Задача 11771 Вычислите производную второго порядка от...

vk393673960

23.11.2016

Вычислите производную второго порядка от функции y=xarccosx

SOVA

23.11.2016

★

y`=(x*arccosx)`=(x)`*arccosx+x*(arccosx)`=
=1*arccosx+x*(-1/sqrt(1-x^2))=
=arccosx-(x/sqrt(1-x^2)).
y``=(y`)`=(arccosx)`-(x/sqrt(1-x^2))`=
=(-1/sqrt(1-x^2))-(x`*sqrt(1-x^2)-x*(sqrt(1-x^2))`)/(1-x^2)=
=(-1/sqrt(1-x^2))-(sqrt(1-x^2)-((x/2(sqrt(1-x^2)))*(1-x^2)`)/(1-x^2)=
=(-1/sqrt(1-x^2))-(sqrt(1-x^2)-(x/2(sqrt(1-x^2))*(-2x))/(1-x^2)=
=(-1/sqrt(1-x^2))-(sqrt(1-x^2)+x^2/(sqrt(1-x^2))))/(1-x^2)=
=(-1/sqrt(1-x^2))-(1-x^2+x^2)/((1-x^2)*(sqrt(1-x^2))=
=(-1+x^2-1)/((1-x^2)*(sqrt(1-x^2))=
=(x^2-2)/(1-x^2)*sqrt(1-x^2)
=2x^2/(1-x^2)*sqrt(1-x^2).