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

1) $$y = 5x^3$$ $$y' = 5 \cdot 3x^{3-1} = 15x^2$$ 2) $$y = 3x^{-\frac{2}{3}}$$ $$y' = 3 \cdot (-\frac{2}{3}) x^{-\frac{2}{3} - 1} = -2x^{-\frac{5}{3}} = -\frac{2}{\sqrt[3]{x^5}}$$ 3) $$y = 4x^2$$ $$y' = 4 \cdot 2x^{2-1} = 8x$$ 4) $$y = \sqrt[3]{x^4} = x^{\frac{4}{3}}$$ $$y' = \frac{4}{3} x^{\frac{4}{3} - 1} = \frac{4}{3} x^{\frac{1}{3}} = \frac{4}{3}\sqrt[3]{x}$$ 5) $$y = -\frac{1}{x^4} = -x^{-4}$$ $$y' = -(-4) x^{-4-1} = 4x^{-5} = \frac{4}{x^5}$$ 6) $$y = x^{\frac{2}{3}}$$ $$y' = \frac{2}{3} x^{\frac{2}{3} - 1} = \frac{2}{3} x^{-\frac{1}{3}} = \frac{2}{3\sqrt[3]{x}}$$ 7) $$y = \sqrt{x} = x^{\frac{1}{2}}$$ $$y' = \frac{1}{2} x^{\frac{1}{2} - 1} = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$$ 8) $$y = \frac{1}{\sqrt[3]{x}} = x^{-\frac{1}{3}}$$ $$y' = -\frac{1}{3} x^{-\frac{1}{3} - 1} = -\frac{1}{3} x^{-\frac{4}{3}} = -\frac{1}{3\sqrt[3]{x^4}}$$