Question

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

Answer 1

$\dfrac{y}{x} = \sin xy$

$\left(\dfrac{y}{x}\right)' = (\sin xy)'$

$\dfrac{y'x-yx'}{x^2} = \cos xy\cdot(xy)'$

$\dfrac{y'x-y}{x^2} = \cos xy\cdot(x'y+xy')$

$\dfrac{y'x}{x^2}-\dfrac{y}{x^2} = \cos xy\cdot(y+xy')$

$\dfrac{1}{x}\cdot y'-\dfrac{y}{x^2} = y\cos xy+x\cos xy\cdot y'$

$\dfrac{1}{x}\cdot y'-x\cos xy\cdot y'= y\cos xy+\dfrac{y}{x^2}$

$\left(\dfrac{1}{x}-x\cos xy\right)\cdot y'=\left(\cos xy+\dfrac{1}{x^2}\right)\cdot y$

$y'=\dfrac{\cos xy+\dfrac{1}{x^2}}{\dfrac{1}{x}-x\cos xy} \cdot y$

$y'=\dfrac{x^2\left(\cos xy+\dfrac{1}{x^2}\right)}{x^2\left(\dfrac{1}{x}-x\cos xy\right)} \cdot y$

$\boxed{y'=\dfrac{1+x^2\cos xy}{1-x^2\cos xy} \cdot \dfrac{y}{x} }$