6. Найдите производную функции: 1) f(x) = 8x⁵ - x³/3 + 3x² + 4; 2) f(x) = (3 - 4x)√x; 3) f(x) = (x² - 2) / x; 4) f(x) = 6/x⁴ - 2/x³

Решение:

1. \( f'(x) = (8x^5 - \frac{x^3}{3} + 3x^2 + 4)' = 8 \cdot 5x^4 - \frac{1}{3} \cdot 3x^2 + 3 \cdot 2x = 40x^4 - x^2 + 6x \)
2. \( f(x) = (3 - 4x)x^{1/2} = 3x^{1/2} - 4x^{3/2} \)
3. \( f'(x) = 3 \cdot \frac{1}{2} x^{-1/2} - 4 \cdot \frac{3}{2} x^{1/2} = \frac{3}{2\sqrt{x}} - 6\sqrt{x} \)
4. \( f(x) = \frac{x^2}{x} - \frac{2}{x} = x - 2x^{-1} \)
5. \( f'(x) = 1 - 2(-1)x^{-2} = 1 + \frac{2}{x^2} \)
6. \( f(x) = 6x^{-4} - 2x^{-3} \)
7. \( f'(x) = 6(-4)x^{-5} - 2(-3)x^{-4} = -24x^{-5} + 6x^{-4} = \frac{-24}{x^5} + \frac{6}{x^4} \)

Ответ: 1) \(40x^4 - x^2 + 6x\); 2) \(\frac{3}{2\sqrt{x}} - 6\sqrt{x}\); 3) \(1 + \frac{2}{x^2}\); 4) \(\frac{6}{x^4} - \frac{24}{x^5}\).