Вариант 1. Задание 1. Найдите производную функции: 1) $$y = 3x^4$$ 2) $$y = 2x^{-3}$$ 3) $$y = 4x^{\frac{1}{2}}$$ 4) $$y = \sqrt[3]{x^2}$$ 5) $$y = \frac{1}{x^3}$$ 6) $$y = x^{\frac{3}{2}}$$ 7) $$y = \frac{1}{\sqrt{x}}$$ 8) $$y = \frac{1}{\sqrt[3]{x^3}}$$

1) $$y = 3x^4$$ $$y' = 3 \cdot 4x^{4-1} = 12x^3$$ 2) $$y = 2x^{-3}$$ $$y' = 2 \cdot (-3)x^{-3-1} = -6x^{-4} = -\frac{6}{x^4}$$ 3) $$y = 4x^{\frac{1}{2}}$$ $$y' = 4 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = 2x^{-\frac{1}{2}} = \frac{2}{\sqrt{x}}$$ 4) $$y = \sqrt[3]{x^2} = x^{\frac{2}{3}}$$ $$y' = \frac{2}{3} x^{\frac{2}{3} - 1} = \frac{2}{3} x^{-\frac{1}{3}} = \frac{2}{3\sqrt[3]{x}}$$ 5) $$y = \frac{1}{x^3} = x^{-3}$$ $$y' = -3x^{-3-1} = -3x^{-4} = -\frac{3}{x^4}$$ 6) $$y = x^{\frac{3}{2}}$$ $$y' = \frac{3}{2} x^{\frac{3}{2} - 1} = \frac{3}{2} x^{\frac{1}{2}} = \frac{3}{2}\sqrt{x}$$ 7) $$y = \frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$$ $$y' = -\frac{1}{2} x^{-\frac{1}{2} - 1} = -\frac{1}{2} x^{-\frac{3}{2}} = -\frac{1}{2\sqrt{x^3}}$$ 8) $$y = \frac{1}{\sqrt[3]{x^3}} = \frac{1}{x} = x^{-1}$$ $$y' = -1 x^{-1-1} = -x^{-2} = -\frac{1}{x^2}$$