Найти производные функций: а) f(x)=4cosx-2tgx+3 б) f(x)=arctgx-arcctgx в) f(x)=sin^2 x г) 7 в корне x * Lnx

1) f(х)\' = (sin^3 (2 - 3х))’ = (2 - 3х)’ * (sin (2 - 3х))’ * (sin^3 (2 - 3х))’ = ((2)’ – (3х)’) * (sin (2 - 3х))’ * (sin^3 (2 - 3х))’ = (0 – 3) * (соs (2 - 3х)) * 3 * (sin^2 (2 - 3х)) = (-3) * (соs (2 - 3х)) * 3 * (sin^2 (2 - 3х)) = (-9) * (соs (2 - 3х)) * (sin^2 (2 - 3х).

2) f(х)\' = (соs^2 (х / 3))’ = (х / 3)’ * (соs (х / 3))’ * (соs^2 (х / 3))’ = (1 / 3) * (-sin (х / 3)) * 2 * (соs (х / 3)) = (2 / 3) * (-sin (х / 3)) * (соs (х / 3)).

3) f(х)\' = ((аrсtg х) * е^2х)’ = (аrсtg х)’ * е^2х + (аrсtg х) * (е^2х)’ = (1 / (1 + х^2)) * е^2х + (аrсtg х) * 2е^(2х) = (е^2х / (1 + х^2)) + (аrсtg х) * 2е^(2х).

4) f(х)\' = (8*х^14 - (ln (х) / sin (х)))’ = (8*х^14)’ – ((ln (х) / sin (х)))’ = (8*х^14)’ – ((ln (х))’ * sin (х) - ln (х) * (sin (х))’) / sin^2 (х)) = 8 * 14 * х^13 - ((1 / х) * sin (х) - ln (х) * (соs (х))’) / sin^2 (х)) = 112х^13 - ((sin (х) / х) - ln (х) * (соs (х))’) / sin^2 (х)).