Упр.180 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

2) корень 5 степени x;
3) 2 корень 6 степени x — корень 3 степени x;
4) 3 корень 6 степени + 7 корень 14 степени x;
5) 2/5 корень x;
6) 1/x корень x;
7) x3+1/x;
8) x4 — корень x/x

1. $$f(x)=\sqrt[3]{x}=x^{\frac13}$$
   $$f'(x)=\frac13x^{-\frac23}=\frac{1}{3\sqrt[3]{x^2}}$$
2. $$f(x)=\sqrt[5]{x}=x^{\frac15}$$
   $$f'(x)=\frac15x^{-\frac45}=\frac{1}{5\sqrt[5]{x^4}}$$
3. $$f(x)=2\sqrt[6]{x}-\sqrt[3]{x}=2x^{\frac16}-x^{\frac13}$$
   $$f'(x)=2\cdot\frac16x^{-\frac56}-\frac13x^{-\frac23}=\frac{1}{3\sqrt[6]{x^5}}-\frac{1}{3\sqrt[3]{x^2}}$$
4. $$f(x)=3\sqrt[6]{x}+7\sqrt[14]{x}=3x^{\frac16}+7x^{\frac1{14}}$$
   $$f'(x)=3\cdot\frac16x^{-\frac56}+7\cdot\frac1{14}x^{-\frac{13}{14}}=\frac{1}{2\sqrt[6]{x^5}}+\frac{1}{2\sqrt[14]{x^{13}}}$$
5. $$f(x)=\frac{2}{5\sqrt{x}}=\frac25x^{-\frac12}$$
   $$f'(x)=\frac25\cdot\left(-\frac12\right)x^{-\frac32}=-\frac{1}{5\sqrt{x^3}}=-\frac{1}{5x\sqrt{x}}$$
6. $$f(x)=\frac{1}{x\sqrt{x}}=\frac{1}{\sqrt{x^3}}=x^{-\frac32}$$
   $$f'(x)=-\frac32x^{-\frac52}=-\frac{3}{2\sqrt{x^5}}=-\frac{3}{2x^2\sqrt{x}}$$
7. $$f(x)=\frac{x^3+1}{x}=x^2+x^{-1}$$
   $$f'(x)=2x-x^{-2}=2x-\frac{1}{x^2}$$
8. $$f(x)=\frac{x^4-\sqrt{x}}{x}=x^3-x^{-\frac12}$$
   $$f'(x)=3x^2+\frac12x^{-\frac32}=3x^2+\frac{1}{2x\sqrt{x}}$$

Ответ

1. $$f'(x)=\frac{1}{3\sqrt[3]{x^2}}$$
2. $$f'(x)=\frac{1}{5\sqrt[5]{x^4}}$$
3. $$f'(x)=\frac{1}{3\sqrt[6]{x^5}}-\frac{1}{3\sqrt[3]{x^2}}$$
4. $$f'(x)=\frac{1}{2\sqrt[6]{x^5}}+\frac{1}{2\sqrt[14]{x^{13}}}$$
5. $$f'(x)=-\frac{1}{5x\sqrt{x}}$$
6. $$f'(x)=-\frac{3}{2x^2\sqrt{x}}$$
7. $$f'(x)=2x-\frac{1}{x^2}$$
8. $$f'(x)=3x^2+\frac{1}{2x\sqrt{x}}$$