Найдите производные следующих функций

1)

2)

3)

4)

5)

6)

1) y=2√x = 2x^ (¹/₂ - 3) = 2x^ (-2.5)

x³

y' = 2 * (-2.5) x^ (-2.5-1) = - 5x^ (-3.5) = _ 5

(x³) √x

2) y = 6∛x = 6x^ (¹/₃ - ¹/₂) = 6x^ (⁻¹/₆)

√x

y' = 6 * (-¹/₆) x^ (-¹/₆ - 1) = - x^ (-⁷/₆) = - 1

x (⁶√x)

3) y=⁴√ (1/x³) = 1 = x^ (-³/₄)

x^ (³/₄)

y' = - ³/₄ x^ (-³/₄ - ⁴/₄) = - ³/₄ x^ (⁻⁷/₄) = - 3

4x (⁴√ (x³))

4) y=∛ (1/x⁻²) = 1 = x^ (²/₃)

x^ (⁻²/₃)

y' = ²/₃ x^ (²/₃ - ³/₃) = ²/₃ x^ (-¹/₃) = 2

3 ∛x

5) f (x) = x⁻² √x ∛x = x^ (-2 + ¹/₂ + ¹/₃) = x^ (-¹²/₆ + ³/₆ + ²/₆) = x^ (⁻⁷/₆)

f (x) ' = - ⁷/₆ x^ (-⁷/₆ - ⁶/₆) = - ⁷/₆ x^ (⁻¹³/₆) = - 7

6x² (⁶√x)

6) S = ∛t ∛ (t²) = t^ (¹/₃ + ²/₃ - 1) = t⁰ = 1

t√1

S' = (1) ' = 0