16. Отыскать производные и дифференциалы: а) y=2^(x+1) arcsin (1/x^2) ,y^39;=?,dy=?

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).

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

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