Решение
3.
а) y = x⁴ - 2x - 1/x
y` = 4x⁴⁻¹ - 2*1 - 1/x² = 4x³ - 1/x² - 2
б) y = x*(x⁴ - 2x - 1) = x⁵ - 2x² - x
y` = 5x⁴ - 4x - 1
в) y = (x⁵ - 2x² - 1)/x = x⁴ - 2x - 1/x
y` = 4x⁴⁻¹ - 2*1 - 1/x² = 4x³ - 1/x² - 2
4.
y = x*tgx
а) y` = (x)` * tgx + x * tg`x = tgx + x/cos²x
б) y = x² / (1 + x²)
y`= [(x²)` * (1 + x²) - (1 + x²)` * (x²)] / (1 + x²)² =
= [(2x) * (1 + x²) - (2x) *(x²)] / (1 + x²)² = (2x + 2x³ - 2x³) / (1 + x²)² =
= 2x / (1 + x²)²
5.
а) y = (x² - x - 1)⁸
y` = 8*(x² - x - 1)⁷ * ((x² - x - 1)` = 8*(x² - x - 1)⁷ * (2x - 1)
б) y = √(x² - x - 1)
y` = [1 / 2√(x² - x - 1)] * (x² - x - 1)` = [1 / 2√(x² - x - 1)] * (2x - 1) =
= (2x - 1) / 2√(x² - x - 1)
в)