Найдите производную функции: 1) y = 8x9 – 3x5 + 6x3 – 2; 1 - 2) y = x10 + 4√x - 3x; 5 2 3) y = x - 2; x 4) y = 2 5 y=; x7 x5 - 1) у = (x² - 5)(x² + 4); 2) y = √x(2x² + 4); 3) y = (√x – 8)(9 – 7√x);

Решение:

1. $$y = 8x^9 - 3x^5 + 6x^3 - 2$$
   $$y' = 8 \cdot 9x^8 - 3 \cdot 5x^4 + 6 \cdot 3x^2 - 0 = 72x^8 - 15x^4 + 18x^2$$
   Ответ: $$y' = 72x^8 - 15x^4 + 18x^2$$
2. $$y = \frac{1}{5}x^{10} + 4\sqrt{x} - 3x$$
   $$y = \frac{1}{5}x^{10} + 4x^{\frac{1}{2}} - 3x$$
   $$y' = \frac{1}{5} \cdot 10x^9 + 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} - 3 = 2x^9 + \frac{2}{\sqrt{x}} - 3$$
   Ответ: $$y' = 2x^9 + \frac{2}{\sqrt{x}} - 3$$
3. $$y = x - \frac{2}{x^2}$$ $$y = x - 2x^{-2}$$ $$y' = 1 - 2 \cdot (-2)x^{-3} = 1 + \frac{4}{x^3}$$ Ответ: $$y' = 1 + \frac{4}{x^3}$$
4. $$y = \frac{5}{x^7} - \frac{2}{x^5}$$ $$y = 5x^{-7} - 2x^{-5}$$ $$y' = 5 \cdot (-7)x^{-8} - 2 \cdot (-5)x^{-6} = -35x^{-8} + 10x^{-6} = -\frac{35}{x^8} + \frac{10}{x^6}$$ Ответ: $$y' = -\frac{35}{x^8} + \frac{10}{x^6}$$
5. $$y = (x^2 - 5)(x^3 + 4)$$ $$y' = (x^2 - 5)'(x^3 + 4) + (x^2 - 5)(x^3 + 4)' = 2x(x^3 + 4) + (x^2 - 5)(3x^2) = 2x^4 + 8x + 3x^4 - 15x^2 = 5x^4 - 15x^2 + 8x$$ Ответ: $$y' = 5x^4 - 15x^2 + 8x$$
6. $$y = \sqrt{x}(2x^2 + 4)$$ $$y = x^{\frac{1}{2}}(2x^2 + 4) = 2x^{\frac{5}{2}} + 4x^{\frac{1}{2}}$$ $$y' = 2 \cdot \frac{5}{2}x^{\frac{3}{2}} + 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} = 5x^{\frac{3}{2}} + 2x^{-\frac{1}{2}} = 5\sqrt{x^3} + \frac{2}{\sqrt{x}}$$ Ответ: $$y' = 5\sqrt{x^3} + \frac{2}{\sqrt{x}}$$
7. $$y = (\sqrt{x} - 8)(9 - 7\sqrt{x})$$ $$y = (x^{\frac{1}{2}} - 8)(9 - 7x^{\frac{1}{2}})$$ $$y' = (x^{\frac{1}{2}} - 8)'(9 - 7x^{\frac{1}{2}}) + (x^{\frac{1}{2}} - 8)(9 - 7x^{\frac{1}{2}})' = \frac{1}{2}x^{-\frac{1}{2}}(9 - 7x^{\frac{1}{2}}) + (x^{\frac{1}{2}} - 8)(-\frac{7}{2}x^{-\frac{1}{2}}) = \frac{9}{2}x^{-\frac{1}{2}} - \frac{7}{2} + (-\frac{7}{2} + \frac{56}{2})x^{-\frac{1}{2}} = \frac{9}{2}x^{-\frac{1}{2}} - \frac{7}{2} - \frac{7}{2} + 28x^{-\frac{1}{2}} = \frac{65}{2}x^{-\frac{1}{2}} - 7 = \frac{65}{2\sqrt{x}} - 7$$ Ответ: $$y' = \frac{65}{2\sqrt{x}} - 7$$