Во всех примерах смотрим таблицу производных
1) y' = (ctg(^3)x -2/x)' = (ctg(^3)x)'*(ctgx)' - (2/x)' = 3*ctg(^2)x*(-1/sin(^2)x) + +2*x(^-2)=- 3*ctg(^2)x / sin(^2)x) + 2 / x(^2)
2) y'=((√x+3)/(sin3x) + 3(^x))' =(√x+3)'*(sin3x) - (√x+3)*(sin3x)'*(3x)' /(sin(^2)3x) +
+(3(^x))' = 1/2*√x+3*(sin3x) - (√x+3)*cos3x *3 /sin(^2)3x + 3(^x)*ln3 =
= (√x+3)*sin3x - 6*(√x+3)*cos3x / 2*sin(^2)3x + 3(^x)*ln3
3) y' = (arccos(^2)5x + e(x(^2))' = (arccos(^2)5x)'* (arccos5x)'*(5x)'+
+(e(x(^2))'*(x(^2))' = 2*arccos5x *(-1/√1-x(^2)) *5 + e(x(^2) *2x =
= - 10 arccos5x /√1-x(^2) +2x* e(x(^2)
4) y' = (arctg (^2)5x +3x(^3) +8x)' = (arctg (^2)5x)'* (arctg 5x)'*(5x)' +
+ (3x(^3))' +(8x)' = 2*arctg 5x*1/1+x(^2) * 5 +9x(^2) +8 =
= 10arctg 5x/1+x(^2) +9x(^2) +8
6) y' =(tg2x - ln2x)' = (tg2x)'*(2x)' - (ln2x)*(2x)' = 1/cos(^2)x *2 -1/2x *2=
=2/cos(^2)x -1/x
7) y' = (√2x+5 + arcsin3x) =(√2x+5)' * (2x+5)' + (arcsin3x)' *(3x)' =
= 1/2√2x+5 * 2 + 1/√1`-x(^2)*3 = 2/√2x+5+3/√1`-x(^2)
9) y' = (arccos(3x+5)*e(^x))' =(arccos(3x+5))'*e(^x)) + (arccos(3x+5)*(e(^x))')=
=e(^x)*(-1/(√1-x(^2)) *3 + e(^x)*arccos(3x+5) =
=e(^x)*arccos(3x+5) - 3e(^x)/(√1-x(^2))
10) y' = (cos(lnx))' = (cos(lnx))'*(lnx)' = -sin(lnx)*1/x = -sin(lnx) / x