Question

Найти производную функций
1)cos(xy) = y/x;

Answer 1

Ответ:

$\underline {1\ sposob.}\ \ \ \ cos(xy)=\dfrac{y}{x}\\\\\\-sin(xy)\cdot (y+xy')=\dfrac{y'x-y}{x^2}\\\\\\-y\cdot sin(xy)-xy'\cdot sin(xy)=\dfrac{y'x-y}{x^2}\\\\\\-x^2y\cdot sin(xy)-x^3y'\cdot sin(xy)=y'x-y\\\\\\y'x+x^3y'\cdot sin(xy)=y-x^2y\cdot sin(xy)\\\\\\y'x\cdot (1+x^2\cdot sin(xy)\, )=y\cdot (1-x^2\cdot sin(xy)\, )\\\\\\y'=\dfrac{y\cdot (1-x^2\cdot sin(xy)\, )}{x\cdot (1+x^2\cdot sin(xy)\, )}$

$\underline {2\ sposob.}\ \ \ \ cos(xy)-\dfrac{y}{x}=0\ \ \ \to \ \ \ F(x,y)=cos(xy)-\dfrac{y}{x}\\\\F'_{x}=-sin(xy)\cdot y-\dfrac{-y}{x^2}=-sin(xy)\cdot y+\dfrac{y}{x^2}=\dfrac{-yx^2\cdot sin(xy)+y}{x^2}\ \ \ ,\\\\\\F'_{y}=-sin(xy)\cdot x-\dfrac{1}{x}=\dfrac{-x^2\cdot sin(xy)-1}{x}\\\\\\y'=-\dfrac{F'_{x}}{F'_{y}}=-\dfrac{-yx^2\cdot sin(xy)+y}{x^2}\cdot \dfrac{x}{-x^2\cdot sin(xy)-1}=\dfrac{y\cdot (1-x^2\cdot sin(xy)\, )}{x\cdot (1+x^2\cdot sin(xy)\, )}$