39.4. Найдите производную функции:

Решение:

1. \( y = (x^3 - 2)(x^2 + 1) \) \(\implies y' = 3x^2(x^2 + 1) + (x^3 - 2)(2x) = 3x^4 + 3x^2 + 2x^4 - 4x = 5x^4 + 3x^2 - 4x\).
2. \( y = (x + 5)\sqrt{x} \) \(\implies y' = 1 \cdot \sqrt{x} + (x + 5)\frac{1}{2\sqrt{x}} = \sqrt{x} + \frac{x+5}{2\sqrt{x}} = \frac{2x + x + 5}{2\sqrt{x}} = \frac{3x + 5}{2\sqrt{x}}\).
3. \( y = x^4 \cos x \) \(\implies y' = 4x^3 \cos x + x^4 (-\sin x) = 4x^3 \cos x - x^4 \sin x\).
4. \( y = x \mathrm{tg} x \) \(\implies y' = 1 \cdot \mathrm{tg} x + x \cdot \frac{1}{\cos^2 x} = \mathrm{tg} x + \frac{x}{\cos^2 x}\).

Ответ: 1) \( 5x^4 + 3x^2 - 4x \) 2) \( \frac{3x + 5}{2\sqrt{x}} \) 3) \( 4x^3 \cos x - x^4 \sin x \) 4) \( \mathrm{tg} x + \frac{x}{\cos^2 x} \).