243. Найти значение производной функции f(x) в точке ₀, если: 1) f(x) = cos x sin x, ₀ = /6; 2) f(x) = ˣ ln x, ₀ = 1; 3) f(x) = 2cos/sin, ₀ = /4; 4) f(x) = /1+ˣ, ₀ = 0.

1) f(x) = cos x sin x, ₀ = /6

f'(x) = cos²x - sin²x = cos(2x)

f'(/6) = cos(/3) = 1/2

2) f(x) = ˣ ln x, ₀ = 1

f'(x) = ˣ ln x + ˣ/x

f'(1) = ¹ ln 1 + ¹/1 = 0 + e = e

3) f(x) = 2cos/sin, ₀ = /4

f(x) = 2 ctg x

f'(x) = -2 / sin²x

f'(/4) = -2 / (1/√2)² = -4

4) f(x) = /(1+ˣ), ₀ = 0

f'(x) = (1 + ˣ - ˣ) / (1 + ˣ)²

f'(0) = (1 + ⁰ - 0) / (1 + ⁰)² = (1 + 1) / (1 + 1)² = 2 / 4 = 1/2