1. Найдите производную функции: 1) f(x) = 6x⁴ - x²/2 - 7x + 10; 2) f(x) = (5x-1)√x; 3) f(x) = x²/x + 3; 4) f(x) = tg⁵ 3x.

Решение:

1. \( f'(x) = (6x^4 - \frac{x^2}{2} - 7x + 10)' = 24x^3 - x - 7 \)
2. \( f(x) = (5x-1)x^{1/2} = 5x^{3/2} - x^{1/2} \)
   \( f'(x) = \frac{3}{2} · 5x^{1/2} - \frac{1}{2}x^{-1/2} = \frac{15}{2}\sqrt{x} - \frac{1}{2\sqrt{x}} \)
3. \( f(x) = \frac{x^2+3}{x} = x + \frac{3}{x} = x + 3x^{-1} \)
   \( f'(x) = 1 - 3x^{-2} = 1 - \frac{3}{x^2} \)
4. \( f(x) = \mathrm{tg}^5(3x) \)
   \( f'(x) = 5 \mathrm{tg}^4(3x) · \frac{1}{\cos^2(3x)} · 3 = \frac{15 \mathrm{tg}^4(3x)}{\cos^2(3x)} \)

Ответ: 1) \( 24x^3 - x - 7 \)
2) \( \frac{15}{2}\sqrt{x} - \frac{1}{2\sqrt{x}} \)
3) \( 1 - \frac{3}{x^2} \)
4) \( \frac{15 \mathrm{tg}^4(3x)}{\cos^2(3x)} \)