Найти производную: 1) y=arcctg 1/x 2) y = (arcsin x) ² 3) y=√1+sin²x 4) y=arctg x/√1+x² 5) y=³√ (2+cosx) ² 6) y=arccos (2^x)

1) y' = (cos (arctg x)) ' = (arctg x) ' * (cos (arctg x)) ' = (1 / (1 + х^2)) * (-sin (arctg x)) = (-sin (arctg x)) / (1 + х^2);

2) y' = (sin (arcctg x)) ' = (arcctg x) ' * (sin (arcctg x)) ' = (1 / (1 + х^2)) * (cos (arcctg x)) = (cos (arcctg x)) / (1 + х^2);

3) y' = (sin (arccos x)) ' = (arccos x) ' * (sin (arccos x)) ' = (-1 / √ (1 - х^2)) * (cos (arccos x)) = (-cos (arccos x)) / √ (1 - х^2);

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

y' = (3sin (x^9 - sin x) + 7) ' = (3sin (x^9 - sin x)) ' + (7) ' = (x^9 - sin x) ' * (3sin (x^9 - sin x)) ' + (7) ' = ((x^9) - (sin x) ') * (3sin (x^9 - sin x)) ' + (7) ' = ((9x^8) - cos x) * 3cos (x^9 - sin x) + 0 = ((9x^8) - cos x) * 3cos (x^9 - sin x).

f (x) ' = (sin^2 (2φ)) ' = (2φ) ' * (sin (2φ)) ' * (sin^2 (2φ)) ' = 2 * (cos (2φ) * 2sin (2φ) = 4 (cos 2φ) (sin 2φ).