Verified answer

1)

[tex]y = (x {}^{3} - 2)(x {}^{2} + 1) =\\ {x}^{5} + {x}^{3} - 2x {}^{2} - 2 \\ y' = 5 {x}^{5 - 1} + 3x {}^{3 - 1} - 2 \times 2x {}^{2 - 1} = \\ 5 {x}^{4} + 3 {x}^{2} - 4x[/tex]

2)

[tex]y = (x + 5) \sqrt{x} \\ y' = (x + 5)' \sqrt{x} + (x + 5)( \sqrt{x} )' = \\ 1 \times \sqrt{x} + (x + 5) \times \frac{1}{ 2\sqrt{x} } = \\ \frac{2 \sqrt{x} \sqrt{x} + x + 5 }{2 \sqrt{x} } = \frac{3x + 5}{2 \sqrt{x} } [/tex]

3)

[tex]y = {x}^{4} \cos(x) \\ y' = ({x}^{4} )' \cos(x) + {x}^{4} ( \cos(x) )' = \\ 4 {x}^{4 - 1} \cos(x) + {x}^{4} \times ( - \sin(x)) = \\ 4 {x}^{3} \cos(x) - {x}^{4} \sin(x) [/tex]

4)

[tex]y = x \tg(x) \\ y' = (x)' \tg(x) + x( \tg(x) )' = \\ \tg(x) + \frac{x}{ \cos {}^{2} (x) } [/tex]