Question

Помагите решить праизводные

Answer 1

Ответ:

1.

$y = \sin(x) - x\cos(x)$

$y '= \cos(x) - (x)' \cos(x) - ( \cos(x)) ' \times x = \\ = \cos(x) - \cos(x) - ( - \sin(x)) \times x = \\ = x \sin(x)$

2.

$y = \frac{ ln(x) }{ ln(x) + 1 }$

$y' = \frac{( ln(x)) '\times (1 + ln(x)) - (1 + ln(x)) ' \times ln(x) }{(1 + ln(x)) {}^{2} } = \\ = \frac{ \frac{1}{x}(1 + ln(x)) - \frac{1}{x} \times ln(x) }{ {(1 + ln(x)) }^{2} } = \\ = \frac{ \frac{1}{x} (1 + ln(x) - ln(x)) }{ {(1 + ln(x)) }^{2} } = \frac{1}{x(1 + ln(x)) {}^{2} }$

3

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

$y' = 3 \times ( - 1) \times {(3 - 4x + 5 {x}^{2} )}^{ - 2} \times (3 - 4x + 5 {x}^{2} )' = \\ = - \frac{3}{ {(3 - 4x + 5 {x}^{2} )}^{2} } \times ( - 4 + 10x) = \frac{12 - 30x}{ {(3 - 4x + 5 {x}^{2}) }^{2} }$