[m]y`=x`+(arctgy)`[/m]
[m]y`=1+\frac{y`}{1+y^2}[/m]
[m]y`-\frac{y`}{1+y^2}=1[/m]
[m]y`\cdot (1-\frac{1}{1+y^2})=1[/m]
[m]y`\cdot \frac{1+y^2-1}{1+y^2}=1[/m]
[m]y`\cdot \frac{y^2}{1+y^2}=1[/m]
[m]y`=\frac{1}{ \frac{y^2}{1+y^2}}[/m]
[m]y`=\frac{1+y^2}{y^2}[/m]