Question

Найти производную неявной функции
y=y(x) : y/x = sin(x y)

Answer 1

Ответ:

$\frac{y}{x} = \sin(xy)$

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