Задача 63068 4.25 y= arctg x Найти производную...

61f4b1cdda08ee262dafd271

09.02.2022 14:37:35

4.25 y= arctg x Найти производную второго порядка

5f3ea7e3faf909182968ddd9

09.02.2022 16:50:26

★

y`=(x)`*arctgx+x*(arctgx)`

y`=arctgx+x*[m]\frac{1}{1+x^2}[/m]


y``=(y`)`=(arctgx+x*[m]\frac{1}{1+x^2})`[/m]=(arctgx)`+(x)`*[m]\frac{1}{1+x^2}[/m]+x*[m](\frac{1}{1+x^2})`=[/m]


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

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


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