Question

Найти производную функции.

Answer 1

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