16. Найти производные и дифференциалы: а) y=2^∛(x+1) ∙ arcsin (1/x^2) ,y^'=?,dy=? б) r=e^(-φ^3/3)∙cos^3 φ + sin√3,dr/dφ=?,dr=?

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 (2φ))’ = (2φ)’ * (sin (2φ))’ * (sin^2 (2φ))’ = 2 * (cos (2φ) * 2sin (2φ) = 4(cos 2φ)(sin 2φ).