Question

Utilizando a regra de derivação, calcule f'(x):
$f(x)= \sqrt[3]{ ( x^{2}+ 1) ^{2} }$

Answer 1

$f(x)=\sqrt[3]{(x^{2}+1)^{2}}\\ \\ f(x)=(x^{2}+1)^{2/3}\\ \\ \\ f'(x)=\dfrac{2}{3}\cdot (x^{2}+1)^{(2/3)-1}\cdot (x^{2}+1)'\\ \\ \\ f'(x)=\dfrac{2}{3}\cdot (x^{2}+1)^{-1/3}\cdot 2x\\ \\ \\ f'(x)=\dfrac{4}{3}\cdot\dfrac{x}{(x^{2}+1)^{1/3}}\\ \\ \\ \boxed{f'(x)=\dfrac{4}{3}\cdot\dfrac{x}{\sqrt[3]{x^{2}+1}}}$