Здравствуйте, Сласти.

1. y’ = ((9 + 6x^3/5)^1/11)’ = 1/11 ∙ (9 + 6x^3/5)^{1/11 – 1} ∙ (9 + 6x^3/5)’ = 1/11 ∙ (9 + 6x^3/5)^-10/11 ∙ 6 ∙ 3/5 ∙ x^{3/5 – 1} =
= 18/55 ∙ x^-2/5 ∙ (9 + 6x^3/5)^-10/11.

2. y’ = (lg (x – cos x))’ = 1/ln 10 ∙ (ln (x – cos x))’ = 1/ln 10 ∙ 1/(x – cos x) ∙ (x – cos x)’ = 1/ln 10 ∙ 1/(x – cos x) ∙ (1 + sin x).

3. y’ = (cos ((arcsin x)/2))’ = -sin ((arcsin x)/2) ∙ ((arcsin x)/2)’ = -1/2 ∙ sin ((arcsin x)/2) ∙ (1 – x²)^-1/2.

4. y = (x + 1)^{1/(x + 1),} ln y = 1/(x + 1) ∙ ln (x + 1), (ln y)’ = (1/(x + 1) ∙ ln (x + 1))’,
y'/y = (1/(x + 1))’ ∙ ln (x + 1) + 1/(x + 1) ∙ (ln (x + 1))’ = -1/(x + 1)² ∙ ln (x + 1) + 1/(x + 1) ∙ 1/(x + 1) =
= 1/(x + 1)² ∙ (1 – ln (x + 1)),
y' = y ∙ 1/(x + 1)² ∙ (1 – ln (x + 1)) = (x + 1)^{1/(x + 1)} ∙ 1/(x + 1)² ∙ (1 – ln (x + 1)) = (x + 1)^{1/(x + 1) – 2} ∙ (1 – ln (x + 1)).

5. x – y = arcsin x – arcsin y, y – arcsin y = x – arcsin x, (y – arcsin y)’ = (x – arcsin x)’,
y' – y’ ∙ (1 – y²)^-1/2 = 1 – (1 – x²)^-1/2, y’(1 – (1 – y²)^-1/2) = 1 – (1 – x²)^-1/2, y’ = (1 – (1 – x²)^-1/2)/(1 – (1 – y²)^-1/2).

6. y’_t = (e^t ∙ cos t)’_t = (e^t)’_t ∙ cos t + e^t ∙ (cos t)’_t = e^t ∙ cos t – e^t ∙ sin t = e^t ∙ (cos t – sin t),
x’_t = (e^t ∙ sin t)’_t = (e^t)’_t ∙ sin t + e^t ∙ (sin t)’_t = e^t ∙ sin t + e^t ∙ cos t = e^t ∙ (cos t + sin t),
y’_x = y’_t/x’_t = e^t ∙ (cos t – sin t)/(e^t ∙ (cos t + sin t)) = (cos t – sin t)/(cos t + sin t).

С уважением.