ПРОИЗВОДНЫЕ 1) y = ctg^2x ; 2) y = (1-tg2x) / (tg2x); 3) y = arccos2/x * x 4) y = arcctg((2x / (1-x^2)) ; 5) y = lntg2x

1) f(x)\' = (x^2 * ctg (x))’ = (x^2)’ * ctg (x) + x^2 * (ctg (x))’ = 2x * ctg (x) + x^2 * (1 / (-sin^2 (x))) = 2xctg (x) - x^2 / (sin^2 (x)).

2) f(x)\' = (x^2 * tg (x))’ = (x^2)’ * tg (x) + x^2 * (tg (x))’ = 2x * tg (x) + x^2 * (1 / (cos^2 (x))) = 2xtg (x) + x^2 / (cos^2 (x)).

3) y’ = (cos (arctg x))’ = (arctg x)’ * (cos (arctg x))’ = (1 / (1 + х^2)) * (-sin (arctg x)) = (-sin (arctg x)) / (1 + х^2).

4) y’ = (sin (arcctg x))’ = (arcctg x)’ * (sin (arcctg x))’ = (1 / (1 + х^2)) * (cos (arcctg x)) = (cos (arcctg x)) / (1 + х^2).

5) y’ = (sin (arccos x))’ = (arccos x)’ * (sin (arccos x))’ = (-1 / √(1 - х^2)) * (cos (arccos x)) = (-cos (arccos x)) / √(1 - х^2).

6) y’ = (cos (arcsin x))’ = (arcsin x)’ * (cos (arcsin x))’ = (1 / √(1 - х^2)) * (-sin (arcsin x)) = (-sin (arcsin x)) / √(1 - х^2).

7) f(x)\' = ((x^2) * (ln x)^5)’ = ((x^2)’ * (ln x)^5) + (x^2) * ((ln x)^5)’ = ((x^2)’ * (ln x)^5) + (x^2) * (ln x)’ * ((ln x)^5)’ = (2x * (ln x)^5) + (x^2) * (1 / х) * (5 * (ln x)^4) = 2x * (ln x)^5 + 5х * (ln x)^4).

8) Это квадрат со стороной «периметр разделить на 2».

9) Это будут числа 8 и 2.