Решение:

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