Question

Найти производные заданных функций

Answer 1

Ответ:

а

$y = {( {x}^{5} - 15x + 8 )}^{ \frac{1}{5} }$

$y' = \frac{1}{5} {( {x}^{5} - 15x + 8)}^{ - \frac{4}{5} } \times ( {x}^{5} - 15x + 8)' = \\ = \frac{1}{5 \sqrt[5]{ {( {x}^{5} - 15x + 8)}^{4} } } \times (5 {x}^{4} - 15) = \\ = \frac{5( {x}^{4} - 3) }{5 \sqrt[5]{ {( {x}^{5} - 15x + 8 )}^{2} } } = \frac{ {x}^{4} - 3}{ \sqrt[5]{ {( {x}^{5} - 15x + 8 )}^{4} } }$

б

$y = ln( \sqrt[8]{ \frac{8 {x}^{2} + 5 }{3 {x}^{2} + 1 } } )$

$y' = \frac{1}{ \sqrt[8]{ \frac{8 {x}^{2} + 5}{3 {x}^{2} + 1} } } \times \frac{1}{8} {( \frac{8 {x}^{2} + 5 }{3 {x}^{2} + 1 } )}^{ - \frac{7}{8} } \times \frac{(8 {x}^{2} + 5)'(3 {x}^{2} + 1) - (3 {x}^{2} + 1)'(8 {x}^{2} + 5) }{ {(3 {x}^{2} + 1) }^{2} } = \\ = \frac{1}{8} \times \sqrt[8]{ \frac{3 {x}^{2} + 1}{8 {x}^{2} + 5 } } \times \sqrt[8]{ {( \frac{3 {x}^{2} + 1}{8 {x}^{2} + 5 } )}^{7} } \times \frac{16x(3 {x}^{2} + 1) - 6x(8 {x}^{2} + 5) }{ {(3 {x}^{2} + 1) }^{2} } = \\ = \frac{1}{8} \times \frac{3 {x}^{2} + 1 }{8 {x}^{2} + 5 } \times \frac{48 {x}^{3} + 16x - 45 {x}^{3} - 30x}{ {(3 {x}^{2} + 1)}^{2} } = \\ = \frac{ - 14x}{8(8 {x}^{2} + 5)(3 {x}^{2} + 1) } = - \frac{7x}{4(8 {x}^{2} + 5)(3 {x}^{2} + 1) }$