• №1 Найдите производную функции: Вариант 2 1)y = x⁶ -7x 2)y = 6x⁴ +9x³-10 3)y = √x-15x 1 4)y = 5x⁻²--+9 x 4 5)y = -tgx x⁵ 6)y = 5sinx + √x 6 7)y = -cosx x⁻³ 8)y = (x⁶ +3)(x⁴ -4) 9)y = √x(6x-1) 6 10)y = x⁶ cosx 11)y=(3/x-4)(3x+6) 4x⁵ 12)y = 2x+5 13)y = 9√x 6+2x⁵ 4x 14)y = - ctgx

Решения заданий Вариант 2:

1. $$y = x^6 - 7x$$ $$y' = 6x^5 - 7$$ Ответ: $$y' = 6x^5 - 7$$
2. $$y = 6x^4 + 9x^3 - 10$$ $$y' = 24x^3 + 27x^2$$ Ответ: $$y' = 24x^3 + 27x^2$$
3. $$y = \sqrt{x} - 15x$$ $$y = x^{\frac{1}{2}} - 15x$$ $$y' = \frac{1}{2}x^{-\frac{1}{2}} - 15$$ $$y' = \frac{1}{2\sqrt{x}} - 15$$ Ответ: $$y' = \frac{1}{2\sqrt{x}} - 15$$
4. $$y = 5x^{-2} - \frac{1}{x} + 9$$ $$y = 5x^{-2} - x^{-1} + 9$$ $$y' = -10x^{-3} + x^{-2}$$ $$y' = -\frac{10}{x^3} + \frac{1}{x^2}$$ Ответ: $$y' = -\frac{10}{x^3} + \frac{1}{x^2}$$
5. $$y = \frac{4}{x^5} - tgx$$ $$y = 4x^{-5} - tgx$$ $$y' = -20x^{-6} - \frac{1}{cos^2x}$$ $$y' = -\frac{20}{x^6} - \frac{1}{cos^2x}$$ Ответ: $$y' = -\frac{20}{x^6} - \frac{1}{cos^2x}$$
6. $$y = 5sinx + \sqrt{x}$$ $$y = 5sinx + x^{\frac{1}{2}}$$ $$y' = 5cosx + \frac{1}{2}x^{-\frac{1}{2}}$$ $$y' = 5cosx + \frac{1}{2\sqrt{x}}$$ Ответ: $$y' = 5cosx + \frac{1}{2\sqrt{x}}$$
7. $$y = \frac{6}{x^{-3}} - cosx$$ $$y = 6x^3 - cosx$$ $$y' = 18x^2 + sinx$$ Ответ: $$y' = 18x^2 + sinx$$
8. $$y = (x^6 + 3)(x^4 - 4)$$ $$y = x^{10} - 4x^6 + 3x^4 - 12$$ $$y' = 10x^9 - 24x^5 + 12x^3$$ Ответ: $$y' = 10x^9 - 24x^5 + 12x^3$$
9. $$y = \sqrt{x}(6x - 1)$$ $$y = 6x^{\frac{3}{2}} - x^{\frac{1}{2}}$$ $$y' = 9x^{\frac{1}{2}} - \frac{1}{2}x^{-\frac{1}{2}}$$ $$y' = 9\sqrt{x} - \frac{1}{2\sqrt{x}}$$ Ответ: $$y' = 9\sqrt{x} - \frac{1}{2\sqrt{x}}$$
10. $$y = x^6 cosx$$ $$y' = 6x^5 cosx - x^6 sinx$$ Ответ: $$y' = 6x^5 cosx - x^6 sinx$$
11. $$y = (\frac{3}{x} - 4)(3x + 6)$$ $$y = (3x^{-1} - 4)(3x + 6)$$ $$y = 9 - 12x - 18x^{-1}$$ $$y' = -12 + 18x^{-2}$$ $$y' = -12 + \frac{18}{x^2}$$ Ответ: $$y' = -12 + \frac{18}{x^2}$$
12. $$y = \frac{4x^5}{2x + 5}$$ $$y' = \frac{20x^4(2x + 5) - 4x^5 \cdot 2}{(2x + 5)^2}$$ $$y' = \frac{40x^5 + 100x^4 - 8x^5}{(2x + 5)^2}$$ $$y' = \frac{32x^5 + 100x^4}{(2x + 5)^2}$$ $$y' = \frac{4x^4(8x + 25)}{(2x + 5)^2}$$ Ответ: $$y' = \frac{4x^4(8x + 25)}{(2x + 5)^2}$$
13. $$y = \frac{9\sqrt{x}}{6 + 2x^5}$$ $$y = \frac{9x^{\frac{1}{2}}}{6 + 2x^5}$$ $$y' = \frac{\frac{9}{2}x^{-\frac{1}{2}}(6 + 2x^5) - 9x^{\frac{1}{2}} \cdot 10x^4}{(6 + 2x^5)^2}$$ $$y' = \frac{\frac{27}{\sqrt{x}} + 9x^{\frac{9}{2}} - 90x^{\frac{9}{2}}}{(6 + 2x^5)^2}$$ $$y' = \frac{\frac{27}{\sqrt{x}} - 81x^{\frac{9}{2}}}{(6 + 2x^5)^2}$$ $$y' = \frac{\frac{27 - 81x^5}{\sqrt{x}}}{(6 + 2x^5)^2}$$ $$y' = \frac{27 - 81x^5}{\sqrt{x}(6 + 2x^5)^2}$$ Ответ: $$y' = \frac{27 - 81x^5}{\sqrt{x}(6 + 2x^5)^2}$$
14. $$y = \frac{4x}{ctgx}$$ $$y' = \frac{4ctgx - 4x(-\frac{1}{sin^2x})}{ctg^2x}$$ $$y' = \frac{4ctgx + \frac{4x}{sin^2x}}{ctg^2x}$$ $$y' = \frac{4ctgx + 4x csc^2x}{ctg^2x}$$ Ответ: $$y' = \frac{4ctgx + 4x csc^2x}{ctg^2x}$$