## 問題の概要

## 問題の概要

与えられた関数 $y$ を $x^n$ の形に変形し、その後微分する問題です。 関数は全部で10個与えられています。

## 解き方の手順

各関数について、$x^n$ の形に変形し、微分を行います。微分には、公式 $\frac{d}{dx}x^n = nx^{n-1}$ を利用します。

(1) $y = x^2 x^3 = x^{2+3} = x^5$

$\frac{dy}{dx} = 5x^{5-1} = 5x^4$

(2) $y = \frac{x^8}{x^2} = x^{8-2} = x^6$

$\frac{dy}{dx} = 6x^{6-1} = 6x^5$

(3) $y = \frac{1}{x^4} = x^{-4}$

$\frac{dy}{dx} = -4x^{-4-1} = -4x^{-5} = -\frac{4}{x^5}$

(4) $y = \sqrt[7]{x} = x^{\frac{1}{7}}$

$\frac{dy}{dx} = \frac{1}{7}x^{\frac{1}{7}-1} = \frac{1}{7}x^{-\frac{6}{7}} = \frac{1}{7\sqrt[7]{x^6}}$

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

$\frac{dy}{dx} = -\frac{4}{3}x^{-\frac{4}{3}-1} = -\frac{4}{3}x^{-\frac{7}{3}} = -\frac{4}{3\sqrt[3]{x^7}}$

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

$\frac{dy}{dx} = \frac{3}{2}x^{\frac{3}{2}-1} = \frac{3}{2}x^{\frac{1}{2}} = \frac{3}{2}\sqrt{x}$

(7) $y = \frac{x^2}{\sqrt[5]{x}} = \frac{x^2}{x^{\frac{1}{5}}} = x^{2-\frac{1}{5}} = x^{\frac{9}{5}}$

$\frac{dy}{dx} = \frac{9}{5}x^{\frac{9}{5}-1} = \frac{9}{5}x^{\frac{4}{5}} = \frac{9}{5}\sqrt[5]{x^4}$

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

$\frac{dy}{dx} = -\frac{5}{3}x^{-\frac{5}{3}-1} = -\frac{5}{3}x^{-\frac{8}{3}} = -\frac{5}{3\sqrt[3]{x^8}}$

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

$\frac{dy}{dx} = -\frac{1}{2}x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}} = -\frac{1}{2\sqrt{x^3}}$

(10) $y = \sqrt[4]{\sqrt{x^9}} = (x^{\frac{9}{2}})^{\frac{1}{4}} = x^{\frac{9}{8}}$

$\frac{dy}{dx} = \frac{9}{8} x^{\frac{9}{8}-1} = \frac{9}{8} x^{\frac{1}{8}} = \frac{9}{8} \sqrt[8]{x}$

## 最終的な答え

(1) $\frac{dy}{dx} = 5x^4$

(2) $\frac{dy}{dx} = 6x^5$

(3) $\frac{dy}{dx} = -\frac{4}{x^5}$

(4) $\frac{dy}{dx} = \frac{1}{7\sqrt[7]{x^6}}$

(5) $\frac{dy}{dx} = -\frac{4}{3\sqrt[3]{x^7}}$

(6) $\frac{dy}{dx} = \frac{3}{2}\sqrt{x}$

(7) $\frac{dy}{dx} = \frac{9}{5}\sqrt[5]{x^4}$

(8) $\frac{dy}{dx} = -\frac{5}{3\sqrt[3]{x^8}}$

(9) $\frac{dy}{dx} = -\frac{1}{2\sqrt{x^3}}$

(10) $\frac{dy}{dx} = \frac{9}{8} \sqrt[8]{x}$