Question

Найти производные функций заданных параметрически и неявно.(2 номер, все 3 функции)​100 баллов

Answer 1

Ответ:

1.

$x = 3 log_{2}(ctgt) \\ y = {5}^{tg(6t + 5)} \: \: \: \: \: \:$

$y'x = \frac{y't}{x't}$

$y't = ln(5) \times {5}^{tg(6t + 5)} \times \frac{1}{ { \cos }^{2}(6t + 5) } \times 6$

$x't= \frac{3}{ ln(2) \times ctgt} \times ( - \frac{1}{ { \sin }^{2}t }) = \\ = \frac{3sint}{ ln(2) \times cost } \times ( - \frac{1}{ { \sin}^{2}t } ) = \\ = - \frac{3}{ ln(2) \times sint \times cost}$

$y'x = \frac{ ln(5) \times {5}^{tg(6t + 5)} \times 6 }{ { \cos }^{2} (6t + 5)} \times ( - \frac{ ln(2) \times sint \times cost }{3} ) = \\ = - \frac{ ln(5) }{ ln(2) } \times \frac{ {5}^{tg(6t + 5)} \times \sin(2t) }{ { \cos }^{2} (6t + 5)}$

2.

$\sin(xy) = x + ln(y)$

$\cos(xy) \times (y + xy') = 1 + \frac{y'}{y} \\ y \cos(xy) + xy'\cos(xy) - \frac{y'}{y} = 1 \\ y'(x \cos(xy) - \frac{1}{y} ) = 1 - y \cos(xy) \\ y' \times \frac{xy \cos(xy) - 1}{y} = 1 - y \cos(xy) \\ y'= \frac{y(1 - y \cos(xy)) }{xy \cos(xy) - 1}$