Question

Найти производную функции.

Answer 1

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