Question

Дам 50 баллов
Помогите найти производную логарифма

Answer 1

Ответ:

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

$y' = (ln(y) )' \times y$

$(ln(y))' = ( ln(( \frac{ ({x}^{3} + 2) \sqrt[3]{x - 1} }{ {(x + 5)}^{4} } )) ' = \\ = ( ln( {x}^{3} + 2) + ln( {(x - 1)}^{ \frac{1}{3} } - ln( {(x + 5)}^{4} ) ) ' = \\ = \frac{3 {x}^{2} }{ {x}^{3} + 2 } + \frac{1}{3} \times \frac{1}{x - 1} - \frac{4}{x + 5} = \\ = \frac{3 {x}^{2} \times 3(x - 1)(x + 5) + ( {x}^{3} + 2)(x + 5) - 4 \times 3(x - 1)( {x}^{3} + 2)}{3( {x}^{3} + 2)(x - 1)(x + 5) } = \\ = \frac{9 {x}^{2}( {x}^{2} + 5x - x - 5) + {x}^{4} + 5 {x}^{3} + 2x + 10 - 12( {x}^{4} + 2x - {x}^{3} - 2) }{3( {x}^{3} + 2)(x - 1)(x + 5) } = \\ = \frac{9 {x}^{4} + 36 {x}^{3} - 45 {x}^{2} + {x}^{4} + 5 {x}^{3} + 2x + 10 - 12 {x}^{4} - 24x + 12 {x}^{3} + 24 }{3( {x}^{3} + 2)(x - 1)(x + 5)} = \\ = \frac{ - 2 {x}^{4} + 53 {x}^{3} - 45 {x}^{2} - 22x + 34 }{3( {x}^ {3} + 2)(x - 1)(x + 5) }$

$y' = \frac{( {x}^{3} + 2) \sqrt[3]{x - 1} }{ {(x + 5)}^{4} } \times \frac{ (- 2 {x}^{4} + 53 {x}^{3} - 45 {x}^{2} - 22x + 34) }{3( {x}^ {3} + 2)(x - 1)(x + 5) } = \\ = - \frac{2 {x}^{4} - 53 {x}^{3} + 45 {x}^{2} + 22x - 34}{ \sqrt[3]{ {(x - 1)}^{2} {( x + 5)}^{5} } }$