Задача 65113 Найти производную [m]y = arcsin...

60b77ffca1c50e1d3043a110

16.06.2022 11:47:28

Найти производную [m]y = arcsin \frac{2x^3}{1+x^6} [/m]

626fab9a354ab65fb2f43abb

16.06.2022 12:50:28

★

[m]y = arcsin \frac{2x^3}{1+x^6} = arcsin(f(x))[/m]
[m]y' = \frac{1}{\sqrt{1 - f^2(x)}}*f'(x) = \frac{1}{\sqrt{1 - \frac{4x^6}{(1+x^6)^2}}}*\frac{6x^2(1+x^6) - 2x^3*6x^5}{(1+x^6)^2} = [/m]
[m]= \frac{\sqrt{(1+x^6)^2}}{(1+x^6)^2 - 4x^6}*\frac{6x^2+6x^8 - 12x^8}{(1+x^6)^2} = \frac{1+x^6}{1+2x^6 + x^{12} - 4x^6}*\frac{6x^2+6x^8 - 12x^8}{(1+x^6)^2} = [/m]
[m]= \frac{1}{1-2x^6 + x^{12}}*\frac{6x^2-6x^8}{1+x^6} = \frac{1}{(1 - x^6)^2}*\frac{6x^2(1 - x^6)}{1+x^6} = \frac{1}{1 - x^6}*\frac{6x^2}{1+x^6} = \frac{6x^2}{(1+x^6)(1 - x^6)} = \frac{6x^2}{1-x^{12}}[/m]
[m]y' = \frac{6x^2}{1-x^{12}}[/m]