1. 3ex + sin x. 3.4 x ln x. 5.4 e5-3г. 7.5 ln (2-3x). 9.4 4 sin 11.5 3e2x - √x. 13.6 ex(x²- 5x + 3). 2.3 cos x 4. 4 tg 3x. 6.5 32x+1. logs x. 8.5 log, (12x + 5). 10.4 cos (-6x + 7). 12.6 el-x x8. 14.7 e2x2x-3. 15.4 sin² x + cos² x. 16. 6 (sin x + cos x)². 17.5 cos² x sin² x. 18. 7 sin² x. 19.8 sin x + cos x 2 sin2 x cos² x. 20.4 f (x) = cos 3x - 元2 π , xo= 3 21.6 f(x) = e3-x + log2 (2x - 3), x = 2. 22.7 f(x) = e3x(3 – 2x), x = 0. 23.5 f(x) = x²e-*.

1. f(x) = eˣ + sin x

Производная суммы равна сумме производных: (u + v)' = u' + v'

Производная eˣ равна eˣ, а производная sin x равна cos x.

f'(x) = (eˣ)' + (sin x)' = eˣ + cos x

2. f(x) = cos x - log₅ x

Производная разности равна разности производных: (u - v)' = u' - v'

Производная cos x равна -sin x, а производная log₅ x равна 1/(x ln 5).

f'(x) = (cos x)' - (log₅ x)' = -sin x - 1/(x ln 5)

3. f(x) = x⁶ ln x

Производная произведения: (uv)' = u'v + uv'

Производная x⁶ равна 6x⁵, а производная ln x равна 1/x.

f'(x) = (x⁶)' ln x + x⁶ (ln x)' = 6x⁵ ln x + x⁶ (1/x) = 6x⁵ ln x + x⁵

4. f(x) = tg 3x

Производная tg x равна 1/cos² x. Используем правило цепочки.

f'(x) = (tg 3x)' = (1/cos² (3x)) ⋅ (3x)' = 3/cos² (3x)

5. f(x) = e^(5-3x)

Используем правило цепочки: (e^u)' = e^u ⋅ u'

f'(x) = (e^(5-3x))' = e^(5-3x) ⋅ (5 - 3x)' = e^(5-3x) ⋅ (-3) = -3e^(5-3x)

6. f(x) = 3^(2x+1)

Производная a^u равна a^u ln a ⋅ u'

f'(x) = (3^(2x+1))' = 3^(2x+1) ln 3 ⋅ (2x + 1)' = 3^(2x+1) ln 3 ⋅ 2 = 2 ⋅ 3^(2x+1) ln 3

7. f(x) = ln(2 - 3x)

Производная ln u равна 1/u ⋅ u'

f'(x) = (ln(2 - 3x))' = (1/(2 - 3x)) ⋅ (2 - 3x)' = (1/(2 - 3x)) ⋅ (-3) = -3/(2 - 3x)

8. f(x) = log₇ (12x + 5)

Производная logₐ u равна 1/(u ln a) ⋅ u'

f'(x) = (log₇ (12x + 5))' = (1/((12x + 5) ln 7)) ⋅ (12x + 5)' = 12/((12x + 5) ln 7)

9. f(x) = sin(π/6 - x)

Производная sin u равна cos u ⋅ u'

f'(x) = (sin(π/6 - x))' = cos(π/6 - x) ⋅ (π/6 - x)' = cos(π/6 - x) ⋅ (-1) = -cos(π/6 - x)

10. f(x) = cos(-6x + 7)

Производная cos u равна -sin u ⋅ u'

f'(x) = (cos(-6x + 7))' = -sin(-6x + 7) ⋅ (-6x + 7)' = -sin(-6x + 7) ⋅ (-6) = 6sin(-6x + 7)

11. f(x) = 3e^(2x) - √x

Производная разности равна разности производных.

Производная 3e^(2x) равна 3e^(2x) ⋅ 2 = 6e^(2x), а производная √x равна 1/(2√x).

f'(x) = (3e^(2x))' - (√x)' = 6e^(2x) - 1/(2√x)

12. f(x) = e^(1-x) ⋅ x⁸

Производная произведения: (uv)' = u'v + uv'

f'(x) = (e^(1-x))' ⋅ x⁸ + e^(1-x) ⋅ (x⁸)' = -e^(1-x) ⋅ x⁸ + e^(1-x) ⋅ 8x⁷ = e^(1-x) (8x⁷ - x⁸)

13. f(x) = eˣ (x² - 5x + 3)

Производная произведения: (uv)' = u'v + uv'

f'(x) = (eˣ)' (x² - 5x + 3) + eˣ (x² - 5x + 3)' = eˣ (x² - 5x + 3) + eˣ (2x - 5) = eˣ (x² - 3x - 2)

14. f(x) = e^(2x) √(2x - 3)

Производная произведения: (uv)' = u'v + uv'

f'(x) = (e^(2x))' √(2x - 3) + e^(2x) (√(2x - 3))' = 2e^(2x) √(2x - 3) + e^(2x) (1/√(2x - 3)) = 2e^(2x) √(2x - 3) + e^(2x) /√(2x - 3)

15. f(x) = sin² x + cos² x

f(x) = 1, так как sin² x + cos² x = 1

f'(x) = (1)' = 0

16. f(x) = (sin x + cos x)²

f(x) = sin² x + 2 sin x cos x + cos² x = 1 + 2 sin x cos x = 1 + sin 2x

f'(x) = (1 + sin 2x)' = 0 + cos 2x ⋅ 2 = 2 cos 2x

17. f(x) = cos² x - sin² x

f(x) = cos 2x, так как cos² x - sin² x = cos 2x

f'(x) = (cos 2x)' = -sin 2x ⋅ 2 = -2 sin 2x

18. f(x) = sin² x

f'(x) = (sin² x)' = 2 sin x ⋅ (sin x)' = 2 sin x cos x = sin 2x

19. f(x) = sin⁴ x + cos⁴ x - 2 sin² x cos² x

f(x) = (sin² x - cos² x)² = (cos² x - sin² x)² = (cos 2x)² = cos² 2x

f'(x) = (cos² 2x)' = 2 cos 2x ⋅ (cos 2x)' = 2 cos 2x ⋅ (-2 sin 2x) = -4 sin 2x cos 2x = -2 sin 4x

20. f(x) = cos(3x - π/2), x₀ = π/3

f'(x) = -sin(3x - π/2) ⋅ 3 = -3 sin(3x - π/2)

f'(π/3) = -3 sin(3(π/3) - π/2) = -3 sin(π - π/2) = -3 sin(π/2) = -3 ⋅ 1 = -3

21. f(x) = e^(3-x) + log₂(2x - 3), x₀ = 2

f'(x) = -e^(3-x) + 2/((2x - 3) ln 2)

f'(2) = -e^(3-2) + 2/((2(2) - 3) ln 2) = -e + 2/ln 2

22. f(x) = e^(3x)(3 - 2x), x₀ = 0

f'(x) = 3e^(3x)(3 - 2x) - 2e^(3x) = e^(3x)(9 - 6x - 2) = e^(3x)(7 - 6x)

f'(0) = e^(3⋅0)(7 - 6⋅0) = e⁰ ⋅ 7 = 1 ⋅ 7 = 7

23. f(x) = x²e^(-x)

f'(x) = 2xe^(-x) - x²e^(-x) = e^(-x)(2x - x²)

f'(x) = 0 ⇒ e^(-x)(2x - x²) = 0

e^(-x) ≠ 0 для любого x, поэтому 2x - x² = 0

x(2 - x) = 0

x = 0 или x = 2