経済学で出る数学

ワークブックでじっくり攻める：問 5.17

1. $\left(x^{-8}\right)^{\prime}=-8x^{-9}=-\dfrac{8}{x^9}$


2. $\left(\dfrac{1}{x^{3}}\right)^{\prime}=\left(x^{-3}\right)^{\prime}=-3x^{-4}=-\dfrac{3}{x^4}$


3. $\left(\dfrac{3}{x^{4}}\right)^{\prime}=\left(3x^{-4}\right)^{\prime}=-12x^{-5} =-\dfrac{12}{x^5}$


4. $\left(-\dfrac{3}{2x^{2}}\right)^{\prime}=\left(-\dfrac{3}{2}x^{-2}\right)^{\prime}=-\dfrac{3}{2}\times (-2)x^{-3} =\dfrac{3}{x^3}$


5. $\left(x^{\frac{1}{3}}\right)^{\prime}=\dfrac{1}{3}x^{\frac{1}{3}-1} =\dfrac{1}{3}x^{-\frac{2}{3}}=\dfrac{1}{3\sqrt[3]{x^2}}$


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


7. $\left(2\sqrt[4]{x}\right)^{\prime}= \left(2x^{\frac{1}{4}}\right)^{\prime}=2\times\dfrac{1}{4}x^{\frac{1}{4}-1} =\dfrac{1}{2}x^{-\frac{3}{4}}=\dfrac{1}{2\sqrt[4]{x^3}}$


8. $\left(\dfrac{3}{\sqrt[3]{x^2}}\right)^{\prime}= \left(3x^{-\frac{2}{3}}\right)^{\prime} =3\times \left(-\dfrac{2}{3}\right)x^{-\frac{2}{3}-1} =-2x^{-\frac{5}{3}}=\dfrac{-2}{\sqrt[3]{x^5}}$