64}}=\frac{1}{4}$