1. f(x) = x⁴ - 5x³ - 7; 2. f(x) = 3/4 x⁸ + 3x⁻² - 1/6 √x 3. f(x) = (x-5)(x+1); 4. f(x) = (2x + 5)/(x-4) 5. f(x) = (x³ - 4x +3)/x 6. f(x) = √8 + 4x .

Решение:

1. f(x) = x⁴ - 5x³ - 7

• Применяем правило степени: \[(x^n)' = nx^{n-1}\]
• Производная константы равна нулю.

\[f'(x) = 4x^{4-1} - 5 \cdot 3x^{3-1} - 0\] \[f'(x) = 4x^3 - 15x^2\]

2. f(x) = \(\frac{3}{4}\)x⁸ + 3x⁻² - \(\frac{1}{6}\)√x

• Применяем правило степени и правило производной для \(\sqrt{x}\): \[(\sqrt{x})' = \frac{1}{2\sqrt{x}}\]

\[f'(x) = \frac{3}{4} \cdot 8x^{8-1} + 3 \cdot (-2)x^{-2-1} - \frac{1}{6} \cdot \frac{1}{2\sqrt{x}}\] \[f'(x) = 6x^7 - 6x^{-3} - \frac{1}{12\sqrt{x}}\]

3. f(x) = (x - 5)(x + 1)

• Применяем правило произведения: \[(uv)' = u'v + uv'\]
• \(u = x - 5\), \(v = x + 1\)
• \(u' = 1\), \(v' = 1\)

\[f'(x) = 1 \cdot (x + 1) + (x - 5) \cdot 1\] \[f'(x) = x + 1 + x - 5\] \[f'(x) = 2x - 4\]

4. f(x) = \(\frac{2x + 5}{x - 4}\)

• Применяем правило частного: \[(\frac{u}{v})' = \frac{u'v - uv'}{v^2}\]
• \(u = 2x + 5\), \(v = x - 4\)
• \(u' = 2\), \(v' = 1\)

\[f'(x) = \frac{2(x - 4) - (2x + 5) \cdot 1}{(x - 4)^2}\] \[f'(x) = \frac{2x - 8 - 2x - 5}{(x - 4)^2}\] \[f'(x) = \frac{-13}{(x - 4)^2}\]

5. f(x) = \(\frac{x³ - 4x + 3}{x}\)

• Разделим каждый член числителя на x:

\[f(x) = x^2 - 4 + \frac{3}{x} = x^2 - 4 + 3x^{-1}\] \[f'(x) = 2x - 0 + 3(-1)x^{-2}\] \[f'(x) = 2x - \frac{3}{x^2}\]

6. f(x) = \(\sqrt{8 + 4x}\)

• Применяем правило цепочки: \[(\sqrt{u})' = \frac{u'}{2\sqrt{u}}\]
• \(u = 8 + 4x\), \(u' = 4\)

\[f'(x) = \frac{4}{2\sqrt{8 + 4x}}\] \[f'(x) = \frac{2}{\sqrt{8 + 4x}}\]