Упр.187 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. $$\left(x+2\right)\sqrt[3]{x}$$
2. $$\left(x+1\right)\sqrt{x}$$
3. $$\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right)^2$$
4. $$x^3+\frac{2}{\sqrt[3]{x}}$$
5. $$\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right)\left(\sqrt[4]{x}-\frac{1}{\sqrt[4]{x}}\right)$$
6. $$\left(\sqrt{x}-\sqrt[3]{x}\right)^2$$

1. $$y=(x+2)\sqrt[3]{x}=x^{4/3}+2x^{1/3}$$

   $$y’=\frac{4}{3}x^{1/3}+\frac{2}{3}x^{-2/3}=\frac{4}{3}\sqrt[3]{x}+\frac{2}{3\sqrt[3]{x^2}}$$

2. $$y=(x+1)\sqrt{x}=x^{3/2}+x^{1/2}$$

   $$y’=\frac{3}{2}x^{1/2}+\frac{1}{2}x^{-1/2}=\frac{3}{2}\sqrt{x}+\frac{1}{2\sqrt{x}}$$

3. $$y=\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right)^2=x^{1/2}+2+x^{-1/2}$$

   $$y’=\frac{1}{2}x^{-1/2}-\frac{1}{2}x^{-3/2}=\frac{1}{2\sqrt{x}}-\frac{1}{2x\sqrt{x}}$$

4. $$y=\frac{x^3+2}{\sqrt[3]{x}}=x^{8/3}+2x^{-1/3}$$

   $$y’=\frac{8}{3}x^{5/3}-\frac{2}{3}x^{-4/3}=\frac{8}{3}x^{5/3}-\frac{2}{3x^{4/3}}$$

5. $$y=\left(\sqrt[4]{x}+\frac{1}{\sqrt[4]{x}}\right)\left(\sqrt[4]{x}-\frac{1}{\sqrt[4]{x}}\right)=x^{1/2}-x^{-1/2}$$

   $$y’=\frac{1}{2}x^{-1/2}+\frac{1}{2}x^{-3/2}=\frac{1}{2\sqrt{x}}+\frac{1}{2x\sqrt{x}}$$

6. $$y=(\sqrt{x}-\sqrt[3]{x})^2=x-2x^{5/6}+x^{2/3}$$

   $$y’=1-2\cdot\frac{5}{6}x^{-1/6}+\frac{2}{3}x^{-1/3}=1-\frac{5}{3\sqrt[6]{x}}+\frac{2}{3\sqrt[3]{x}}$$

Ответ

1. $$y’=\frac{4}{3}\sqrt[3]{x}+\frac{2}{3\sqrt[3]{x^2}}$$
2. $$y’=\frac{3}{2}\sqrt{x}+\frac{1}{2\sqrt{x}}$$
3. $$y’=\frac{1}{2\sqrt{x}}-\frac{1}{2x\sqrt{x}}$$
4. $$y’=\frac{8}{3}x^{5/3}-\frac{2}{3x^{4/3}}$$
5. $$y’=\frac{1}{2\sqrt{x}}+\frac{1}{2x\sqrt{x}}$$
6. $$y’=1-\frac{5}{3\sqrt[6]{x}}+\frac{2}{3\sqrt[3]{x}}$$