Упр.790 ГДЗ Алимов 10-11 класс (Алгебра)
Воспользуемся формулой производной степенной функции $$\left(x^p\right)’=px^{p-1}.$$
$$f(x)=\frac{1}{x^5}=x^{-5}$$
$$f'(x)=\left(x^{-5}\right)’=-5x^{-6}=-\frac{5}{x^6}.$$
$$f(x)=\frac{1}{x^9}=x^{-9}$$
$$f'(x)=\left(x^{-9}\right)’=-9x^{-10}=-\frac{9}{x^{10}}.$$
$$f(x)=\sqrt[4]{x} = x^{\frac14}$$
$$f'(x)=\left(x^{\frac14}\right)’=\frac14 x^{-\frac34}=\frac{1}{4\sqrt[4]{x^3}}.$$
$$f(x)=\sqrt[3]{x^2}=x^{\frac23}$$
$$f'(x)=\left(x^{\frac23}\right)’=\frac23 x^{-\frac13}=\frac{2}{3\sqrt[3]{x}}.$$
$$f(x)=\frac{1}{\sqrt[3]{x}}=x^{-\frac13}$$
$$f'(x)=\left(x^{-\frac13}\right)’=-\frac13 x^{-\frac43}=-\frac{1}{3x\sqrt[3]{x}}.$$
$$f(x)=\frac{1}{\sqrt[4]{x^3}}=x^{-\frac34}$$
$$f'(x)=\left(x^{-\frac34}\right)’=-\frac34 x^{-\frac74}=-\frac{3}{4\sqrt[4]{x^7}}.$$
Ответ: $$-\frac{5}{x^6},\ -\frac{9}{x^{10}},\ \frac{1}{4\sqrt[4]{x^3}},\ \frac{2}{3\sqrt[3]{x}},\ -\frac{1}{3x\sqrt[3]{x}},\ -\frac{3}{4\sqrt[4]{x^7}}.$$
