Verified answer

Ответ:

1. а)

[tex]y' = 4[/tex]

б)

[tex]y' = \frac{6}{ {x}^{2} } [/tex]

в)

[tex]y = {x}^{3} (2x - x) = 2 {x}^{4} - {x}^{4} [/tex]

[tex]y' = 8 {x}^{3} - 4 {x}^{3} [/tex]

д)

[tex]y' = \cos(x) + \frac{1}{4 \sqrt{x} } [/tex]

2.

[tex]f'(x) = 2x - \frac{1}{ \sqrt{x} } [/tex]

[tex]f'(x_{0}) = 2 \times 64 - \frac{1}{ \sqrt{64} } = 128 - \frac{1}{8} = 127 \frac{7}{8} [/tex]

3.

[tex]s' = v[/tex]

[tex]s' = 6 {t}^{5} - 16 {t}^{3} [/tex]

[tex]v = 6 \times {2}^{5} - 16 \times {2}^{3} = 6 \times 32 - 16 \times 8 = 192 - 128 = 64[/tex]

4. Уравнение касательной:

[tex]y = f(x_{0}) + f'(x_{0} )(x - x_{0} )[/tex]

[tex]f(x_{0}) = 4 \times \sqrt{4} = 4 \times 2 = 8[/tex]

[tex]f'(x) = \frac{2}{ \sqrt{x} } [/tex]

[tex]f'(x _{0}) = \frac{2}{ \sqrt{4} } = \frac{2}{2} = 1[/tex]

[tex]y = 8 + 1(x - 4) \\ y = 8 + x - 4 \\ y = x - 4[/tex]