Найдите производную функции: 1) $$y=x^3+2x$$ 2) $$y=\sqrt{x} - 5x^2$$ 3) $$y = \frac{1}{x} - 8x$$ 4) $$y = 3tgx + 4x - 9$$ 5) $$y = 4x^7 - 19x^{11} + 1$$ 6) $$y = (x^2 + 9)(x^6 - 10)$$ 7) $$y = (x^4 + 3)\sqrt{x}$$ 8) $$y = (x^2 + 1)cosx$$ 9) $$y = (8 - \frac{1}{x})(5x + 4)$$ 10) $$y = \frac{3x-7}{4x+5}$$ 11) $$y = \frac{8x}{2x^2 - 3}$$ 12) $$y = \frac{sin x}{2x}$$

Решения: 1) $$y = x^3 + 2x$$ $$y' = 3x^2 + 2$$ 2) $$y = \sqrt{x} - 5x^2$$ $$y' = \frac{1}{2\sqrt{x}} - 10x$$ 3) $$y = \frac{1}{x} - 8x$$ $$y' = -\frac{1}{x^2} - 8$$ 4) $$y = 3tgx + 4x - 9$$ $$y' = \frac{3}{cos^2x} + 4$$ 5) $$y = 4x^7 - 19x^{11} + 1$$ $$y' = 28x^6 - 209x^{10}$$ 6) $$y = (x^2 + 9)(x^6 - 10)$$ $$y' = (2x)(x^6 - 10) + (x^2 + 9)(6x^5) = 2x^7 - 20x + 6x^7 + 54x^5 = 8x^7 + 54x^5 - 20x$$ 7) $$y = (x^4 + 3)\sqrt{x}$$ $$y' = (4x^3)\sqrt{x} + (x^4 + 3)(\frac{1}{2\sqrt{x}}) = 4x^{7/2} + \frac{x^4}{2\sqrt{x}} + \frac{3}{2\sqrt{x}} = 4x^{7/2} + \frac{1}{2}x^{7/2} + \frac{3}{2\sqrt{x}} = \frac{9}{2}x^{7/2} + \frac{3}{2\sqrt{x}}$$ 8) $$y = (x^2 + 1)cosx$$ $$y' = (2x)cosx + (x^2 + 1)(-sinx) = 2xcosx - (x^2 + 1)sinx$$ 9) $$y = (8 - \frac{1}{x})(5x + 4)$$ $$y' = (\frac{1}{x^2})(5x + 4) + (8 - \frac{1}{x})(5) = \frac{5x}{x^2} + \frac{4}{x^2} + 40 - \frac{5}{x} = \frac{5}{x} + \frac{4}{x^2} + 40 - \frac{5}{x} = \frac{4}{x^2} + 40$$ 10) $$y = \frac{3x-7}{4x+5}$$ $$y' = \frac{(3)(4x+5) - (3x-7)(4)}{(4x+5)^2} = \frac{12x + 15 - 12x + 28}{(4x+5)^2} = \frac{43}{(4x+5)^2}$$ 11) $$y = \frac{8x}{2x^2 - 3}$$ $$y' = \frac{(8)(2x^2 - 3) - (8x)(4x)}{(2x^2 - 3)^2} = \frac{16x^2 - 24 - 32x^2}{(2x^2 - 3)^2} = \frac{-16x^2 - 24}{(2x^2 - 3)^2}$$ 12) $$y = \frac{sin x}{2x}$$ $$y' = \frac{(cos x)(2x) - (sin x)(2)}{(2x)^2} = \frac{2x cos x - 2 sin x}{4x^2} = \frac{x cos x - sin x}{2x^2}$$