Задача 14418 найти производную x/(sqrt(1+arcsinx^2) *...

u193341603

19.03.2017

найти производную x/(sqrt(1+arcsinx^2) * sqrt(1-x^2)

SOVA

19.03.2017

★

По формуле (u/v)`=(u`*v-u*v`)/v^2

u=x
v=sqrt(1+arcsinx^2)*sqrt(1-x^2)

u`=x`=1
v`=(sqrt(1+arcsinx^2)*sqrt(1-x^2))`=
=(sqrt(1+arcsinx^2))`*sqrt(1-x^2)+sqrt(1+arcsinx^2)*(sqrt(1-x^2))`=

=(1/(2*sqrt(1+arcsinx^2)))*(1+arcsinx^2)`*sqrt(1-x^2)+sqrt(1+arcsinx^2)*(1/2sqrt(1-x^2))(1-x^2)`=

=(sqrt(1-x^2)*(x^2)`)/(2*sqrt(1+arcsinx^2)*sqrt(1-(x^2))^2)+

+sqrt(1+arcsinx^2)*(1/(2*sqrt(1-x^2)))*(1-x^2)`=

=(x*sqrt(1-x^2))/(sqrt(1+arcsinx^2)*sqrt(1-(x^2))^2)+

+sqrt(1+arcsinx^2)*(1/(2*sqrt(1-x^2)))*(1-x^2)`=

=(x*sqrt(1-x^2))/(sqrt(1+arcsinx^2)*sqrt(1-(x^2))^2)+

-x*sqrt(1+arcsinx^2)*(1/sqrt(1-x^2))

О т в е т.

(sqrt(1+arcsinx^2)*sqrt(1-x^2)-x^2*sqrt(1-x^2))/(sqrt(1+arcsinx^2)*sqrt(1-(x^2))^2)+

+x^2sqrt(1+arcsinx^2)*(1/sqrt(1-x^2)) и это все
делим на
v^2=(1+arcsinx^2)*(1-x^2)