Вариант 3 1)y = 7x+x⁴ 2)y=3x⁷+10x²-13 3)y = 4x + √x 1 4)y = -+3x³-35 x 6 5)y=--ctgx x⁸ 6)y = √x-3cosx 4 7)y = --sinx x⁵ 8)y = (x⁹-1)(10+x²) 9)y = √x(3x-4) 4 10)y = x⁴ sin x 11)y = (2/x-5) (3x+7) 5x³ 12)y = 2x-7 -2√x 13)y = x⁶-2 tgx 14)y =

Решения заданий Вариант 3:

1. $$y = 7x + x^4$$ $$y' = 7 + 4x^3$$ Ответ: $$y' = 7 + 4x^3$$
2. $$y = 3x^7 + 10x^2 - 13$$ $$y' = 21x^6 + 20x$$ Ответ: $$y' = 21x^6 + 20x$$
3. $$y = 4x + \sqrt{x}$$ $$y = 4x + x^{\frac{1}{2}}$$ $$y' = 4 + \frac{1}{2}x^{-\frac{1}{2}}$$ $$y' = 4 + \frac{1}{2\sqrt{x}}$$ Ответ: $$y' = 4 + \frac{1}{2\sqrt{x}}$$
4. $$y = \frac{1}{x} + 3x^3 - 35$$ $$y = x^{-1} + 3x^3 - 35$$ $$y' = -x^{-2} + 9x^2$$ $$y' = -\frac{1}{x^2} + 9x^2$$ Ответ: $$y' = -\frac{1}{x^2} + 9x^2$$
5. $$y = -\frac{6}{x^8} - ctgx$$ $$y = -6x^{-8} - ctgx$$ $$y' = 48x^{-9} + \frac{1}{sin^2x}$$ $$y' = \frac{48}{x^9} + \frac{1}{sin^2x}$$ Ответ: $$y' = \frac{48}{x^9} + \frac{1}{sin^2x}$$
6. $$y = \sqrt{x} - 3cosx$$ $$y = x^{\frac{1}{2}} - 3cosx$$ $$y' = \frac{1}{2}x^{-\frac{1}{2}} + 3sinx$$ $$y' = \frac{1}{2\sqrt{x}} + 3sinx$$ Ответ: $$y' = \frac{1}{2\sqrt{x}} + 3sinx$$
7. $$y = -\frac{4}{x^5} - sinx$$ $$y = -4x^{-5} - sinx$$ $$y' = 20x^{-6} - cosx$$ $$y' = \frac{20}{x^6} - cosx$$ Ответ: $$y' = \frac{20}{x^6} - cosx$$
8. $$y = (x^9 - 1)(10 + x^2)$$ $$y = x^{11} + 10x^9 - x^2 - 10$$ $$y' = 11x^{10} + 90x^8 - 2x$$ Ответ: $$y' = 11x^{10} + 90x^8 - 2x$$
9. $$y = \sqrt{x}(3x - 4)$$ $$y = 3x^{\frac{3}{2}} - 4x^{\frac{1}{2}}$$ $$y' = \frac{9}{2}x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}$$ $$y' = \frac{9}{2}\sqrt{x} - \frac{2}{\sqrt{x}}$$ Ответ: $$y' = \frac{9}{2}\sqrt{x} - \frac{2}{\sqrt{x}}$$
10. $$y = x^4 sinx$$ $$y' = 4x^3 sinx + x^4 cosx$$ Ответ: $$y' = 4x^3 sinx + x^4 cosx$$
11. $$y = (\frac{2}{x} - 5)(3x + 7)$$ $$y = (2x^{-1} - 5)(3x + 7)$$ $$y = 6 - 15x - 35 + 14x^{-1}$$ $$y = -15x - 29 + 14x^{-1}$$ $$y' = -15 - 14x^{-2}$$ $$y' = -15 - \frac{14}{x^2}$$ Ответ: $$y' = -15 - \frac{14}{x^2}$$
12. $$y = \frac{5x^3}{2x - 7}$$ $$y' = \frac{15x^2(2x - 7) - 5x^3 \cdot 2}{(2x - 7)^2}$$ $$y' = \frac{30x^3 - 105x^2 - 10x^3}{(2x - 7)^2}$$ $$y' = \frac{20x^3 - 105x^2}{(2x - 7)^2}$$ $$y' = \frac{5x^2(4x - 21)}{(2x - 7)^2}$$ Ответ: $$y' = \frac{5x^2(4x - 21)}{(2x - 7)^2}$$
13. $$y = \frac{-2\sqrt{x}}{x^6 - 2}$$ $$y = \frac{-2x^{\frac{1}{2}}}{x^6 - 2}$$ $$y' = \frac{-x^{-\frac{1}{2}}(x^6 - 2) + 2x^{\frac{1}{2}} \cdot 6x^5}{(x^6 - 2)^2}$$ $$y' = \frac{\frac{-x^6 + 2}{\sqrt{x}} + 12x^{\frac{11}{2}}}{(x^6 - 2)^2}$$ $$y' = \frac{\frac{-x^6 + 2 + 12x^{12}}{\sqrt{x}}}{(x^6 - 2)^2}$$ $$y' = \frac{-x^6 + 2 + 12x^{12}}{\sqrt{x}(x^6 - 2)^2}$$ Ответ: $$y' = \frac{12x^{12} - x^6 + 2}{\sqrt{x}(x^6 - 2)^2}$$
14. $$y = \frac{tgx}{2x^2}$$ $$y = \frac{1}{2} \cdot \frac{tgx}{x^2}$$ $$y' = \frac{\frac{1}{cos^2x} \cdot x^2 - tgx \cdot 2x}{2x^4}$$ $$y' = \frac{\frac{x^2}{cos^2x} - 2x tgx}{2x^4}$$ Ответ: $$y' = \frac{\frac{x^2}{cos^2x} - 2x tgx}{2x^4}$$