Question

8) Calcule a derivada de:
ajudem ai galera preciso urgenteee​

Answer 1

a)

$\sqrt[5]{x} = {x}^{ \frac{1}{5} }$

$y = {x}^{ \frac{1}{5} } \\ \\ y' = \frac{1}{5} \: \: {x}^{ \frac{1}{5} - 1 } \\ \\ y' = \frac{1}{5} \: \: {x}^{ \frac{1}{5} - \frac{5}{5} } \\ \\ y' = \frac{1}{5} \: \: {x}^{ - \frac{4}{5} } \\ \\ y' = \frac{1}{5} \times ( \frac{1}{x} ) {}^{ \frac{4}{5} } \\ \\ y' = \frac{1}{5} \times \sqrt[5]{( \frac{1}{x} ) {}^{4} } \\ \\ y' = \frac{1}{5} \times \sqrt[5]{ \frac{1}{ {x}^{4} } } \\ \\ y' = \frac{1}{5} \times \frac{ \sqrt[5]{1} }{ \sqrt[5]{ {x}^{4} } } \\ \\ y' = \frac{1}{5} \times \frac{1}{ \sqrt[5]{ {x}^{4} } } \\ \\ y' = \frac{1}{5 \sqrt[5]{ {x}^{4} } }$

b)

$\sqrt[5]{3 {x}^{2} } = 3x {}^{ \frac{2}{5} }$

$y' = 3 \times \frac{2}{5} \: \: {x}^{ \frac{2}{5} - 1 } \\ \\ y' = 3 \times \frac{2}{5} \: \: {x}^{ \frac{2}{5} - \frac{5}{5} } \\ \\ y' = 3 \times \frac{2}{5} \: \: {x}^{ - \frac{3}{5} } \\ \\ \\ y' =3 \times \frac{2}{5} \times ( \frac{1}{x} ) {}^{ \frac{3}{5} } \\ \\ y' =3 \times \frac{2}{5} \times \frac{1 {}^{ \frac{3}{5} } }{x {}^{ \frac{3}{5} } } \\ \\ y' = 3 \times \frac{2}{5} \times \frac{1}{ \sqrt[5]{ {x}^{3} } } \\ \\ y' = \frac{6}{5 \sqrt[5]{ {x}^{3} } }$