Question

Найти производную функции

Answer 1

$y=arccos(\frac{x}{1+x^2}) \\ \\y'=(\frac{x}{1+x^2} )'*(arccos(\frac{x}{1+x^2}))'=\\ \\ \\=\frac{(x)'(1+x^2)-(x)(1+x^2)'}{(1+x^2)^2}*\frac{-1}{\sqrt{1-(\frac{x}{1+x^2} )^2}}  =\frac{1+x^2-2x^2}{(1+x^2)^2}*\frac{-1}{\sqrt{\frac{(1+x^2)^2-x^2}{(1+x^2)^2} }} = \\ \\ \\=\frac{x^2-1}{\frac{\sqrt{(1+x^2)^2-x^2}}{1+x^2}*(1+x^2)^2 } =\frac{x^2-1}{\sqrt{x^4+x^2+1}(1+x^2)}$