39.6. Найдите производную функции:

Решение:

1. \( y = \frac{3x + 5}{x - 8} \) \(\implies y' = \frac{3(x - 8) - (3x + 5)(1)}{(x - 8)^2} = \frac{3x - 24 - 3x - 5}{(x - 8)^2} = \frac{-29}{(x - 8)^2}\).
2. \( y = \frac{2x^2}{1 - 6x} \) \(\implies y' = \frac{4x(1 - 6x) - 2x^2(-6)}{(1 - 6x)^2} = \frac{4x - 24x^2 + 12x^2}{(1 - 6x)^2} = \frac{4x - 12x^2}{(1 - 6x)^2} = \frac{4x(1 - 3x)}{(1 - 6x)^2}\).
3. \( y = \frac{\sin x}{x} \) \(\implies y' = \frac{\cos x \cdot x - \sin x \cdot 1}{x^2} = \frac{x \cos x - \sin x}{x^2}\).
4. \( y = \frac{x^2 - 1}{x^2 + 1} \) \(\implies y' = \frac{2x(x^2 + 1) - (x^2 - 1)(2x)}{(x^2 + 1)^2} = \frac{2x^3 + 2x - 2x^3 + 2x}{(x^2 + 1)^2} = \frac{4x}{(x^2 + 1)^2}\).

Ответ: 1) \( \frac{-29}{(x - 8)^2} \) 2) \( \frac{4x(1 - 3x)}{(1 - 6x)^2} \) 3) \( \frac{x \cos x - \sin x}{x^2} \) 4) \( \frac{4x}{(x^2 + 1)^2} \).