Найдите производную функции у [197, 198). 197.1) y = 4x³; 2) y = 3x 4; 3) y = 4x³/⁴; 4) y = ³√x²; 5) y = ³√8x; 6) y = 1/2 ³√x². 198.1) y = 2/x³; 2) ㅂ= √x/2; 3) y = 1/³√x ; 4) y = 6/³√x²; 5) y = 1/√4x³; 6) y = 1/√x²/³.

Решим данное задание, используя правила нахождения производной функции.

197.1) $$y = 4x^3$$

$$y' = 4 \cdot 3x^{3-1} = 12x^2$$

Ответ: $$y' = 12x^2$$

2) $$y = 3x^4$$

$$y' = 3 \cdot 4x^{4-1} = 12x^3$$

Ответ: $$y' = 12x^3$$

3) $$y = 4x^{\frac{3}{4}}$$

$$y' = 4 \cdot \frac{3}{4}x^{\frac{3}{4}-1} = 3x^{-\frac{1}{4}} = \frac{3}{\sqrt[4]{x}}$$

Ответ: $$y' = \frac{3}{\sqrt[4]{x}}$$

4) $$y = \sqrt[3]{x^2} = x^{\frac{2}{3}}$$

$$y' = \frac{2}{3}x^{\frac{2}{3}-1} = \frac{2}{3}x^{-\frac{1}{3}} = \frac{2}{3\sqrt[3]{x}}$$

Ответ: $$y' = \frac{2}{3\sqrt[3]{x}}$$

5) $$y = \sqrt[3]{8x} = 2\sqrt[3]{x} = 2x^{\frac{1}{3}}$$

$$y' = 2 \cdot \frac{1}{3}x^{\frac{1}{3}-1} = \frac{2}{3}x^{-\frac{2}{3}} = \frac{2}{3\sqrt[3]{x^2}}$$

Ответ: $$y' = \frac{2}{3\sqrt[3]{x^2}}$$

6) $$y = \frac{1}{2}\sqrt[3]{x^2} = \frac{1}{2}x^{\frac{2}{3}}$$

$$y' = \frac{1}{2} \cdot \frac{2}{3}x^{\frac{2}{3}-1} = \frac{1}{3}x^{-\frac{1}{3}} = \frac{1}{3\sqrt[3]{x}}$$

Ответ: $$y' = \frac{1}{3\sqrt[3]{x}}$$

198.1) $$y = \frac{2}{x^3} = 2x^{-3}$$

$$y' = 2 \cdot (-3)x^{-3-1} = -6x^{-4} = -\frac{6}{x^4}$$

Ответ: $$y' = -\frac{6}{x^4}$$

2) $$y = \frac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}$$

$$y' = 2 \cdot (-\frac{1}{2})x^{-\frac{1}{2}-1} = -x^{-\frac{3}{2}} = -\frac{1}{\sqrt{x^3}} = -\frac{1}{x\sqrt{x}}$$

Ответ: $$y' = -\frac{1}{x\sqrt{x}}$$

3) $$y = \frac{1}{\sqrt[3]{x}} = x^{-\frac{1}{3}}$$

$$y' = -\frac{1}{3}x^{-\frac{1}{3}-1} = -\frac{1}{3}x^{-\frac{4}{3}} = -\frac{1}{3\sqrt[3]{x^4}} = -\frac{1}{3x\sqrt[3]{x}}$$

Ответ: $$y' = -\frac{1}{3x\sqrt[3]{x}}$$

4) $$y = \frac{6}{\sqrt[3]{x^2}} = 6x^{-\frac{2}{3}}$$

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

Ответ: $$y' = -\frac{4}{x\sqrt[3]{x^2}}$$

5) $$y = \frac{1}{\sqrt{4x^3}} = \frac{1}{2\sqrt{x^3}} = \frac{1}{2}x^{-\frac{3}{2}}$$

$$y' = \frac{1}{2} \cdot (-\frac{3}{2})x^{-\frac{3}{2}-1} = -\frac{3}{4}x^{-\frac{5}{2}} = -\frac{3}{4\sqrt{x^5}} = -\frac{3}{4x^2\sqrt{x}}$$

Ответ: $$y' = -\frac{3}{4x^2\sqrt{x}}$$

6) $$y = \frac{1}{\sqrt{x^{\frac{2}{3}}}} = \frac{1}{x^{\frac{1}{3}}} = x^{-\frac{1}{3}}$$

$$y' = -\frac{1}{3}x^{-\frac{1}{3}-1} = -\frac{1}{3}x^{-\frac{4}{3}} = -\frac{1}{3\sqrt[3]{x^4}} = -\frac{1}{3x\sqrt[3]{x}}$$

Ответ: $$y' = -\frac{1}{3x\sqrt[3]{x}}$$