Question

sobre derivadas f(x)=(x+5)^-5/³

Answer 1

Função:

$f_{(x)}=(x+5)^{-\frac{5}{3}}$


Resolvendo a derivada utilizando a regra da cadeia:

$f'_{(x)}=(u)'.u'\ \ \ \ (\ Derivada\ de\ fora\ vezes\ derivada\ de\ dentro\ )$

$f'_{(x)}=\left(-\dfrac{5}{3}(x+5)^{-\frac{5}{3}-1}\right) \times (1)\\\\\\ f'_{(x)}=-\dfrac{5}{3}(x+5)^{-\frac{8}{3}}\\\\\\ f'_{(x)}=-\dfrac{5}{3}\dfrac{1}{(x+5)^{\frac{8}{3}}}\\\\\\ f'_{(x)}=-\dfrac{5}{3(x+5)^{\frac{8}{3}}}\\\\\\ f'_{(x)}=-\dfrac{5}{3\sqrt[3]{(x+5)^{8}}}\\\\\\ f'_{(x)}=-\dfrac{5}{3\sqrt[3]{(x+5)^3 \times (x+5)^3 \times (x+5)^2}}\\\\\\ f'_{(x)}=-\dfrac{5}{3\sqrt[3]{(x+5)^{\not 3} \times (x+5)^{\not 3} \times (x+5)^2}}$


$f'_{(x)}=-\dfrac{5}{3(x+5)^2\sqrt[3]{(x+5)^2}}\\\\\\ Racionalizando:\\\\ f'_{(x)}=-\dfrac{5}{3(x+5)^2\sqrt[3]{(x+5)^2}}\times \dfrac{\sqrt[3]{(x+5)}}{\sqrt[3]{(x+5)}}\\\\\\ f'_{(x)}=-\dfrac{5\sqrt[3]{(x+5)}}{3(x+5)^2\sqrt[3]{(x+5)^3}}\\\\\\ f'_{(x)}=-\dfrac{5\sqrt[3]{(x+5)}}{3(x+5)^2\sqrt[\not 3]{(x+5)^{\not 3}}}\\\\\\ \boxed{\boxed{f'_{(x)}=-\dfrac{5\sqrt[3]{(x+5)}}{3(x+5)^3}}}$


Bons estudos!