Здравствуйте, Влвдимир1601.

1. y = 2x² – x – (1/3)(2 + 4x)^-1/2,
y' = (2x² – x – (1/3)(2 + 4x)^-1/2)’ = 4x – 1 – (1/3) ∙ (-1/2) ∙ 4 ∙ (2 + 4x)^-3/2 = 4x – 1 + 2/3 ∙ (2 + 4x)^-3/2.

2. y = ((1 + x²)/(1 – x²))^1/2 – e^{1 – ln 2x},
y₁ = ((1 + x²)/(1 – x²))^1/2,
y₂ = e^{1 – ln 2x},
ln y₁ = (1/2)ln ((1 + x²)/(1 – x²)),
(ln y₁)’ = ((1/2)ln ((1 + x²)/(1 – x²)))’,
(y₁)’/y₁ = (1/2)(ln (1 + x²) – ln (1 – x²))’ = (1/2)(2x/(1 + x²) + 2x/(1 – x²)) = x(1/(1 + x²) + 1/(1 – x²)) =
= 2x/((1 – x²)(1 + x²)),
(y₁)’ = 2xy₁/((1 – x²)(1 + x²)) = 2x/((1 – x²)(1 + x²)) ∙ ((1 + x²)/(1 – x²))^1/2 = 2x/((1 – x²)³(1 + x²))^1/2,
(y₂)’ = (e^{1 – ln 2x})’ = e^{1 – ln 2x}(1 – ln 2x)’ = e^{1 – ln 2x} ∙ (-1/(2x)) ∙ 2 = -1/x ∙ e^{1 – ln 2x},
y' = (y₁)’ + (y₂)’ = 2x/((1 – x²)³(1 + x²))^1/2 – 1/x ∙ e^{1 – ln 2x}.

3. y = ln arccos x^-1/2,
y' = (ln arccos x^-1/2)’ = 1/arccos x^-1/2 ∙ (arccos x^-1/2)’ = 1/arccos x^-1/2 ∙ -1/(1 – (x^-1/2)²)^1/2 ∙ (x^-1/2)’ =
= 1/arccos x^-1/2 ∙ -1/(1 – x^-1)^1/2 ∙ -1/2 ∙ x^-3/2 = 1/arccos x^-1/2 ∙ 1/(2(x³(1 – x^-1))^1/2).

4. y = (arcsin x)^{e^x},
ln y = e^x ∙ ln arcsin x,
(ln y)’ = (e^x ∙ ln arcsin x)’,
y'/y = (e^x)’ ∙ ln arcsin x + e^x ∙ (ln arcsin x)’ = e^x ∙ ln arcsin x + e^x ∙ 1/arcsin x ∙ (arcsin x)’ =
= e^x ∙ ln arcsin x + e^x ∙ 1/arcsin x ∙ 1/(1 – x²) = e^x(ln arcsin x + 1/(arcsin x ∙ (1 – x²)),
y' = ye^x(ln arcsin x + 1/(arcsin x ∙ (1 – x²)) = (arcsin x)^{e^x}e^x(ln arcsin x + 1/(arcsin x ∙ (1 – x²)).

С уважением.